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Prof. Dr. Daniel Loss

Contact Department of Physics University of Basel Klingelbergstrasse 82 CH-4056 Basel, Switzerland office: office 4.12 email: view address tel: ++41 (0)61 207 37 49 (office) +41 (0)61 207 37 83 (administrator) fax: xx

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CV and Research Interests

Daniel Loss received a Ph.D. in Theoretical Physics at the University of Zurich in 1985 under the supervision of Prof. A. Thellung. He stayed there as postdoctoral researcher for four more years before moving to the US in 1989. From 1989 to 1991 he worked as postdoctoral researcher in the group of Prof. A. J. Leggett, Urbana, and from 1991 to 1993 at IBM Research Center, NY (USA). In 1993 he moved to Vancouver (Canada) to become Assistant and then Associate Professor of Physics at Simon Fraser University. In 1996 he returned to Switzerland to become full Professor of Theoretical Physics at the University of Basel. Loss is director of the Basel Center for Quantum Computing and Quantum Coherence (QC2), was co-director (2006-2013) of the Swiss National Center of Competence and Research (NCCR) in Nanoscale Science, co-director (2006-2021) of the Swiss Nanoscience Institute at the University of Basel, and since 2020 he is co-director (and founding member) of the NCCR SPIN QUBITS IN SILICON. He received several prestigious fellowships, is a Fellow of the American Physical Society, a member of the European Academy of Sciences and of the German National Academy of Sciences Leopoldina. He has been awarded the Humboldt Research Prize in 2005, the Marcel Benoist Prize in 2010; the most prestigious science prize in Switzerland (see www.admin.ch and www.marcel-benoist.ch), the Blaise Pascal Medal in Physics 2014 from the European Academy of Sciences, and the King Faisal International Prize in Science 2017 (see King Faisal Prize and Uni news). In 2021, he has been elected as External Scientific Member of the Max Planck Society (see here and Uni news). In 2024 he received the European Materials Research Society (E-MRS) Czochralski Award Gold Metal in recognition and acknowledgement of his lifetime achievements in Materials Science (see also Uni News).

Loss' research interests include many aspects of the quantum theory of condensed matter systems with a particular focus on spin-dependent and phase-coherent phenomena ("mesoscopics") in semiconducting nanostructures and molecular magnets. A major portion of Loss' current research involves the theory of spin dynamics, spin coherence, spintronics in two-dimensional electron gases, and spin-related phenomena in semiconducting quantum dots--artificial atoms and molecules. Part of this work is related to quantum information processing (QIP)--quantum computing and quantum communication in solid state systems with focus on spin qubits, where Loss and collaborators made pioneering and seminal contributions. Their theoretical predictions and proposals have stimulated many further investigations, and in particular many experimental programs on spin qubits worldwide. Current research includes spin relaxation and decoherence in quantum dots due to spin-orbit and hyperfine interaction; non-Markovian spin dynamics in bosonic and nuclear spin environments; generation and characterization of non-local entanglement with quantum dots, superconductors, Luttinger liquids or Coulomb scattering in interacting 2DEGs; spin currents in magnetic insulators and in semiconductors; spin Hall effect in disordered systems; spin orbit effects in transport and noise; asymmetric quantum shot noise in quantum dots; entanglement transfer from electron spins to photons; QIP with electron and hole spin qubits in Si and Ge quantum dots and in molecular magnets; macroscopic quantum phenomena (spin tunneling and coherence) in molecular and nanoscale magnetism; topological excitations in magnetic systems such as domain walls, skyrmions, merons etc.; topological quantum computing and Majorana fermions and parafermions; superconducting and semiconducting hybrid systems; non-Fermi liquid effects in interacting d-dimensional systems etc. [see list of publications below].

Extended CV

Download an extended CV here.

Open Positions

We are constantly looking for outstanding, highly motivated, and enthusiastic graduate students and/or postdoctoral fellows.

PhD Candidates need to hold a Master's (or equivalent) degree in theoretical condensed matter physics or similar. Postdoc Candidates should have a PhD in theoretical condensed matter physics or similar.

To apply please submit the following documents online (PhD / Postdoc)

1. a curriculum vitae
2. a list of publications
3. your academic records (Bachelor's, Master's or PhD diploma)
4. a short statement of your research interests and how they relate to the work of our group
5. please arrange for 2-3 letters of recommendation

Publications

Citation record.

Show all abstracts.

Interband contributions to nonlinear transport in semiconductor nanostructures

Kazuki Nakazawa, Henry F. Legg, Jelena Klinovaja, Daniel Loss arXiv:2410.16639 Spin-orbit interaction (SOI) is a crucial ingredient for many potential applications of quantum devices, such as the use of semiconductor nanostructures for quantum computing. It is known that nonlinear conductivities are sensitive to the strength and type of SOI, however, many calculations of nonlinear transport coefficients are based on the semiclassical Boltzmann theory and make simplifying assumptions about scattering effects due to disorder. In this paper we develop and employ a microscopic theory based on the Keldysh formalism that goes beyond simple semiclassical approximations. This approach, for instance, naturally takes into account the effects of interband transitions, Berry curvature, and allows for a more precise treatment of impurity scattering. As a test of this formalism, we consider the nonlinear transport properties in an effective two-band model of one-dimensional nanowires (1DNWs) and two-dimensional hole gases (2DHGs) in the presence of a magnetic field causing Zeeman splittings of the spin states. We find that the small energy scales in nanostructures mean that interband contributions can be relevant, especially in the dirty limit, and therefore could modify qualitative features found using a purely semiclassical approach. Nonetheless, we find that different types of SOI (linear or cubic) still result in remarkably pronounced different in-plane field angle dependences, which survive even when interband effects are relevant. Our results provide a detailed understanding of when interband effects become important for nonlinear transport and can serve as the basis for a microscopic description to predict other nonlinear transport effects in materials and devices.

Topological Spin Textures Enabling Quantum Transmission

Ji Zou, Stefano Bosco, Jelena Klinovaja, Daniel Loss arXiv:2409.14373 Quantum spintronics is an emerging field focused on developing novel applications by utilizing the quantum coherence of magnetic systems. Robust information transmission is crucial for achieving fully scalable spintronic devices. However, despite its significance, reliable long-range quantum information transmission in magnetic systems remains a challenging frontier. Here, we introduce a long-distance quantum transmission scheme based on topological spin textures in a hybrid system of a magnetic racetrack and spin qubits. We demonstrate this principle by employing the domain wall (DW)-the most fundamental texture-to transport quantum signal between distant qubits. In particular, we propose and analyze an efficient measurement-free protocol that leverages the DW mobility to enable robust entanglement generation between distant qubits. Strikingly, we find that this remote entanglement generation is not only fast and of high fidelity, but also remarkably tunable, capable of being easily turned on and off on-demand by switching the DW initial state. Furthermore, we show that, interestingly, spin qubits can serve as quantum stations for the racetrack, facilitating state transfer among fast-moving DWs on the same track. Our study provides a hybrid quantum platform for scalable quantum computation and opens the door to future explorations of topological textures for quantum information processing in spintronics architectures.

The origins of noise in the Zeeman splitting of spin qubits in natural-silicon devices

Juan S. Rojas-Arias, Yohei Kojima, Kenta Takeda, Peter Stano, Takashi Nakajima, Jun Yoneda, Akito Noiri, Takashi Kobayashi, Daniel Loss, Seigo Tarucha arXiv:2408.13707 We measure and analyze noise-induced energy-fluctuations of spin qubits defined in quantum dots made of isotopically natural silicon. Combining Ramsey, time-correlation of single-shot measurements, and CPMG experiments, we cover the qubit noise power spectrum over a frequency range of nine orders of magnitude without any gaps. We find that the low-frequency noise spectrum is similar across three different devices suggesting that it is dominated by the hyperfine coupling to nuclei. The effects of charge noise are smaller, but not negligible, and are device dependent as confirmed from the noise cross-correlations. We also observe differences to spectra reported in GaAs {[Phys. Rev. Lett. 118, 177702 (2017), Phys. Rev. Lett. 101, 236803 (2008)]}, which we attribute to the presence of the valley degree of freedom in silicon. Finally, we observe T∗2 to increase upon increasing the external magnetic field, which we speculate is due to the increasing field-gradient of the micromagnet suppressing nuclear spin diffusion.

From perfect to imperfect poor man's Majoranas in minimal Kitaev chains

Melina Luethi, Henry F. Legg, Daniel Loss, Jelena Klinovaja arXiv:2408.03071 Poor man's Majoranas (PMMs) hold the promise to engineer Majorana bound states in a highly tunable setup consisting of a chain of quantum dots that are connected via superconductors. Due to recent progress in controlling the amplitudes of elastic cotunneling (ECT) and crossed Andreev reflection (CAR), two vital ingredients for PMMs, experimental investigations of PMMs have gained significant interest. Previously, analytic conditions for the "sweet spots" that result in PMMs have focused on systems with infinite Zeeman energy. Here, we derive analytically a sweet spot condition for PMMs in a system with finite Zeeman energy in the absence of Coulomb interaction. We then consider two numerical models, one in which ECT and CAR are transmitted via superconducting bulk states and one in which they are transmitted via an Andreev bound state. In both models we find numerically near zero-energy states that are highly localized, even in the presence of finite Coulomb interaction. These states can be related to the analytic sweet spot derived in this work. However, we also find that in these more realistic models such states do not perfectly satisfy the characteristics of PMMs, rather they exist only within a "region of threshold" where PMM characteristics are approximately achieved.

Long-range crossed Andreev reflection in topological insulator nanowires proximitized by Nb

Junya Feng, Henry F. Legg, Mahasweta Bagchi, Daniel Loss, Jelena Klinovaja, Yoichi Ando arXiv:2407.02383 Crossed Andreev reflection (CAR) is a nonlocal transport phenomenon that creates/detects Cooper-pair correlations between distant places. It is also the basis of Cooper-pair splitting to generate remote entanglement. Although CAR has been extensively studied in semiconductors proximity-coupled to a superconductor, it has been very difficult to observe it in a topological insulator (TI). Here we report the first observation of CAR in a proximitized TI nanowire (TINW). We performed local and nonlocal conductance spectroscopy on mesoscopic TINW devices in which superconducting (Nb) and metallic (Pt/Au) contacts are made on a bulk-insulating TINW. The local conductance detected a hard gap, accompanied by the appearance of Andreev bound states that can reach zero-bias, while a negative nonlocal conductance was occasionally observed upon sweeping the chemical potential, giving evidence for CAR. Surprisingly, the CAR signal was detected even over 1.5 μm, which implies that pair correlations extend over a length scale much longer than the expected superconducting coherence length of either Nb or the proximitised TINW. Such a long-range CAR effect is possibly due to an intricate role of disorder in proximitized nanowires. Also, our 0.9-μm device presented a decent Cooper-pair splitting efficiency of up to 0.5.

Quantum computation with hybrid parafermion-spin qubits

Denis V. Kurlov, Melina Luethi, Anatoliy I. Lotkov, Katharina Laubscher, Jelena Klinovaja, Daniel Loss arXiv:2405.20950 We propose a universal set of single- and two-qubit quantum gates acting on a hybrid qubit formed by coupling a quantum dot spin qubit to a Z2m parafermion qubit with arbitrary integer m. The special case m=1 reproduces the results previously derived for Majorana qubits. Our formalism utilizes Fock parafermions, facilitating a transparent treatment of hybrid parafermion-spin systems. Furthermore, we highlight the previously overlooked importance of particle-hole symmetry in these systems. We give concrete examples how the hybrid qubit system could be realized experimentally for Z4 and Z6 parafermions. In addition, we discuss a simple readout scheme for the fractional parafermion charge via the measurement of the spin qubit resonant frequency.

Heavy-hole--light-hole mixing at the zone center

Peter Stano, Daniel Loss arXiv:2405.16844 We investigate heavy-hole--light-hole mixing in quasi-two-dimensional semiconducting structures. We propose and analyze several measures that characterize the mixing quantitatively in Ge, Si, and GaAs. We find that a mixing that is `weak' when considering the light-hole content can be `strong' when considering the induced spin-orbit interactions (SOIs). We identify two different types of hole-Hamiltonian terms concerning mixing. The first type changes the direction of the pure spinors, without admixing light-hole components. It is efficient for Rabi driving the heavy-hole spin. The second type induces mixing and changes the eigenvalues of the g tensor. We restrict ourselves to the zone center, what leads to a simple description, analytical results, and physical insights. We show how the model can be used to investigate heavy-hole spin qubit g tensor, dephasing, relaxation, and Rabi frequencies.

A coherence sweet spot with enhanced dipolar coupling

Jann H. Ungerer, Alessia Pally, Stefano Bosco, Artem Kononov, Deepankar Sarmah, Sebastian Lehmann, Claes Thelander, Ville F. Maisi, Pasquale Scarlino, Daniel Loss, Andreas Baumgartner, Christian Schönenberger arXiv:2405.10796 Qubits require a compromise between operation speed and coherence. Here, we demonstrate a compromise-free singlet-triplet (ST) qubit, where the qubit couples maximally to the driving field while simultaneously coupling minimally to the dominant noise sources. The qubit is implemented in a crystal-phase defined double-quantum dot in an InAs nanowire. Using a superconducting resonator, we measure the spin-orbit interaction (SOI) gap, the spin-photon coupling strength and the qubit decoherence rate as a function of the in-plane magnetic-field orientation. We demonstrate a spin qubit sweet spot maximizing the dipolar coupling and simultaneously minimizing the decoherence. Our theoretical description postulates phonons as the most likely dominant noise source. The compromise-free sweet spot originates from the SOI suggesting that it is not restricted to this material platform, but might find applications in any material with SOI. These findings pave the way for enhanced engineering of these nanomaterials for next-generation qubit technologies.

Coherence limit due to hyperfine interaction with nuclei in the barrier material of Si spin qubits

Lukas Cvitkovich, Peter Stano, Christoph Wilhelmer, Dominic Waldhör, Daniel Loss, Yann-Michel Niquet, Tibor Grasser arXiv:2405.10667 On the quest to understand and reduce environmental noise in Si spin qubits, hyperfine interactions between electron and nuclear spins impose a major challenge. Silicon is a promising host material because one can enhance the spin coherence time by removing spinful 29Si isotopes. As more experiments rely on isotopic purification of Si, the role of other spinful atoms in the device should be clarified. This is not a straightforward task, as the hyperfine interactions with atoms in the barrier layers are poorly understood. We utilize density functional theory to determine the hyperfine tensors of both Si and Ge in a crystalline epitaxial Si/SiGe quantum well as well as Si and O atoms in an amorphous Si/SiO2 (MOS) interface structure. Based on these results, we estimate the dephasing time T∗2 due to magnetic noise from the spin bath and show that the coherence is limited by interactions with non-Si barrier atoms to a few \textmu s in Si/SiGe (for non-purified Ge) and about 100\,\textmu s in Si-MOS. Expressing these numbers alternatively, in Si/SiGe the interactions with Ge dominate below 1000\,ppm of 29Si content, and, due to low natural concentration of the spinful oxygen isotopes, the interactions with oxygen in Si-MOS become significant only below 1\,ppm of 29Si content.

Cavity-enhanced superconductivity via band engineering

Valerii K. Kozin, Even Thingstad, Daniel Loss, Jelena Klinovaja arXiv:2405.08642 We consider a two-dimensional electron gas interacting with a quantized cavity mode. We find that the coupling between the electrons and the photons in the cavity enhances the superconducting gap. Crucially, all terms in the Peierls phase are kept, in contrast to more naive approaches, which may result in spurious superradiant phase transitions. We use a mean-field theory to show that the gap increases approximately linearly with the cavity coupling strength. The effect can be observed locally as an increase in the gap size via scanning tunneling microscopy (STM) measurements for a flake of a 2D material (or for a Moiré system where the enhancement is expected to be more pronounced due to a large lattice constant) interacting with a locally-structured electromagnetic field formed by split-ring resonators. Our results are also relevant for quantum optics setups with cold atoms interacting with the cavity mode, where the lattice geometry and system parameters can be tuned in a vast range.

Topological Interlayer Superconductivity in a van der Waals Heterostructure

Even Thingstad, Joel Hutchinson, Daniel Loss, Jelena Klinovaja arXiv:2405.07927 We show that when a honeycomb antiferromagnetic insulator (AFMI) is sandwiched between two transition metal dichalcogenide (TMD) monolayers in a commensurate way, magnons in the AFMI can mediate an interaction between electrons in the TMDs that gives rise to interlayer Cooper pairing. This interaction opens coexisting extended s-wave and chiral p-wave superconducting gaps in the energy spectrum of the coupled system, and the latter give rise to topological Majorana edge modes.

Tunable Ultrafast Dynamics of Antiferromagnetic Vortices in Nanoscale Dots

Ji Zou, Even Thingstad, Se Kwon Kim, Jelena Klinovaja, Daniel Loss arXiv:2404.18306 Topological vortex textures in magnetic disks have garnered great attention due to their interesting physics and diverse applications. However, up to now, the vortex state has mainly been studied in microsize ferromagnetic disks, which have oscillation frequencies confined to the GHz range. Here, we propose an experimentally feasible ultrasmall and ultrafast vortex state in an antiferromagnetic nanodot surrounded by a heavy metal, which is further harnessed to construct a highly tunable vortex network. We theoretically demonstrate that, interestingly, the interfacial Dzyaloshinskii-Moriya interaction (iDMI) induced by the heavy metal at the boundary of the dot acts as an effective chemical potential for the vortices in the interior. Mimicking the creation of a superfluid vortex by rotation, we show that a magnetic vortex state can be stabilized by this iDMI. Subjecting the system to an electric current can trigger vortex oscillations via spin-transfer torque, which reside in the THz regime and can be further modulated by external magnetic fields. Furthermore, we show that coherent coupling between vortices in different nanodisks can be achieved via an antiferromagnetic link. Remarkably, this interaction depends on the vortex polarity and topological charge and is also exceptionally tunable through the vortex resonance frequency. This opens up the possibility for controllable interconnected networks of antiferromagnetic vortices. Our proposal therefore introduces a new avenue for developing high-density memory, ultrafast logic devices, and THz signal generators, which are ideal for compact integration into microchips.

Relations between normal state nonreciprocal transport and the superconducting diode effect in the trivial and topological phases

Georg Angehrn, Henry F. Legg, Daniel Loss, Jelena Klinovaja Appl. Phys. Lett. 125, 023507 (2024); arXiv:2404.17501. Nonreciprocal transport effects can occur in the normal state of conductors and in superconductors when both inversion and time-reversal symmetry are broken. Here, we consider systems where magnetochiral anisotropy (MCA) of the energy spectrum due to an externally applied magnetic field results in a rectification effect in the normal state and a superconducting (SC) diode effect when the system is proximitised by a superconductor. Focussing on nanowire systems, we obtain analytic expressions for both normal state rectification and SC diode effects that reveal the commonalities - as well as differences - between these two phenomena. Furthermore, we consider the nanowire brought into an (almost) helical state in the normal phase or a topological superconducting phase when proximitised. In both cases this reveals that the topology of the system considerably modifies its nonreciprocal transport properties. Our results provide new insights into how to determine the origin of nonreciprocal effects and further evince the strong connection of nonreciprocal transport with the topological properties of a system

Spin-resolved nonlocal transport in proximitized Rashba nanowires

Paweł Szumniak, Daniel Loss, Jelena Klinovaja arXiv:2404.08527 Non-equilibrium transport in hybrid semiconductor-superconductor nanowires is crucial for many quantum phenomena such as generating entangled states via cross Andreev reflection (CAR) processes, detecting topological superconductivity, reading out Andreev spin qubits, coupling spin qubits over long distances and so on. Here, we investigate numerically transport properties of a proximitized Rashba nanowire that hosts spin-polarized low-energy quasiparticle states. We show that the spin polarization in such one-dimensional Andreev bands, extended over the entire nanowire length, can be detected in nonlocal transport measurements with tunnel-coupled side leads that are spin polarized. Remarkably, we find an exact correspondence between the sign of the nonlocal conductance and the spin density of the superconducting quasiparticles at the side lead position. We demonstrate that this feature is robust to moderate static disorder. As an example, we show that such a method can be used to detect spin inversion of the bands, accompanying the topological phase transition (TPT) for realistic system parameters. Furthermore, we show that such effects can be used to switch between CAR and elastic cotunneling (ECT) processes by tuning the strength of either the electric or the magnetic field. These findings hold significant practical implications for state-of-the-art transport experiments in such hybrid systems.

Magnonic φ Josephson Junctions and Synchronized Precession

Kouki Nakata, Ji Zou, Jelena Klinovaja, Daniel Loss arXiv:2403.01625 There has been a growing interest in non-Hermitian physics. One of its main goals is to engineer dissipation and to explore ensuing functionality. In magnonics, the effect of dissipation due to local damping on magnon transport has been explored. However, the effects of non-local damping on the magnonic analog of the Josephson effect remain missing, despite that non-local damping is inevitable and has been playing a central role in magnonics. Here, we uncover theoretically that a surprisingly rich dynamics can emerge in magnetic junctions due to intrinsic non-local damping, using analytical and numerical methods. In particular, under microwave pumping, we show that coherent spin precession in the right and left insulating ferromagnet (FM) of the junction becomes synchronized by non-local damping and thereby a magnonic analog of the φ Josephson junction emerges, where φ stands here for the relative precession phase of right and left FM in the stationary limit. Remarkably, φ decreases monotonically from π to π/2 as the magnon-magnon interaction, arising from spin anisotropies, increases. Moreover, we also find a magnonic diode effect giving rise to rectification of magnon currents. Our predictions are readily testable with current device and measurement technologies at room temperatures.

Fractional Spin Quantum Hall Effect in Weakly Coupled Spin Chain Arrays

Even Thingstad, Pierre Fromholz, Flavio Ronetti, Daniel Loss, Jelena Klinovaja arXiv:2402.10849 Topological magnetic insulators host chiral gapless edge modes. In the presence of strong interaction effects, the spin of these modes may fractionalize. Studying a 2D array of coupled insulating spin-1/2 chains, we show how spatially modulated magnetic fields and Dzyaloshinskii-Moriya interactions can be exploited to realize chiral spin liquids or integer and fractional spin quantum Hall effect phases. These are characterized by a gapped bulk spectrum and gapless chiral edge modes with fractional spin. The spin fractionalization is manifested in the quantized spin conductance, which can be used to probe the fractional spin quantum Hall effect. We analyze the system via bosonization and perturbative renormalization group techniques that allow us to identify the most relevant terms induced by the spin-spin interactions that open gaps and render the system topological under well-specified resonance conditions. We show explicitly that the emerging phase is a genuine chiral spin liquid. We suggest that the phases can be realized experimentally in synthetic spin chains and ultracold atom systems.

A Qubit with Simultaneously Maximized Speed and Coherence

Miguel J. Carballido, Simon Svab, Rafael S. Eggli, Taras Patlatiuk, Pierre Chevalier Kwon, Jonas Schuff, Rahel M. Kaiser, Leon C. Camenzind, Ang Li, Natalia Ares, Erik P.A.M Bakkers, Stefano Bosco, J. Carlos Egues, Daniel Loss, Dominik M. Zumbühl arXiv:2402.07313 To overcome the threshold for fault-tolerant quantum computation, qubits have to be protected from their noisy environment to attain the necessary high fidelities. Recent experiments discovered sweet spots with strongly enhanced coherence. However, decoupling a qubit from its surroundings also limits the control over the qubit's state, typically leading to either coherent but slow or fast but short-lived qubits. This trade-off appears to be a severe fundamental limitation hampering the performance of qubits. Here, we show how this can be circumvented by demonstrating a simultaneously fast and coherent tunable regime in a hole spin qubit. In this regime, we can triple the operation speed, while simultaneously quadrupling the coherence time when tuning a local electric field, demonstrating that the qubit speed and coherence scale together without compromise. This relies on strong, quasi 1D confinement providing a local maximum in drive strength, where charge fluctuations are decoupled and thus the coherence is enhanced, yet the drive speed is maximal. A Ge/Si core/shell nanowire, operated at 1.5 K, provides the strong confinement. The driving mechanism here is the strong and tunable direct Rashba spin-orbit interaction, achieving a maximal strength at finite electrical field due to gate-dependent heavy-hole light-hole mixing. Breaking the speed-coherence trade-off makes it possible to boost fidelity and speed of one- and two-qubit gates. This concept can be expanded to planar arrays of hole or electron spin qubits as well. In this regime, the coupling to a microwave resonator is also predicted to be both strong and coherent. Altogether, this is opening a new path towards fault-tolerant quantum computation.

Microscopic Mechanism of Pair-, Charge- and Spin-Density-Wave Instabilities in Interacting D-Dimensional Fermi Liquids

Dmitry Miserev, Herbert Schoeller, Jelena Klinovaja, Daniel Loss Phys. Rev. B 110, 125128 (2024); arXiv:2312.17208. We present an analytic theory unraveling the microscopic mechanism of instabilities within interacting D-dimensional Fermi liquid. Our model consists of a D-dimensional electron gas subject to an instantaneous electron-electron interaction of a finite range exceeding the average interparticle distance. Pair, charge, and spin susceptibilities are evaluated via the one-loop renormalization group theory and via the bosonization approach, giving identical results. In case of a repulsive interaction, we identify an intrinsic Fermi-liquid instability towards insulating spin- and charge-density-wave order when the interaction coupling strength reaches a universal critical value. If both electron and hole pockets of the same size are present, the ground state is an excitonic insulator at arbitrarily small repulsive interaction. If the interaction is attractive, the ground state is a singlet non-BCS superconductor with a uniform condensate. In case if both electron and hole Fermi surfaces are present, we predict an instability towards the interpocket pair-density-wave ordering at the critical coupling. This prediction lends strong theoretical support to the pair-density-wave scenario of superconductivity in cuprate materials. Due to its simple and universal nature, presented microscopic mechanism of intrinsic instabilities of interacting D-dimensional Fermi liquids constitutes a solid theoretical ground for understanding quantum phase transitions in a variety of quantum materials, from ultraclean semiconductor quantum wells to high-temperature superconductors.

High-fidelity spin qubit shuttling via large spin-orbit interaction

Stefano Bosco, Ji Zou, Daniel Loss PRX Quantum 5, 020353 (2024); arXiv:2311.15970. Shuttling spins with high fidelity is a key requirement to scale up semiconducting quantum computers, enabling qubit entanglement over large distances and favoring the integration of control electronics on-chip. To decouple the spin from the unavoidable charge noise, state-of-the-art spin shuttlers try to minimize the inhomogeneity of the Zeeman field. However, this decoupling is challenging in otherwise promising quantum computing platforms such as hole spin qubits in silicon and germanium, characterized by a large spin-orbit interaction and electrically-tunable qubit frequency. In this work, we show that, surprisingly, the large inhomogeneity of the Zeeman field stabilizes the coherence of a moving spin state, thus enabling high-fidelity shuttling also in these systems. We relate this enhancement in fidelity to the deterministic dynamics of the spin which filters out the dominant low-frequency contributions of the charge noise. By simulating several different scenarios and noise sources, we show that this is a robust phenomenon generally occurring at large field inhomogeneity. By appropriately adjusting the motion of the quantum dot, we also design realistic protocols enabling faster and more coherent spin shuttling. Our findings are generally applicable to a wide range of setups and could pave the way toward large-scale quantum processors.

Strong hole-photon coupling in planar Ge: probing the charge degree and Wigner molecule states

Franco De Palma, Fabian Oppliger, Wonjin Jang, Stefano Bosco, Marián Janík, Stefano Calcaterra, Georgios Katsaros, Giovanni Isella, Daniel Loss, Pasquale Scarlino arXiv:2310.20661 Semiconductor quantum dots (QDs) in planar germanium (Ge) heterostructures have emerged as frontrunners for future hole-based quantum processors. Notably, the large spin-orbit interaction of holes offers rapid, coherent electrical control of spin states, which can be further beneficial for interfacing hole spins to microwave photons in superconducting circuits via coherent charge-photon coupling. Here, we present strong coupling between a hole charge qubit, defined in a double quantum dot (DQD) in a planar Ge, and microwave photons in a high-impedance (Z_r=1.3 kΩ) superconducting quantum interference device (SQUID) array resonator. Our investigation reveals vacuum-Rabi splittings with coupling strengths up to g_0/2\pi=260 MHz, and a cooperativity of C∼100, dependent on DQD tuning, confirming the strong charge-photon coupling regime within planar Ge. Furthermore, utilizing the frequency tunability of our resonator, we explore the quenched energy splitting associated with strongly-correlated Wigner molecule (WM) states that emerge in Ge QDs. The observed enhanced coherence of the WM excited state signals the presence of distinct symmetries within related spin functions, serving as a precursor to the strong coupling between photons and spin-charge hybrid qubits in planar Ge. This work paves the way towards coherent quantum connections between remote hole qubits in planar Ge, required to scale up hole-based quantum processors.

Gate-tunable topological superconductivity in a supramolecular electron spin lattice

Jung-Ching Liu, Chao Li, Richard Hess, Hongyan Chen, Carl Drechsel, Ping Zhou, Robert Häner, Ulrich Aschauer, Thilo Glatzel, Silvio Decurtins, Daniel Loss, Jelena Klinovaja, Shi-Xia Liu, Wulf Wulfhekel, Ernst Meyer, Rémy Pawlak arXiv:2310.18134 Topological superconductivity emerges in chains or arrays of magnetic atoms coupled to a superconductor. However, the external controllability of such systems with gate voltages is detrimental for their future implementation in a topological quantum computer. Here we showcase the supramolecular assembly of radical molecules on Pb(111), whose discharge is controlled by the tip of a scanning tunneling microscope. Charged molecules carry a spin-1/2 state, as confirmed by observing Yu-Shiba-Rusinov in-gap states by tunneling spectroscopy at millikelvin temperature. Low energy modes are localized at island boundaries with a long decay towards the interior, whose spectral signature is consistent with Majorana zero modes protected by mirror symmetry. Our results open up a vast playground for the synthesis of gate-tunable organic topological superconductors.

Quantum phase transitions and cat states in cavity-coupled quantum dots

Valerii K. Kozin, Dmitry Miserev, Daniel Loss, Jelena Klinovaja Phys. Rev. Research 6, 033188 (2024); Editors' Suggestion; arXiv:2310.15167. We study double quantum dots coupled to a quasistatic cavity mode with high mode-volume compression allowing for strong light-matter coupling. Besides the cavity-mediated interaction, electrons in different double quantum dots interact with each other via dipole-dipole (Coulomb) interaction. For attractive dipolar interaction, a cavity-induced ferroelectric quantum phase transition emerges leading to ordered dipole moments. Surprisingly, we find that the phase transition can be either continuous or discontinuous, depending on the ratio between the strengths of cavity-mediated and Coulomb interactions. We show that, in the strong coupling regime, both the ground and the first excited states of an array of double quantum dots are squeezed Schrödinger cat states. Such states are actively discussed as high-fidelity qubits for quantum computing, and thus our proposal provides a platform for semiconductor implementation of such qubits. We also calculate gauge-invariant observables such as the net dipole moment, the optical conductivity, and the absorption spectrum beyond the semiclassical approximation.

Spin susceptibility in interacting two-dimensional semiconductors and bilayer systems at first order: Kohn anomalies and spin density wave ordering

Joel Hutchinson, Dmitry Miserev, Jelena Klinovaja, Daniel Loss Phys. Rev. B 109, 075139 (2024); arXiv:2310.05555. This work is an analytic theoretical study of a 2D semiconductor with a Fermi surface that is split by the Zeeman coupling of electron spins to an external magnetic field in the presence of electron-electron interactions. For the first time, we calculate the spin susceptibility for long-range and finite-range interactions diagrammatically, and find a resonant peak structure at the Kohn anomaly already in first-order perturbation theory. In contrast to the density-density correlator that is suppressed due to the large electrostatic energy required to stabilize charge density order, the spin susceptibility does not suffer from electrostatic screening effects, thus favouring spin-density-wave order in 2D semiconductors. Our results impose significant consequences for determining magnetic phases in 2D semiconductors. For example, a strongly enhanced Kohn anomaly may result in helical ordering of magnetic impurities due to the RKKY interaction. Furthermore, the spin degree of freedom can equally represent a layer pseudospin in the case of bilayer materials. In this case, the external "magnetic field" is a combination of layer bias and interlayer hopping. The sharp peak of the 2D static spin susceptibility may then be responsible for dipole-density-wave order in bilayer materials at large enough electron-phonon coupling.

Valley-free silicon fins by shear strain

Christoph Adelsberger, Stefano Bosco, Jelena Klinovaja, Daniel Loss Phys. Rev. Lett. 133, 037001 (2024); arXiv:2308.13448. Electron spins confined in silicon quantum dots are promising candidates for large-scale quantum computers. However, the degeneracy of the conduction band of bulk silicon introduces additional levels dangerously close to the window of computational energies, where the quantum information can leak. The energy of the valley states - typically 0.1 meV - depends on hardly controllable atomistic disorder and still constitutes a fundamental limit to the scalability of these architectures. In this work, we introduce designs of CMOS-compatible silicon fin field-effect transistors that enhance the energy gap to non-computational states by more than one order of magnitude. Our devices comprise realistic silicon-germanium nanostructures with a large shear strain, where troublesome valley degrees of freedom are completely removed. The energy of non-computational states is therefore not affected by unavoidable atomistic disorder and can further be tuned in-situ by applied electric fields. Our design ideas are directly applicable to a variety of setups and will offer a blueprint towards silicon-based large-scale quantum processors.

Josephson transistor from the superconducting diode effect in domain wall and skyrmion magnetic racetracks

Richard Hess, Henry F. Legg, Daniel Loss, Jelena Klinovaja Phys. Rev. B 108, 174516 (2023); arXiv:2308.04817. In superconductors, the combination of broken time-reversal and broken inversion symmetries can result in a critical current being dependent on the direction of current flow. This phenomenon is known as superconducting diode effect (SDE) and has great potential for applications in future low-temperature electronics. Here, we investigate how magnetic textures such as domain walls or skyrmions on a racetrack can be used to control the SDE in a Josephson junction and how the SDE can be used as a low-temperature read-out of the data in racetrack memory devices. First, we consider a two-dimensional electron gas (2DEG) with strong spin-orbit-interaction (SOI) coupled to a magnetic racetrack, which forms the weak-link in a Josephson junction. In this setup, the exchange coupling between the magnetic texture and the itinerant electrons in the 2DEG breaks time-reversal symmetry and enables the SDE. When a magnetic texture, such as a domain wall or skyrmion enters the Josephson junction, the local exchange field within the junction is changed and, consequently, the strength of the SDE is altered. In particular, depending on the position and form of the magnetic texture, moving the magnetic texture can cause the SDE coefficient to change its sign, enabling a Josephson transistor effect with potentially fast switching frequencies. Further, we find that the SDE is enhanced if the junction length-scales are comparable with the length-scale of the magnetic texture. Furthermore, we show that, under certain circumstances, the symmetry breaking provided by particular magnetic textures, such as skyrmions, can lead to an SDE even in the absence of Rashba SOI in the 2DEG. Our results provide a basis for new forms of readout in low-temperature memory devices as well as demonstrating how a Josephson transistor effect can be achieved even in the absence of an external magnetic field and intrinsic Rashba SOI.

Spatially correlated classical and quantum noise in driven qubits

Ji Zou, Stefano Bosco, Daniel Loss npj Quantum Inf 10, 46 (2024); arXiv:2308.03054. Correlated noise across multiple qubits poses a significant challenge for achieving scalable and fault-tolerant quantum processors. Despite recent experimental efforts to quantify this noise in various qubit architectures, a comprehensive understanding of its role in qubit dynamics remains elusive. Here, we present an analytical study of the dynamics of driven qubits under spatially correlated noise, including both Markovian and non-Markovian noise. Surprisingly, we find that, while correlated classical noise only leads to correlated decoherence without increasing the quantum coherence in the system, the correlated quantum noise can be exploited to generate entanglement. In particular, we reveal that, in the quantum limit, pure dephasing noise induces a coherent long-range two-qubit Ising interaction that correlates distant qubits. In contrast, for purely transverse noise when qubits are subjected to coherent drives, the correlated quantum noise induces both coherent symmetric exchange and Dzyaloshinskii-Moriya interaction between the qubits, as well as correlated relaxation, both of which give rise to significant entanglement. Remarkably, in this case, we uncover that the system exhibits distinct dynamical phases in different parameter regimes. Finally, we reveal the impact of spatio-temporally correlated 1/f noise on the decoherence rate, and how its temporal correlations restore lost entanglement. Our analysis not only offers critical insights into designing effective error mitigation strategies to reduce harmful effects of correlated noise, but also enables tailored protocols to leverage and harness noise-induced correlations for quantum information processing.

Reply to "Comment on 'Trivial Andreev Band Mimicking Topological Bulk Gap Reopening in the Nonlocal Conductance of Long Rashba Nanowires'"

Richard Hess, Henry F. Legg, Daniel Loss, Jelena Klinovaja arXiv:2306.16853 (reply 1a); Phys. Rev. Lett. 132, 099602 (2024) (reply 1b); arXiv:2308.10669 (reply 2). Reply 1a:In this Reply we respond to the comment by Das Sarma and Pan [1] on Hess et al., Phys. Rev. Lett. 130, 207001, "Trivial Andreev Band Mimicking Topological Bulk Gap Reopening in the Nonlocal Conductance of Long Rashba Nanowires" [2]. First, we note that Das Sarma and Pan reproduce the key results of Hess et al., substantiating that our findings are entirely valid. Next, we clarify the incorrect statement by Das Sarma and Pan that the main result of Hess et al. requires a "contrived periodic pristine system", pointing out the extensive discussion of positional disorder in the Hess et al. We also demonstrate that nonlocal conductance features are generically reduced by disorder. This applies to both an Andreev band and to a genuine topological bulk gap reopening signature (BRS). In fact, the suppression of nonlocal conductance of a genuine BRS by disorder was discussed in, e.g., Pan, Sau, Das Sarma, PRB 103, 014513 (2021) [3]. We conclude that, contrary to the claims of Das Sarma and Pan, the minimal model of Hess et al. is relevant to current realistic nanowire devices where only a few overlapping ABSs would be required to mimic a BRS. Reply 2: In this Reply we respond to the comment by Antipov et al. from Microsoft Quantum on Hess et al., PRL 130, 207001 (2023). Antipov et al. reported only a single simulation and claimed it did not pass the Microsoft Quantum topological gap protocol (TGP). They have provided no parameters or data for this simulation (despite request). Regardless, in this reply we demonstrate that the trivial bulk gap reopening mechanism outlined in Hess et al., in combination with trivial ZBPs, passes the TGP and therefore can result in TGP false positives.

Dissipative Spin-wave Diode and Nonreciprocal Magnonic Amplifier

Ji Zou, Stefano Bosco, Even Thingstad, Jelena Klinovaja, Daniel Loss Phys. Rev. Lett. 132, 036701 (2024); arXiv:2306.15916. We propose an experimentally feasible dissipative spin-wave diode comprising two magnetic layers coupled via a non-magnetic spacer. We theoretically demonstrate that the spacer mediates not only coherent interactions but also dissipative coupling. Interestingly, an appropriately engineered dissipation engenders a nonreciprocal device response, facilitating the realization of a spin-wave diode. This diode permits wave propagation in one direction alone, given that the coherent Dzyaloshinskii- Moriya (DM) interaction is balanced with the dissipative coupling. The polarity of the diode is determined by the sign of the DM interaction. Furthermore, we show that when the magnetic layers undergo incoherent pumping, the device operates as a unidirectional spin-wave amplifier. The amplifier gain is augmented by cascading multiple magnetic bilayers. By extending our model to a one-dimensional ring structure, we establish a connection between the physics of spin-wave amplification and non-Hermitian topology. Our proposal opens up a new avenue for harnessing inherent dissipation in spintronic applications.

Dynamical nuclear spin polarization in a quantum dot with an electron spin driven by electric dipole spin resonance

Peter Stano, Takashi Nakajima, Akito Noiri, Seigo Tarucha, Daniel Loss Phys. Rev. B 108, 155306 (2023); Editor's suggestion; arXiv:2306.11253. We analyze the polarization of nuclear spins in a quantum dot induced by a single-electron spin that is electrically driven to perform coherent Rabi oscillations. We derive the associated nuclear-spin polarization rate and analyze its dependence on the accessible control parameters, especially the detuning of the driving frequency from the electron Larmor frequency. The arising nuclear-spin polarization is related to the Hartmann-Hahn effect known from the NMR literature with two important differences. First, in quantum dots, one typically uses a micromagnet, leading to a small deflection of the quantization axes of the electron and nuclear spins. Second, the electric driving wiggles the electron with respect to the atomic lattice. The two effects, absent in the traditional Hartmann-Hahn scenario, give rise to two mechanisms of nuclear-spin polarization in gated quantum dots. The arising nuclear-spin polarization is a resonance phenomenon, achieving maximal efficiency at the resonance of the electron Rabi and nuclear Larmor frequency (typically a few or a few tens of MHz). As a function of the driving frequency, the polarization rate can develop sharp peaks and reach large values at them. Since the nuclear polarization is experimentally detected as changes of the electron Larmor frequency, we often convert the former to the latter in our formulas and figures. In these units, the polarization can reach hundreds of MHz/s in GaAs quantum dots and at least tens of kHz/s in Si quantum dots. We analyze possibilities to exploit the resonant polarization effects for achieving large nuclear polarization and for stabilizing the Overhauser field through feedback.

Microscopic analysis of proximity-induced superconductivity and metallization effects in superconductor-germanium hole nanowires

Christoph Adelsberger, Henry F. Legg, Daniel Loss, Jelena Klinovaja Phys. Rev. B 108, 155433 (2023); arXiv:2306.06944. Low-dimensional Ge hole devices are promising systems with many potential applications such as hole spin qubits, Andreev spin qubits, Josephson junctions, and can serve as a basis for the realization of topological superconductivity. This vast array of potential uses for Ge largely stems from the exceptionally strong and controllable spin-orbit interaction (SOI), ultralong mean free paths, long coherence times, and CMOS compatibility. However, when brought into proximity with a superconductor (SC), metallization normally diminishes many useful properties of a semiconductor, for instance, typically reducing the g factor and SOI energy, as well as renormalizing the effective mass. In this paper we consider metallization of a Ge nanowire (NW) in proximity to a SC, explicitly taking into account the 3D geometry of the NW. We find that proximitized Ge exhibits a unique phenomenology of metallization effects, where the 3D cross section plays a crucial role. For instance, in contrast to expectations, we find that SOI can be enhanced by strong coupling to the superconductor. We also show that the thickness of the NW plays a critical role in determining both the size of the proximity induced pairing potential and metallization effects, since the coupling between NW and SC strongly depends on the distance of the NW wave function from the interface with the SC. In the absence of electrostatic effects, we find that a sizable gap opens only in thin NWs (d≲3 nm). In thicker NWs, the wave function must be pushed closer to the SC by electrostatic effects in order to achieve a sizable proximity gap such that the required electrostatic field strength can simultaneously induce a strong SOI. The unique and sometimes beneficial nature of metallization effects in SC-Ge NW devices evinces them as ideal platforms for future applications in quantum information processing.

Majorana Bound States in Germanium Josephson Junctions via Phase Control

Melina Luethi, Henry F. Legg, Katharina Laubscher, Daniel Loss, Jelena Klinovaja Phys. Rev. B 108, 195406 (2023); arXiv:2304.12689. We consider superconductor-normal-superconductor-normal-superconductor (SNSNS) planar Josephson junctions in hole systems with spin-orbit interaction that is cubic in momentum (CSOI). Utilizing only the superconducting phase difference, we find parameter `sweet spots' for reasonable junction transparencies that result in a topological region of phase space, within which Majorana bound states (MBSs) appear at the ends of the junction. In planar germanium hetereostructures CSOI can be the dominant form of SOI and extremely strong. We show analytically and numerically that, within experimental regimes, our results provide an achievable roadmap for a new MBS platform with low disorder, minimal magnetic fields, and very strong spin-orbit interaction, overcoming many of the key deficiencies that have so far prevented the conclusive observation of MBSs.

Cavity-induced charge transfer in periodic systems: length-gauge formalism

Ekaterina Vlasiuk, Valerii K. Kozin, Jelena Klinovaja, Daniel Loss, Ivan V. Iorsh, Ilya V. Tokatly Phys. Rev. B 108, 085410 (2023); arXiv:2304.11364. We develop a length-gauge formalism for treating one-dimensional periodic lattice systems in the presence of a photon cavity inducing light-matter interaction. The purpose of the formalism is to remove mathematical ambiguities that occur when defining the position operator in the context of the Power-Zienau-Woolley Hamiltonian. We then use a diagrammatic approach to analyze perturbatively the interaction between an electronic quantum system and a photonic cavity mode of long wavelength. We illustrate the versatility of the formalism by studying the cavity-induced electric charge imbalance and polarization in the Rice-Mele model with broken inversion symmetry.

Dimensional reduction of the Luttinger-Ward functional for spin-degenerate D-dimensional electron gases

D. Miserev, J. Klinovaja, D. Loss Phys. Rev. B 108, 235116 (2023); arXiv:2303.16732. We consider an isotropic spin-degenerate interacting uniform D-dimensional electron gas (DDEG) with D>1 within the Luttinger-Ward (LW) formalism. We derive the asymptotically exact semiclassical/infrared limit of the LW functional at large distances, r≫λF, and large times, τ≫1/EF, where λF and EF are the Fermi wavelength and the Fermi energy, respectively. The LW functional is represented by skeleton diagrams, each skeleton diagram consists of appropriately connected dressed fermion loops. First, we prove that every D-dimensional skeleton diagram consisting of a single fermion loop is reduced to a one-dimensional (1D) fermion loop with the same diagrammatic structure, which justifies the name dimensional reduction. This statement, combined with the fermion loop cancellation theorem (FLCT), agrees with results of multidimensional bosonization. Here we show that the backscattering and the spectral curvature, both explicitly violate the FLCT and both are irrelevant for a 1DEG, become relevant at D>1 and D>2, respectively. The reason for this is a strong infrared divergence of the skeleton diagrams containing multiple fermion loops at D>1. These diagrams, which are omitted within the multidimensional bosonization approaches, account for the non-collinear scattering processes. Thus, the dimensional reduction provides the framework to go beyond predictions of the multidimensional bosonization. A simple diagrammatic structure of the reduced LW functional is another advantage of our approach. The dimensional reduction technique is also applicable to the thermodynamic potential and various approximations, from perturbation theory to self-consistent approaches.

High-fidelity two-qubit gates of hybrid superconducting-semiconducting singlet-triplet qubits

Maria Spethmann, Stefano Bosco, Andrea Hofmann, Jelena Klinovaja, Daniel Loss Phys. Rev. B 109, 085303 (2024); arXiv:2304.05086. Hybrid systems comprising superconducting and semiconducting materials are promising architectures for quantum computing. Superconductors induce long-range interactions between the spin degrees of freedom of semiconducting quantum dots. These interactions are widely anisotropic when the semiconductor material has strong spin-orbit interactions. We show that this anisotropy is tunable and enables fast and high-fidelity two-qubit gates between singlet-triplet (ST) spin qubits. Our design is immune to leakage of the quantum information into non-computational states and removes always-on interactions between the qubits, thus resolving key open challenges for these architectures. Our ST qubits do not require additional technologically-demanding components nor fine-tuning of parameters. They operate at low magnetic fields of a few milli Tesla and are fully compatible with superconductors. In realistic devices, we estimate infidelities below 10−3, that could pave the way toward large-scale hybrid superconducting-semiconducting quantum processors.

Phase driving hole spin qubits

Stefano Bosco, Simon Geyer, Leon C. Camenzind, Rafael S. Eggli, Andreas Fuhrer, Richard J. Warburton, Dominik M. Zumbühl, J. Carlos Egues, Andreas V. Kuhlmann, Daniel Loss Phys. Rev. Lett. 131, 197001 (2023); Editor's suggestion; arXiv:2303.03350. The spin-orbit interaction in spin qubits enables spin-flip transitions, resulting in Rabi oscillations when an external microwave field is resonant with the qubit frequency. Here, we introduce an alternative driving mechanism of hole spin qubits, where a far-detuned oscillating field couples to the qubit phase. Phase driving at radio frequencies, orders of magnitude slower than the microwave qubit frequency, induces highly non-trivial spin dynamics, violating the Rabi resonance condition. By using a qubit integrated in a silicon fin field-effect transistor (Si FinFET), we demonstrate a controllable suppression of resonant Rabi oscillations, and their revivals at tunable sidebands. These sidebands enable alternative qubit control schemes using global fields and local far-detuned pulses, facilitating the design of dense large-scale qubit architectures with local qubit addressability. Phase driving also decouples Rabi oscillations from noise, an effect due to a gapped Floquet spectrum and can enable Floquet engineering high-fidelity gates in future quantum processors.

General scatterings and electronic states in the quantum-wire network of moiré systems

Chen-Hsuan Hsu, Daniel Loss, Jelena Klinovaja Phys. Rev. B 108, L121409 (2023); arXiv:2303.00759. We investigate electronic states in a two-dimensional network consisting of interacting quantum wires, a model adopted for twisted bilayer systems. We construct general operators which describe various scattering processes in the system. In a twisted bilayer structure, the moiré periodicity allows for generalized umklapp scatterings, leading to a class of correlated states at fractional fillings. We identify scattering processes which can lead to an insulating bulk with gapless chiral edge modes at certain fractional fillings, resembling the quantum anomalous Hall effect recently observed in twisted bilayer graphene. Finally, we propose experimental setups to detect and characterize the edge modes through spectroscopic and transport measurements.

Spatial noise correlations beyond nearest-neighbor in 28Si/SiGe spin qubits

Juan S. Rojas-Arias, Akito Noiri, Peter Stano, Takashi Nakajima, Jun Yoneda, Kenta Takeda, Takashi Kobayashi, Amir Sammak, Giordano Scappucci, Daniel Loss, Seigo Tarucha Phys. Rev. Applied 20, 054024 (2023); arXiv:2302.11717. We detect correlations in qubit-energy fluctuations of non-neighboring qubits defined in isotopically purified Si/SiGe quantum dots. At low frequencies (where the noise is strongest), the correlation coefficient reaches 10% for a next-nearest-neighbor qubit-pair separated by 200 nm. Assigning the observed noise to be of electrical origin, a simple theoretical model quantitatively reproduces the measurements and predicts a polynomial decay of correlations with interqubit distance. Our results quantify long-range correlations of noise dephasing quantum-dot spin qubits arranged in arrays, essential for scalability and fault-tolerance of such systems.

Parity protected superconducting diode effect in topological Josephson junctions

Henry F. Legg, Katharina Laubscher, Daniel Loss, Jelena Klinovaja Phys. Rev. B 108, 214520 (2023); arXiv:2301.13740. In bulk superconductors or Josephson junctions formed in materials with spin-orbit interaction, the critical current can depend on the direction of current flow and applied magnetic field, an effect known as the superconducting (SC) diode effect. Here, we consider the SC diode effect in Josephson junctions in nanowire devices. We find that the 4π-periodic contribution of Majorana bound states (MBSs) to the current phase relation (CPR) of single junctions results in a significant enhancement of the SC diode effect when the device enters the topological phase. Crucially, this enhancement of the SC diode effect is independent of the parity of the junction and therefore protected from parity altering events, such as quasiparticle poisoning, which have hampered efforts to directly observe the 4π-periodic CPR of MBSs. We show that this effect can be generalized to SQUIDs and that, in such devices, the parity-protected SC diode effect can provide a highly controllable probe of the topology in a Josephson junction.

Enhancement of the Kondo effect in a quantum dot formed in a full-shell nanowire

Aleksandr E. Svetogorov, Daniel Loss, Jelena Klinovaja Phys. Rev. B 107, 134505 (2023); arXiv:2301.12442. We analyze results of a recent experiment [D. Razmadze et al., Phys. Rev. Lett., 125, 116803 (2020)] on transport through a quantum dot between two full-shell nanowires and show that the observed effects are caused by the Kondo effect enhancement due to a nontrivial geometry (magnetic flux in a full-shell nanowire) rather than the presence of Majorana bound states. Moreover, we propose that such a setup presents a unique and convenient system to study the competition between superconductivity and the Kondo effect and has significant advantages in comparison to other known approaches, as the important parameter is controlled by the magnetic flux through the full-shell nanowire, which can be significantly varied with small changes of magnetic field, and does not require additional gates. This competition is of fundamental interest as it results in a quantum phase transition between an unscreened doublet and a many-body Kondo singlet ground states of the system.

Domain wall qubits on magnetic racetracks

Ji Zou, Stefano Bosco, Banabir Pal, Stuart S. P. Parkin, Jelena Klinovaja, Daniel Loss Phys. Rev. Research 5, 033166 (2023); arXiv:2212.12019. We propose a scalable implementation of a quantum computer based on hardware-efficient mobile domain walls on magnetic racetracks. In our proposal, the quantum information is encoded in the chirality of the spin structure of nanoscale domain walls. We estimate that these qubits are long-lived and could be operated at sweet spots reducing possible noise sources. Single-qubit gates are implemented by controlling the movement of the walls in magnetic nanowires, and two-qubit entangling gates take advantage of naturally emerging interactions between different walls. These gates are sufficient for universal quantum computing and are fully compatible with current state-of-the-art experiments on racetrack memories. Possible schemes for qubit readout and initialization are also discussed.

Realization of a three-dimensional quantum Hall effect in a Zeeman-induced second order topological insulator on a torus

Zhe Hou, Clara S. Weber, Dante M. Kennes, Daniel Loss, Herbert Schoeller, Jelena Klinovaja, Mikhail Pletyukhov Phys. Rev. B 107, 075437 (2023); arXiv:2212.09053. We propose a realization of a quantum Hall effect (QHE) in a second-order topological insulator (SOTI) in three dimensions (3D), which is mediated by hinge states on a torus surface. It results from the nontrivial interplay of the material structure, Zeeman effect, and the surface curvature. In contrast to the conventional 2D- and 3D-QHE, we show that the 3D-SOTI QHE is not affected by orbital effects of the applied magnetic field and exists in the presence of a Zeeman term only, induced e.g. by magnetic doping. To explain the 3D-SOTI QHE, we analyze the boundary charge for a 3D-SOTI and establish its universal dependence on the Aharonov-Bohm flux threading through the torus hole. Exploiting the fundamental relation between the boundary charge and the Hall conductance, we demonstrate the universal quantization of the latter, as well as its stability against random disorder potentials and continuous deformations of the torus surface.

Anisotropic exchange interaction of two hole-spin qubits

Simon Geyer, Bence Hetényi, Stefano Bosco, Leon C. Camenzind, Rafael S. Eggli, Andreas Fuhrer, Daniel Loss, Richard J. Warburton, Dominik M. Zumbühl, and Andreas V. Kuhlmann Nature Physics (2024); arXiv:2212.02308. Semiconductor spin qubits offer the potential to employ industrial transistor technology to produce large-scale quantum computers. Silicon hole spin qubits benefit from fast all-electrical qubit control and sweet spots to counteract charge and nuclear spin noise. However, the demonstration of a two-qubit interaction has remained an open challenge. One missing factor is an understanding of the exchange coupling in the presence of a strong spin–orbit interaction. Here we study two hole-spin qubits in a silicon fin field-effect transistor, the workhorse device of today’s semiconductor industry. We demonstrate electrical tunability of the exchange splitting from above 500 MHz to close-to-off and perform a conditional spin-flip in 24 ns. The exchange is anisotropic because of the spin–orbit interaction. Upon tunnelling from one quantum dot to the other, the spin is rotated by almost 180 degrees. The exchange Hamiltonian no longer has the Heisenberg form and can be engineered such that it enables two-qubit controlled rotation gates without a trade-off between speed and fidelity. This ideal behaviour applies over a wide range of magnetic field orientations, rendering the concept robust with respect to variations from qubit to qubit, indicating that it is a suitable approach for realizing a large-scale quantum computer.

Second-order topology and supersymmetry in two-dimensional topological insulators

Clara S. Weber, Mikhail Pletyukhov, Zhe Hou, Dante M. Kennes, Jelena Klinovaja, Daniel Loss, and Herbert Schoeller Phys. Rev. B 107, 235402 (2023); arXiv:2212.01307. We unravel a fundamental connection between a supersymmetry and a wide class of two dimensional second-order topological insulators (SOTI). This particular supersymmetry is induced by applying a half-integer Aharonov-Bohm flux f=Φ/Φ0=1/2 through a hole in the system. Here, three symmetries are essential to establish this fundamental link: chiral symmetry, inversion symmetry, and mirror symmetry. At such a flux of half-integer value the mirror symmetry anticommutes with the inversion symmetry leading to a nontrivial n=1-SUSY representation for the absolute value of the Hamiltonian in each chiral sector, separately. This implies that a unique zero-energy state and an exact 2-fold degeneracy of all eigenstates with non-zero energy is found even at finite system size. For arbitrary smooth surfaces the link between 2D-SOTI and SUSY can be described within a universal low-energy theory in terms of an effective surface Hamiltonian which encompasses the whole class of supersymmetric periodic Witten models. Applying this general link to the prototypical example of a Bernevig-Hughes-Zhang-model with an in-plane Zeeman field, we analyse the entire phase diagram and identify a gapless Weyl phase separating the topological from the non-topological gapped phase. Surprisingly, we find that topological states localized at the outer surface remain in the Weyl phase, whereas topological hole states move to the outer surface and change their spatial symmetry upon approaching the Weyl phase. Therefore, the topological hole states can be tuned in a versatile manner opening up a route towards magnetic-field-induced topological engineering in multi-hole systems. Finally, we demonstrate the stability of localized states against deviation from half-integer flux, flux penetration into the sample, surface distortions, and random impurities for impurity strengths up to the order of the surface gap.

RKKY interaction in one-dimensional flat band lattices

Katharina Laubscher, Clara S. Weber, Maximilian Hünenberger, Herbert Schoeller, Dante M. Kennes, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 108, 155429 (2023); arXiv:2210.10025. We study the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between two classical magnetic impurities in one-dimensional lattice models with flat bands. As two representative examples, we pick the stub lattice and the diamond lattice at half filling. We first calculate the exact RKKY interaction numerically and then compare our data to results obtained via different analytical techniques. In both our examples, we find that the RKKY interaction exhibits peculiar features that can directly be traced back to the presence of a flat band. Importantly, these features are not captured by the conventional RKKY approximation based on non-degenerate perturbation theory. Instead, we find that degenerate perturbation theory correctly reproduces our exact results if there is an energy gap between the flat and the dispersive bands, while a non-perturbative approach becomes necessary in the absence of a gap.

Determination of spin-orbit interaction in semiconductor nanostructures via non-linear transport

Renato M. A. Dantas, Henry F. Legg, Stefano Bosco, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 107, L241202 (2023); arXiv:2210.05429. We investigate non-linear transport signatures stemming from linear and cubic spin-orbit interactions in one- and two-dimensional systems. The analytical zero-temperature response to external fields is complemented by finite temperature numerical analysis, establishing a way to distinguish between linear and cubic spin-orbit interactions. We also propose a protocol to determine the relevant material parameters from transport measurements attainable in realistic conditions, illustrated by values for Ge heterostructures. Our results establish a method for the fast benchmarking of spin-orbit properties in semiconductor nanostructures.

Trivial Andreev band mimicking topological bulk gap reopening in the non-local conductance of long Rashba nanowires

Richard Hess, Henry F. Legg, Daniel Loss, and Jelena Klinovaja Phys. Rev. Lett. 130, 207001 (2023); Editor's Suggestion; arXiv:2210.03507. We consider a one-dimensional Rashba nanowire in which multiple Andreev bound states in the bulk of the nanowire form an Andreev band. We show that, under certain circumstances, this trivial Andreev band can produce an apparent closing and reopening signature of the bulk band gap in the non-local conductance of the nanowire. Furthermore, we show that the existence of the trivial bulk reopening signature (BRS) in non-local conductance is essentially unaffected by the additional presence of trivial zero-bias peaks (ZBPs) in the local conductance at either end of the nanowire. The simultaneous occurrence of a trivial BRS and ZBPs mimics the basic features required to pass the so-called "topological gap protocol". Our results therefore provide a topologically trivial minimal model by which the applicability of this protocol can be benchmarked.

Planar Josephson junctions in germanium: Effect of cubic spin-orbit interaction

Melina Luethi, Katharina Laubscher, Stefano Bosco, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 107, 035435 (2023); arXiv:2209.12745. Planar Josephson junctions comprising semiconductors with strong spin-orbit interaction (SOI) are promising platforms to host Majorana bound states (MBSs). In this context, two-dimensional hole gases in germanium (Ge) are favorable candidates due to their particularly large SOI. In contrast to electron gases, where the SOI is a linear function of momentum, the SOI is cubic in momentum for a hole gas in planar Ge. Using a discretized model, we numerically simulate a Ge planar Josephson junction and show that it can host MBSs. Interestingly, we find that the cubic SOI yields an asymmetric phase diagram as a function of the superconducting phase difference across the junction. We also find that trivial Andreev bound states can imitate the signatures of MBSs in a Ge planar Josephson junction, therefore making the experimental detection of MBSs difficult. We use experimentally realistic parameters to assess if the topological phase is accessible within experimental limitations. Our analysis shows that two-dimensional Ge is an auspicious candidate for topological phases.

Noise-correlation spectrum for a pair of spin qubits in silicon

J. Yoneda, J. S. Rojas-Arias, P. Stano, K. Takeda, A. Noiri, T. Nakajima, D. Loss, and S. Tarucha Nat. Phys. 19, 1793 (2023); arXiv:2208.14150. Semiconductor qubits are appealing for building quantum processors as they may be densely integrated due to small footprint. However, a high density raises the issue of noise correlated across different qubits, which is of practical concern for scalability and fault tolerance. Here, we analyse and quantify in detail the degree of noise correlation in a pair of neighbouring silicon spin qubits ~100 nm apart. We evaluate all a-priori independent auto- and cross- power spectral densities of noise as a function of frequency. We reveal strong inter-qubit noise correlation with a correlation strength as large as ~0.7 at ~1 Hz (70% of the maximum in-phase correlation), even in the regime where the spin-spin exchange interaction contributes negligibly. We furthermore find that fluctuations of single-spin precession rates are strongly correlated with exchange noise, giving away their electrical origin. Noise cross-correlations have thus enabled us to pinpoint the most influential noise in the present device among compelling mechanisms including nuclear spins. Our work presents a powerful tool set to assess and identify the noise acting on multiple qubits and highlights the importance of long-range electric noise in densely packed silicon spin qubits.

Enhanced orbital magnetic field effects in Ge hole nanowires

Christoph Adelsberger, Stefano Bosco, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 106, 235408 (2022); arXiv:2207.12050. Hole semiconductor nanowires (NW) are promising platforms to host spin qubits and Majorana bound states for topological qubits because of their strong spin-orbit interactions (SOI). The properties of these systems depend strongly on the design of the cross section and on strain, as well as on external electric and magnetic fields. In this work, we analyze in detail the dependence of the SOI and g factors on the orbital magnetic field. We focus on magnetic fields aligned along the axis of the NW, where orbital effects are enhanced and result in a renormalization of the effective g factor up to 400%, even at small values of magnetic field. We provide an exact analytical solution for holes in Ge NWs and we derive an effective low-energy model that enables us to investigate the effect of electric fields applied perpendicular to the NW. We also discuss in detail the role of strain, growth direction, and high energy valence bands in different architectures, including Ge/Si core/shell NWs, gate-defined one-dimensional channels in planar Ge, and curved Ge quantum wells. In core/shell NWs grown along the [110] direction the g factor can be twice larger than for other growth directions which makes this growth direction advantageous for Majorana bound states. Also curved Ge quantum wells feature large effective g factors and SOI, again ideal for hosting Majorana bound states. Strikingly, because these quantities are independent of the electric field, hole spin qubits encoded in curved quantum wells are to good approximation not susceptible to charge noise, significantly boosting their coherence time.

Shor–Laflamme distributions of graph states and noise robustness of entanglement

Daniel Miller, Daniel Loss, Ivano Tavernelli, Hermann Kampermann, Dagmar Bruß, and Nikolai Wyderka J. Phys. A: Math. Theor. 56, 335303 (2023); arXiv:2207.07665. The sector length distribution (SLD) of a quantum state is a collection of local unitary invariants that quantify k-body correlations. We show that the SLD of graph states can be derived by solving a graph-theoretical problem. In this way, the mean and variance of the SLD are obtained as simple functions of efficiently computable graph properties. Furthermore, this formulation enables us to derive closed expressions of SLDs for some graph state families. For cluster states, we observe that the SLD is very similar to a binomial distribution, and we argue that this property is typical for graph states in general. Finally, we derive an SLD-based entanglement criterion from the majorization criterion and apply it to derive meaningful noise thresholds for entanglement.

Long-distance coupling of spin qubits via topological magnons

Bence Hetényi, Alexander Mook, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 106, 235409 (2022); arXiv:2207.01264. We consider two distant spin qubits in quantum dots, both coupled to a two-dimensional topological ferromagnet hosting chiral magnon edge states at the boundary. The chiral magnon is used to mediate entanglement between the spin qubits, realizing a fundamental building block of scalable quantum computing architectures: a long-distance two-qubit gate. Previous proposals for long-distance coupling with magnons involved off-resonant coupling, where the detuning of the spin-qubit frequency from the magnonic band edge provides protection against spontaneous relaxation. The topological magnon mode, on the other hand, lies in-between two magnonic bands far away from any bulk magnon resonances, facilitating strong and highly tuneable coupling between the two spin qubits. Even though the coupling between the qubit and the chiral magnon is resonant for a wide range of qubit splittings, we find that the magnon-induced qubit relaxation is vastly suppressed if the coupling between the qubit and the ferromagnet is antiferromagnetic. A fast and high-fidelity long-distance coupling protocol is presented capable of achieving spin-qubit entanglement over micrometer distances with 1MHz gate speed and up to 99.9% fidelities. The resulting spin-qubit entanglement may be used as a probe for the long-sought detection of topological edge magnons.

Superconducting diode effect due to magnetochiral anisotropy in topological insulator and Rashba nanowires

Henry F. Legg, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 106, 104501 (2022); arXiv:2205.12939. The critical current of a superconductor can depend on the direction of current flow due to magnetochiral anisotropy when both inversion and time-reversal symmetry are broken, an effect known as the superconducting (SC) diode effect. Here, we consider one-dimensional (1D) systems in which superconductivity is induced via the proximity effect. In both topological insulator and Rashba nanowires, the SC diode effect due to a magnetic field applied along the spin-polarization axis and perpendicular to the nanowire provides a measure of inversion symmetry breaking in the presence of a superconductor. Furthermore, a strong dependence of the SC diode effect on an additional component of magnetic field applied parallel to the nanowire as well as on the position of the chemical potential can be used to detect that a device is in the region of parameter space where the phase transition to topological superconductivity is expected to arise.

Coupled superconducting spin qubits with spin-orbit interaction

Maria Spethmann, Xian-Peng Zhang, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 106, 115411 (2022); arXiv:2205.03843. Superconducting spin qubits, also known as Andreev spin qubits, promise to combine the benefits of superconducting qubits and spin qubits defined in quantum dots. While most approaches to control these qubits rely on controlling the spin degree-of-freedom via the supercurrent, superconducting spin qubits can also be coupled to each other via the superconductor to implement two-qubit quantum gates. We theoretically investigate the interaction between superconducting spin qubits in the weak tunneling regime and concentrate on the effect of spin-orbit interaction (SOI), which can be large in semiconductor-based quantum dots and thereby offers an additional tuning parameter for quantum gates. We find analytically that the effective interaction between two superconducting spin qubits consists of Ising, Heisenberg, and Dzyaloshinskii-Moriya interactions and can be tuned by the superconducting phase difference, the tunnel barrier strength, or the SOI parameters. The Josephson current becomes dependent on SOI and spin orientations. We demonstrate that this interaction can be used for fast controlled phase-flip gates with a fidelity >99.99%. We propose a scalable network of superconducting spin qubits which is suitable for implementing the surface code.

Bayesian estimation of correlation functions

Angel Gutierrez-Rubio, Daniel Loss, and Peter Stano Phys. Rev. Research 4, 043166 (2022); arXiv:2205.03611. We apply Bayesian statistics to the estimation of correlation functions. We give the probability distributions of auto and cross correlations as functions of the data. Our procedure uses the measured data optimally and informs about the certainty level of the estimation. Our results apply to general stationary processes and their essence is a non-parametric estimation of spectra. It allows one to understand better the noise statistical fluctuations, assess the correlation between two variables, and postulate parametric models of spectra that can be further tested. We also propose a method to numerically generate correlated noise with a given spectrum.

Anomalous zero-field splitting for hole spin qubits in Si and Ge quantum dots

Bence Hetényi, Stefano Bosco, and Daniel Loss Phys. Rev. Lett. 129, 116805 (2022); arXiv:2205.02582. An anomalous energy splitting of spin triplet states at zero magnetic field has recently been measured in germanium quantum dots. This zero-field splitting could crucially alter the coupling between tunnel-coupled quantum dots, the basic building blocks of state-of-the-art spin-based quantum processors, with profound implications for semiconducting quantum computers. We develop an analytical model linking the zero-field splitting to spin-orbit interactions that are cubic in momentum. Such interactions naturally emerge in hole nanostructures, where they can also be tuned by external electric fields, and we find them to be particularly large in silicon and germanium, resulting in a significant zero-field splitting in the μeV range. We confirm our analytical theory by numerical simulations of different quantum dots, also including other possible sources of zero-field splitting. Our findings are applicable to a broad range of current architectures encoding spin qubits and provide a deeper understanding of these materials, paving the way towards the next generation of semiconducting quantum processors.

Hole spin qubits in thin curved quantum wells

Stefano Bosco and Daniel Loss Phys. Rev. Applied 18, 044038 (2022); arXiv:2204.08212. Hole spin qubits are frontrunner platforms for scalable quantum computers because of their large spin-orbit interaction which enables ultrafast all-electric qubit control at low power. The fastest spin qubits to date are defined in long quantum dots with two tight confinement directions, when the driving field is aligned to the smooth direction. However, in these systems the lifetime of the qubit is strongly limited by charge noise, a major issue in hole qubits. We propose here a different, scalable qubit design, compatible with planar CMOS technology, where the hole is confined in a curved germanium quantum well surrounded by silicon. This design takes full advantage of the strong spin-orbit interaction of holes, and at the same time suppresses charge noise in a wide range of configurations, enabling highly coherent, ultrafast qubit gates. While here we focus on a Si/Ge/Si curved quantum well, our design is also applicable to different semiconductors. Strikingly, these devices allow for ultrafast operations even in short quantum dots by a transversal electric field. This additional driving mechanism relaxes the demanding design constraints, and opens up a new way to reliably interface spin qubits in a single quantum dot to microwave photons. By considering state-of-the-art high-impedance resonators and realistic qubit designs, we estimate interaction strengths of a few hundreds of MHz, largely exceeding the decay rate of spins and photons. Reaching such a strong coupling regime in hole spin qubits will be a significant step towards high-fidelity entangling operations between distant qubits, as well as fast single-shot readout, and will pave the way towards the implementation of a large-scale semiconducting quantum processor.

Observation of fractional spin textures in a Heusler material

Jagannath Jena, Börge Göbel, Tomoki Hirosawa, Sebastian A. Diaz, Daniel Wolf, Taichi Hinokihara, Vivek Kumar, Ingrid Mertig, Claudia Felser, Axel Lubk, Daniel Loss, and Stuart S. P. Parkin Nat Commun 13, 2348 (2022) Recently a zoology of non-collinear chiral spin textures has been discovered, most of which, such as skyrmions and antiskyrmions, have integer topological charges. Here we report the experimental real-space observation of the formation and stability of fractional antiskyrmions and fractional elliptical skyrmions in a Heusler material. These fractional objects appear, over a wide range of temperature and magnetic field, at the edges of a sample, whose interior is occupied by an array of nano-objects with integer topological charges, in agreement with our simulations. We explore the evolution of these objects in the presence of magnetic fields and show their interconversion to objects with integer topological charges. This means the topological charge can be varied continuously. These fractional spin textures are not just another type of skyrmion, but are essentially a new state of matter that emerges and lives only at the boundary of a magnetic system. The coexistence of both integer and fractionally charged spin textures in the same material makes the Heusler family of compounds unique for the manipulation of the real-space topology of spin textures and thus an exciting platform for spintronic and magnonic applications.

Prevalence of trivial zero-energy sub-gap states in non-uniform helical spin chains on the surface of superconductors

Richard Hess, Henry F. Legg, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 106, 104503 (2022); arXiv:2204.02324. Helical spin chains, consisting of magnetic (ad-)atoms, on the surface of bulk superconductors are predicted to host Majorana bound states (MBSs) at the ends of the chain. Here, we investigate the prevalence of trivial zero-energy bound states in these helical spin chain systems. First, we show that the Hamiltonian of a helical spin chain on a superconductor can be mapped to an effective Hamiltonian reminiscent of a semiconductor nanowire with strong Rashba spin-orbit coupling. In particular, we show that a varying rotation rate between neighbouring magnetic moments maps to smooth non-uniform potentials in the effective nanowire Hamiltonian. Previously it has been found that trivial zero-energy states are abundant in nanowire systems with smooth potentials. Therefore, we perform an extensive search for zero-energy bound states in helical spin chain systems with varying rotation rates. Although bound states with near zero-energy do exist for certain dimensionalities and rotation profiles, we find that zero-energy bound states are far less prevalent than in semiconductor nanowire systems with equivalent non-uniformities. In particular, utilising varying rotation rates, we do not find zero-energy bound states in the most experimentally relevant setup consisting of a one-dimensional helical spin chain on the surface of a three-dimensional superconductor, even for profiles that produce near zero-energy states in equivalent one- and two- dimensional systems. Although our findings do not rule them out, the much reduced prevalence of zero-energy bound states in long non-uniform helical spin chains compared with equivalent semi-conductor nanowires, as well as the ability to measure states locally via STM, should reduce the experimental barrier to identifying MBSs in such systems.

Fully tunable longitudinal spin-photon interactions in Si and Ge quantum dots

Stefano Bosco, Pasquale Scarlino, Jelena Klinovaja, and Daniel Loss Phys. Rev. Lett. 129, 066801 (2022); arXiv:2203.17163. Spin qubits in silicon and germanium quantum dots are promising platforms for quantum computing, but entangling spin qubits over micrometer distances remains a critical challenge. Current prototypical architectures maximize transversal interactions between qubits and microwave resonators, where the spin state is flipped by nearly resonant photons. However, these interactions cause back-action on the qubit, that yield unavoidable residual qubit-qubit couplings and significantly affect the gate fidelity. Strikingly, residual couplings vanish when spin-photon interactions are longitudinal and photons couple to the phase of the qubit. We show that large longitudinal interactions emerge naturally in state-of-the-art hole spin qubits. These interactions are fully tunable and can be parametrically modulated by external oscillating electric fields. We propose realistic protocols to measure these interactions and to implement fast and high-fidelity two-qubit entangling gates. These protocols work also at high temperatures, paving the way towards the implementation of large-scale quantum processors.

Magnons, magnon bound pairs, and their hybrid spin-multipolar topology

Alexander Mook, Rhea Hoyer, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 107, 064429 (2023); arXiv:2203.12374. We consider quantum condensed matter systems without particle-number conservation. Since the particle number is not a good quantum number, states belonging to different particle-number sectors can hybridize, which causes topological anticrossings in the spectrum. The resulting spectral gaps support chiral edge excitations whose wavefunction is a superposition of states in the two hybridized sectors. This situation is realized in fully saturated spin-anisotropic quantum magnets without spin conservation, in which single magnons hybridize with magnon bound pairs, i.e., two-magnon bound states. The resulting chiral edge excitations are exotic composites that carry mixed spin-multipolar character, inheriting spin-dipolar and spin-quadrupolar character from their single-particleness and two-particleness, respectively. In contrast to established topological magnons, the topological effects discussed here are of genuine quantum mechanical origin and vanish in the classical limit. We discuss implications for intrinsic anomalous Hall-type transport and estimate that the thermal Hall conductivity brought about by the hybridization of magnons and magnon bound pairs can be as large as that of magnons with other magnons. We conclude that fully polarized quantum magnets are a promising platform for topology caused by hybridizations between particle-number sectors, complementing the field of ultracold atoms working with a conserved number of particles.

RKKY interaction at helical edges of topological superconductors

Katharina Laubscher, Dmitry Miserev, Vardan Kaladzhyan, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 107, 115421 (2023); arXiv:2203.08137. We study spin configurations of magnetic impurities placed close to the edge of a two-dimensional topological superconductor both analytically and numerically. First, we demonstrate that the spin of a single magnetic impurity close to the edge of a topological superconductor tends to align along the edge. The strong easy-axis spin anisotropy behind this effect originates from the interaction between the impurity and the gapless helical Majorana edge states. We then compute the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between two magnetic impurities placed close to the edge. We show that, in the limit of large interimpurity distances, the RKKY interaction between the two impurities is mainly mediated by the Majorana edge states and leads to a ferromagnetic alignment of both spins along the edge. This effect can be used to detect helical Majorana edge states.

Crossed Andreev reflection in spin-polarized chiral edge states due to Meissner effect

Tamás Haidekker Galambos, Flavio Ronetti, Bence Hetényi, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 106, 075410 (2022); arXiv:2203.05894. We consider a hybrid quantum Hall-superconductor system, where a superconducting finger with oblique profile is wedged into a two-dimensional electron gas in the presence of a perpendicular magnetic field, as considered by Lee et al., Nat. Phys. 13, 693 (2017). The electron gas is in the quantum Hall regime at filling factor ν=1. Due to the Meissner effect, the perpendicular magnetic field close to the quantum Hall-superconductor boundary is distorted and gives rise to an in-plane component of the magnetic field. This component enables non-local crossed Andreev reflection between the spin-polarized chiral edge states running on opposite sides of the superconducting finger, thus opening a gap in the spectrum of the edge states without the need of spin-orbit interaction or non-trivial magnetic textures. We compute numerically the transport properties of this setup and show that a negative resistance exists as consequence of non-local Andreev processes. We also obtain numerically the zero-energy local density of states, which systematically shows peaks stable to disorder. The latter result is compatible with the emergence of Majorana bound states.

Magnetoelectric Cavity Magnonics in Skyrmion Crystals

Tomoki Hirosawa, Alexander Mook, Jelena Klinovaja, and Daniel Loss PRX Quantum 3, 040321 (2022); arXiv:2203.03241. We present a theory of magnetoelectric magnon-photon coupling in cavities hosting noncentrosymmetric magnets. Analogously to nonreciprocal phenomena in multiferroics, the magnetoelectric coupling is time-reversal and inversion asymmetric. This asymmetry establishes a means for exceptional tunability of magnon-photon coupling, which can be switched on and off by reversing the magnetization direction. Taking the multiferroic skyrmion-host Cu2OSeO3 as an example, we reveal the electrical activity of skyrmion eigenmodes and propose it for magnon-photon splitting of "magnetically dark" elliptic modes. Furthermore, we predict a cavity-induced magnon-magnon coupling between magnetoelectrically active skyrmion excitations. Our study highlights magnetoelectric cavity magnonics as a novel platform for realizing quantum-hybrid systems and the quantum control of topological magnetic textures.

Quantum-Coherent Nanoscience

Andreas J. Heinrich, William D. Oliver, Lieven Vandersypen, Arzhang Ardavan, Roberta Sessoli, Daniel Loss, Ania Bleszynski Jayich, Joaquin Fernandez-Rossier, Arne Laucht, and Andrea Morello Nature Nanotechnology 16, 1318 (2021); arXiv:2202.01431. For the past three decades, nanoscience has widely affected many areas in physics, chemistry, and engineering, and has led to numerous fundamental discoveries as well as applications and products. Concurrently, quantum science and technology has developed into a cross-disciplinary research endeavour connecting these same areas and holds a burgeoning commercial promise. Although quantum physics dictates the behaviour of nanoscale objects, quantum coherence, which is central to quantum information, communication and sensing has not played an explicit role in much of nanoscience. This Review describes fundamental principles and practical applications of quantum coherence in nanoscale systems, a research area we call quantum-coherent nanoscience. We structure this manuscript according to specific degrees of freedom that can be quantum-coherently controlled in a given nanoscale system such as charge, spin, mechanical motion, and photons. We review the current state of the art and focus on outstanding challenges and opportunities unlocked by the merging of nanoscience and coherent quantum operations.

Instability of the ferromagnetic quantum critical point in strongly interacting 2D and 3D electron gases with arbitrary spin-orbit splitting

Dmitry Miserev, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 106, 134417 (2022); arXiv:2201.10995. In this work we revisit itinerant ferromagnetism in 2D and 3D electron gases with arbitrary spin-orbit splitting and strong electron-electron interaction. We identify the resonant scattering processes close to the Fermi surface that are responsible for the instability of the ferromagnetic quantum critical point at low temperatures. In contrast to previous theoretical studies, we show that such processes cannot be fully suppressed even in presence of arbitrary spin-orbit splitting. A fully self-consistent non-perturbative treatment of the electron-electron interaction close to the phase transition shows that these resonant processes always destabilize the ferromagnetic quantum critical point and lead to a first-order phase transition. Characteristic signatures of these processes can be measured via the non-analytic dependence of the spin susceptibility on magnetic field both far away or close to the phase transition

Metallization and proximity superconductivity in topological insulator nanowires

Henry F. Legg, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 105, 155413 (2022); arXiv:2201.02918. A heterostructure consisting of a topological insulator (TI) nanowire brought into proximity with a superconducting layer provides a promising route to achieve topological superconductivity and associated Majorana bound states (MBSs). Here, we study effects caused by such a coupling between a thin layer of an s-wave superconductor and a TI nanowire. We show that there is a distinct phenomenology arising from the metallization of states in the TI nanowire by the superconductor. In the strong coupling limit, required to induce a large superconducting pairing potential, we find that metallization results in a shift of the TI nanowire subbands (∼20 meV) as well as it leads to a small reduction in the size of the subband gap opened by a magnetic field applied parallel to the nanowire axis. Surprisingly, we find that metallization effects in TI nanowires can also be beneficial. Most notably, coupling to the superconductor induces a potential in the portion of the TI nanowire close to the interface with the superconductor, this breaks inversion symmetry and at finite momentum lifts the spin degeneracy of states within a subband. As such coupling to a superconductor can create or enhance the subband splitting that is key to achieving topological superconductivity. This is in stark contrast to semiconductors, where it has been shown that metallization effects always reduce the equivalent subband-splitting caused by spin-orbit coupling. We also find that in certain geometries metallization effects can reduce the critical magnetic required to enter the topological phase. We conclude that, unlike in semiconductors, the metallization effects that occur in TI nanowires can be relatively easily mitigated, for instance by modifying the geometry of the attached superconductor or by compensation of the TI material.

Charge-noise induced dephasing in silicon hole-spin qubits

Ognjen Malkoc, Peter Stano, and Daniel Loss Phys. Rev. Lett. 129, 247701 (2022); arXiv:2201.06181. We investigate theoretically charge-noise induced spin dephasing of a hole confined in a quasi-two-dimensional silicon quantum dot. Central to our treatment is accounting for higher-order corrections to the Luttinger Hamiltonian. Using experimentally reported parameters, we find that the new terms give rise to sweet-spots for the hole-spin dephasing, which are sensitive to device details: dot size and asymmetry, growth direction, and applied magnetic and electric fields. Furthermore, we estimate that the dephasing time at the sweet-spots is boosted by several orders of magnitude, up to order of milliseconds.

Non-Majorana zero energy modes in diluted spin chains proximitized to a superconductor

Felix Küster, Sascha Brinker, Richard Hess, Daniel Loss, Stuart Parkin, Jelena Klinovaja, Samir Lounis, and Paolo Sessi PNAS 2022 Vol. 119 No. 42 e2210589119; arXiv:2112.05708. Spin chains proximitized with superconducting condensates have emerged as one of the most promising platforms for the realization of Majorana modes. The recent use of atomic manipulation techniques raised great expectations for successfully creating and controlling such chains. Here, we craft diluted spin chains atom-by-atom following seminal theoretical proposal suggesting indirect coupling mechanisms as a viable route to trigger topological superconductivity. We demonstrate that, starting from deep Shiba states, it is possible to cross the quantum phase transition, a necessary condition for the emergence of topological superconductivity, for very short chains. This transition is associated with the emergence of highly localized zero energy end modes. The use of a substrate with highly anisotropic Fermi surface enables to create spin chains characterized by distinct magnetic configurations along various crystallographic directions. By scrutinizing a large set of parameters we reveal the ubiquitous existence of zero energy boundary modes. Although mimicking signatures generally assigned to Majorana modes, the end modes are identified as topologically trivial Shiba states. These results highlight the important role of the local environment, showing that it cannot be completely eliminated also in diluted systems where the effect is expected to be minimized. Our work demonstrates that zero energy modes in spin chains proximitized to supercondcutors are not necessarily a link to Majorana modes while simultaneously identifying new experimental platforms, driving mechanisms, and test protocols for the determination of topologically non trivial superconducting phases.

Quasiparticle poisoning in trivial and topological Josephson junctions

Aleksandr E. Svetogorov, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 105, 174519 (2022); arXiv:2112.05650. We study theoretically a short single-channel Josephson junction between superconductors in the trivial and topological phases. The junction is assumed to be biased by a small current and subjected to quasiparticle poisoning. We find that the presence of quasiparticles leads to a voltage signal from the Josephson junction that can be observed both in the trivial and in the topological phase. Quite remarkably, these voltage signatures are sufficiently different in the two phases such that they can serve as means to clearly distinguish between trivial Andreev and topological Majorana bound states in the system. Moreover, these voltage signatures, in the trivial and topological phase, would allow one to measure directly the quasiparticle poisoning rates and to test various approaches for protection against quasiparticle poisoning.

Hole Spin Qubits in Ge Nanowire Quantum Dots: Interplay of Orbital Magnetic Field, Strain, and Growth Direction

Christoph Adelsberger, Mónica Benito, Stefano Bosco, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 105, 075308 (2022); arXiv:2110.15039. Hole spin qubits in quasi one-dimensional structures are a promising platform for quantum information processing because of the strong spin-orbit interaction (SOI). We present analytical results and discuss device designs that optimize the SOI in Ge semiconductors. We show that at the magnetic field values at which qubits are operated, orbital effects of magnetic fields can strongly affect the response of the spin qubit. We study one-dimensional hole systems in Ge under the influence of electric and magnetic fields applied perpendicularly to the device. In our theoretical description, we include these effects exactly. The orbital effects lead to a strong renormalization of the g-factor. We find a sweet-spot of the nanowire (NW) g-factor where charge noise is strongly suppressed and present an effective low-energy model that captures the dependence of the SOI on the electromagnetic fields. Moreover, we compare properties of NWs with square and circular cross-sections with ones of gate-defined one-dimensional channels in two-dimensional Ge heterostructures. Interestingly, the effective model predicts a flat band ground state for fine-tuned electric and magnetic fields. By considering a quantum dot (QD) harmonically confined by gates, we demonstrate that the NW g-factor sweet spot is retained in the QD. Our calculations reveal that this sweet spot can be designed to coincide with the maximum of the SOI, yielding highly coherent qubits with large Rabi frequencies. We also study the effective g-factor of NWs grown along different high symmetry axes and find that our model derived for isotropic semiconductors is valid for the most relevant growth directions of non-isotropic Ge NWs. Moreover, a NW grown along one of the three main crystallographic axes shows the largest SOI. Our results show that the isotropic approximation is not justified in Ge in all cases.

Fractional spin excitations and conductance in the spiral staircase Heisenberg ladder

Flavio Ronetti, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 105, 134413 (2022); arXiv:2110.02130. We investigate theoretically the spiral staircase Heisenberg spin-1/2 ladder in the presence of antiferromagnetic long-range spin interactions and a uniform magnetic field. As a special case we also consider the Kondo necklace model. If the magnetizations of the two chains forming the ladder satisfy a certain resonance condition, involving interchain couplings as perturbations, the system is in a partially gapped magnetic phase hosting excitations characterized by fractional spins, whose values can be changed by the magnetic field. We show that these fractional spin excitations can be probed by spin currents in a transport setup with a spin conductance that reveals the fractionalized spin. In some special cases, the spin conductance reaches universal values in units of (gμ_B)^2/h, where g is the g-factor, μ_B the Bohr magneton, and h the Planck constant. We obtain our results with the help of bosonization and numerical density matrix renormalization group methods.

Giant magnetochiral anisotropy from quantum confined surface states of topological insulator nanowires

Henry F. Legg, Matthias Rößler, Felix Münning, Dingxun Fan, Oliver Breunig, Andrea Bliesener, Gertjan Lippertz, Anjana Uday, A. A. Taskin, Daniel Loss, Jelena Klinovaja, and Yoichi Ando Nature Nanotechnology 17, 696 (2022); arXiv:2109.05188. Wireless technology relies on the conversion of alternating electromagnetic fields to direct currents, a process known as rectification. While rectifiers are normally based on semiconductor diodes, quantum mechanical non-reciprocal transport effects that enable highly controllable rectification have recently been discovered. One such effect is magnetochiral anisotropy (MCA), where the resistance of a material or a device depends on both the direction of current flow and an applied magnetic field. However, the size of rectification possible due to MCA is usually extremely small, because MCA relies on electronic inversion symmetry breaking which typically stems from intrinsic spin-orbit coupling - a relativistic effect - in a non-centrosymmetric environment. Here, to overcome this limitation, we artificially break inversion symmetry via an applied gate voltage in thin topological insulator (TI) nanowire heterostructures and theoretically predict that such a symmetry breaking can lead to a giant MCA effect. Our prediction is confirmed via experiments on thin bulk-insulating (Bi1−xSbx)2Te3 TI nanowires, in which we observe the largest ever reported size of MCA rectification effect in a normal conductor - over 10000 times greater than in a typical material with a large MCA - and its behaviour is consistent with theory. Our findings present new opportunities for future technological applications of topological devices.

Laser-controlled real- and reciprocal-space topology in multiferroic insulators

Tomoki Hirosawa, Jelena Klinovaja, Daniel Loss, and Sebastian A. Diaz Phys. Rev. Lett. 128, 037201 (2022); arXiv:2108.06535. Magnetic materials in which it is possible to control the topology of their magnetic order in real space or the topology of their magnetic excitations in reciprocal space are highly sought-after as platforms for alternative data storage and computing architectures. Here we show that multiferroic insulators, owing to their magneto-electric coupling, offer a natural and advantageous way to address these two different topologies using laser fields. We demonstrate that via a delicate balance between the energy injection from a high-frequency laser and dissipation, single skyrmions---archetypical topological magnetic textures---can be set into motion with a velocity and propagation direction that can be tuned by the laser field amplitude and polarization, respectively. Moreover, we uncover an ultrafast Floquet magnonic topological phase transition in a laser-driven skyrmion crystal and we propose a new diagnostic tool to reveal it using the magnonic thermal Hall conductivity.

Helical Liquids in Semiconductors

Chen-Hsuan Hsu, Peter Stano, Jelena Klinovaja, and Daniel Loss Semicond. Sci. Technol. 36, 123003 (2021); arXiv:2107.13553. One-dimensional helical liquids can appear at boundaries of certain condensed matter systems. Two prime examples are the edge of a quantum spin Hall insulator, also known as a two-dimensional topological insulator, and the hinge of a three-dimensional second-order topological insulator. For these materials, the presence of a helical state at the boundary serves as a signature of their nontrivial bulk topology. Additionally, these boundary states are of interest themselves, as a novel class of strongly correlated low-dimensional systems with interesting potential applications. Here, we review existing results on such helical liquids in semiconductors. Our focus is on the theory, though we confront it with existing experiments. We discuss various aspects of the helical states, such as their realization, topological protection and stability, or possible experimental characterization. We lay emphasis on the hallmark of these states, being the prediction of a quantized electrical conductance. Since so far reaching a well-quantized conductance remained challenging experimentally, a large part of the review is a discussion of various backscattering mechanisms which have been invoked to explain this discrepancy. Finally, we include topics related to proximity-induced topological superconductivity in helical states, as an exciting application towards topological quantum computation with the resulting Majorana bound states.

Review of performance metrics of spin qubits in gated semiconducting nanostructures

Peter Stano and Daniel Loss Nat Rev Phys 4, 672 (2022); arXiv:2107.06485. This Technical Review collects values of selected performance characteristics of semiconductor spin qubits defined in electrically controlled nanostructures. The characteristics are envisaged to serve as a community source for the values of figures of merit with agreed definitions allowing the comparison of different spin- qubit platforms. We include characteristics on the qubit coherence, speed, fidelity and qubit size of multi- qubit devices. The focus is on collecting and curating the values of these characteristics as reported in the literature, rather than on their motivation or significance.

Fully tunable hyperfine interactions of hole spin qubits in Si and Ge quantum dots

Stefano Bosco and Daniel Loss Phys. Rev. Lett. 127, 190501 (2021); arXiv:2106.13744. Hole spin qubits are frontrunner platforms for scalable quantum computers, but state-of-the-art devices suffer from noise originating from the hyperfine interactions with nuclear defects. We show that these interactions have a highly tunable anisotropy that is controlled by device design and external electric fields. This tunability enables sweet spots where the hyperfine noise is suppressed by an order of magnitude and is comparable to isotopically purified materials. We identify surprisingly simple designs where the qubits are highly coherent and are largely unaffected by both charge and hyperfine noise. We find that the large spin-orbit interaction typical of elongated quantum dots not only speeds up qubit operations, but also dramatically renormalizes the hyperfine noise, altering qualitatively the dynamics of driven qubits and enhancing the fidelity of qubit gates. Our findings serve as guidelines to design high performance qubits for scaling up quantum computers.

Local and non-local quantum transport due to Andreev bound states in finite Rashba nanowires with superconducting and normal sections

Richard Hess, Henry F. Legg, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 104, 075405 (2021) (Editors' Suggestion); arXiv:2105.02791. We analyze Andreev bound states (ABSs) that form in normal sections of a Rashba nanowire that is only partially covered by a superconducting layer. These ABSs are localized close to the ends of the superconducting section and can be pinned to zero energy over a wide range of magnetic field strengths even if the nanowire is in the non-topological regime. For finite-size nanowires (typically ≲1 μm in current experiments), the ABS localization length is comparable to the length of the nanowire. The probability density of an ABS is therefore non-zero throughout the nanowire and differential-conductance calculations reveal a correlated zero-bias peak (ZBP) at both ends of the nanowire. When a second normal section hosts an additional ABS at the opposite end of the superconducting section, the combination of the two ABSs can mimic the closing and reopening of the bulk gap in local and non-local conductances accompanied by the appearance of the ZBP. These signatures are reminiscent of those expected for Majorana bound states (MBSs) but occur here in the non-topological regime. Our results demonstrate that conductance measurements of correlated ZBPs at the ends of a typical superconducting nanowire or an apparent closing and reopening of the bulk gap in the local and non-local conductance are not conclusive indicators for the presence of MBSs.

Squeezed hole spin qubits in Ge quantum dots with ultrafast gates at low power

Stefano Bosco, Mónica Benito, Christoph Adelsberger, and Daniel Loss Phys. Rev. B 104, 115425 (2021); arXiv:2103.16724. Hole spin qubits in planar Ge heterostructures are one of the frontrunner platforms for scalable quantum computers. In these systems, the spin-orbit interactions permit efficient all-electric qubit control. We propose a minimal design modification of planar devices that enhances these interactions by orders of magnitude and enables low power ultrafast qubit operations in the GHz range. Our approach is based on an asymmetric potential that strongly squeezes the quantum dot in one direction. This confinement-induced spin-orbit interaction does not rely on microscopic details of the device such as growth direction or strain, and could be turned on and off on demand in state-of-the-art qubits.

Tuning interactions between spins in a superconductor

Hao Ding, Yuwen Hu, Mallika T. Randeria, Silas Hoffman, Oindrila Deb, Jelena Klinovaja, Daniel Loss, and Ali Yazdani Proc. Natl. Acad. Sci. (PNAS) 118, e2024837118 (2021); arXiv:2103.14656. Novel many-body and topological electronic phases can be created in assemblies of interacting spins coupled to a superconductor, such as one-dimensional topological superconductors with Majorana zero modes (MZMs) at their ends. Understanding and controlling interactions between spins and the emergent band structure of the in-gap Yu-Shiba-Rusinov (YSR) states they induce in a superconductor are fundamental for engineering such phases. Here, by precisely positioning magnetic adatoms with a scanning tunneling microscope (STM), we demonstrate both the tunability of exchange interaction between spins and precise control of the hybridization of YSR states they induce on the surface of a bismuth (Bi) thin film that is made superconducting with the proximity effect. In this platform, depending on the separation of spins, the interplay between Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, spin-orbit coupling, and surface magnetic anisotropy stabilizes different types of spin alignments. Using high-resolution STM spectroscopy at millikelvin temperatures, we probe these spin alignments through monitoring the spin-induced YSR states and their energy splitting. Such measurements also reveal a quantum phase transition between the ground states with different electron number parity for a pair of spins in a superconductor tuned by their separation. Experiments on larger assemblies show that spin-spin interactions can be mediated in a superconductor over long distances. Our results show that controlling hybridization of the YSR states in this platform provides the possibility of engineering the band structure of such states for creating topological phases.

Majorana bound states in topological insulators without a vortex

Henry F. Legg, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 104, 165405 (2021); arXiv:2103.13412. We consider a three-dimensional topological insulator (TI) wire with a non-uniform chemical potential induced by gating across the cross-section. This inhomogeneity in chemical potential lifts the degeneracy between two one-dimensional surface state subbands. A magnetic field applied along the wire, due to orbital effects, breaks time-reversal symmetry and lifts the Kramers degeneracy at zero-momentum. If placed in proximity to an s-wave superconductor, the system can be brought into a topological phase at relatively weak magnetic fields. Majorana bound states (MBSs), localized at the ends of the TI wire, emerge and are present for an exceptionally large region of parameter space in realistic systems. Unlike in previous proposals, these MBSs occur without the requirement of a vortex in the superconducting pairing potential, which represents a significant simplification for experiments. Our results open a pathway to the realisation of MBSs in present day TI wire devices.

Majorana Bound States Induced by Antiferromagnetic Skyrmion Textures

Sebastián A. Díaz, Jelena Klinovaja, Daniel Loss, and Silas Hoffman Phys. Rev. B 104, 214501 (2021); arXiv:2102.03423. Majorana bound states are zero-energy states predicted to emerge in topological superconductors and intense efforts seeking a definitive proof of their observation are still ongoing. A standard route to realize them involves antagonistic orders: a superconductor in proximity to a ferromagnet. Here we show this issue can be resolved using antiferromagnetic rather than ferromagnetic order. We propose to use a chain of antiferromagnetic skyrmions, in an otherwise collinear antiferromagnet, coupled to a bulk conventional superconductor as a novel platform capable of supporting Majorana bound states that are robust against disorder. Crucially, the collinear antiferromagnetic region neither suppresses superconductivity nor induces topological superconductivity, thus allowing for Majorana bound states localized at the ends of the chain. Our model introduces a new class of systems where topological superconductivity can be induced by editing antiferromagnetic textures rather than locally tuning material parameters, opening avenues for the conclusive observation of Majorana bound states.

Fractional boundary charges with quantized slopes in interacting one- and two-dimensional systems

Katharina Laubscher, Clara S. Weber, Dante M. Kennes, Mikhail Pletyukhov, Herbert Schoeller, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 104, 035432 (2021); arXiv:2101.10301. We study fractional boundary charges (FBCs) for two classes of strongly interacting systems. First, we study strongly interacting nanowires subjected to a periodic potential with a period that is a rational fraction of the Fermi wavelength. For sufficiently strong interactions, the periodic potential leads to the opening of a charge density wave gap at the Fermi level. The FBC then depends linearly on the phase offset of the potential with a quantized slope determined by the period. Furthermore, different possible values for the FBC at a fixed phase offset label different degenerate ground states of the system that cannot be connected adiabatically. Next, we turn to the fractional quantum Hall effect (FQHE) at odd filling factors ν=1/(2l+1), where l is an integer. For a Corbino disk threaded by an external flux, we find that the FBC depends linearly on the flux with a quantized slope that is determined by the filling factor. Again, the FBC has 2l+1 different branches that cannot be connected adiabatically, reflecting the (2l+1)-fold degeneracy of the ground state. These results allow for several promising and strikingly simple ways to probe strongly interacting phases via boundary charge measurements.

Yu-Shiba-Rusinov States and Ordering of Magnetic Impurities Near the Boundary of a Superconducting Nanowire

Oindrila Deb, Silas Hoffman, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 103, 165403 (2021); arXiv:2012.15205. We theoretically study the spectrum induced by one and two magnetic impurities near the boundary of a one-dimensional nanowire in proximity to a conventional s-wave superconductor and extract the ground state magnetic configuration. We show that the energies of the subgap states, supported by the magnetic impurities, are strongly affected by the boundary for distances less than the superconducting coherence length. In particular, when the impurity is moved towards the boundary, multiple quantum phase transitions periodically occur in which the parity of the superconducting condensate oscillates between even and odd. We find that the magnetic ground state configuration of two magnetic impurities depends not only on the distance between them but also explicitly on their distance away from the boundary of the nanowire. As a consequence, the magnetic ground state can switch from ferromagnetic to antiferromagnetic while keeping the inter-impurity distance unaltered by simultaneously moving both impurities away from the boundary. The ground state magnetic configuration of two impurities is found analytically in the weak coupling regime and exactly for an arbitrary impurity coupling strength using numerical tight-binding simulations.

Insulating regime of an underdamped current-biased Josephson junction supporting Z3 and Z4 parafermions

Aleksandr E. Svetogorov, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 103, L180505 (2021); arXiv:2012.09062. We study analytically a current-biased topological Josephson junction supporting Zn parafermions. First, we show that in an infinite-size system a pair of parafermions on the junction can be in n different states; the (2pi)n periodicity of the phase potential of the junction results in a significant suppression of the maximal current Im for an insulating regime of the underdamped junction. Second, we study the behaviour of a realistic finite-size system with avoided level crossings characterized by splitting delta. We consider two limiting cases: when the phase evolution may be considered adiabatic, which results in decreased periodicity of the effective potential, and the opposite case, when Landau-Zener transitions restore the (2pi)n periodicity of the phase potential. The resulting current Im is exponentially different in the opposite limits, which allows us to propose a new detection method to establish the appearance of parafermions in the system experimentally, based on measuring Im at different values of the splitting delta.

Hole spin qubits in Si FinFETs with fully tunable spin-orbit coupling and sweet spots for charge noise

Stefano Bosco, Bence Hetényi, and Daniel Loss PRX Quantum 2, 010348 (2021); arXiv:2011.09417. The strong spin-orbit coupling in hole spin qubits enables fast and electrically tunable gates, but at the same time enhances the susceptibility of the qubit to charge noise. Suppressing this noise is a significant challenge in semiconductor quantum computing. Here, we show theoretically that hole Si FinFETs are not only very compatible with modern CMOS technology, but they present operational sweet spots where the charge noise is completely removed. The presence of these sweet spots is a result of the interplay between the anisotropy of the material and the triangular shape of the FinFET cross-section, and it does not require an extreme fine-tuning of the electrostatics of the device. We present how the sweet spots appear in FinFETs grown along different crystallographic axes and we study in detail how the behaviour of these devices change when the cross-section area and aspect ratio are varied. We identify designs that maximize the qubit performance and could pave the way towards a scalable spin-based quantum computer.

Interaction-stabilized topological magnon insulator in ferromagnets

Alexander Mook, Kirill Plekhanov, Jelena Klinovaja, and Daniel Loss Phys. Rev. X 11, 021061 (2021); arXiv:2011.06543. Condensed matter systems admit topological collective excitations above a trivial ground state, an example being Chern insulators formed by Dirac bosons with a gap at finite energies. However, in contrast to electrons, there is no particle-number conservation law for collective excitations. This gives rise to particle number-nonconserving many-body interactions whose influence on single-particle topology is an open issue of fundamental interest in the field of topological quantum materials. Taking magnons in ferromagnets as an example, we uncover topological magnon insulators that are stabilized by interactions through opening Chern-insulating gaps in the magnon spectrum. This can be traced back to the fact that the particle-number nonconserving interactions break the effective time-reversal symmetry of the harmonic theory. Hence, magnon-magnon interactions are a source of topology that can introduce chiral edge states, whose chirality depends on the magnetization direction. Importantly, interactions do not necessarily cause detrimental damping but can give rise to topological magnons with exceptionally long lifetimes. We identify two mechanisms of interaction-induced topological phase transitions---one driven by an external field, the other by temperature---and show that they cause unconventional sign reversals of transverse transport signals, in particular of the thermal Hall conductivity. We identify candidate materials where this many-body mechanism is expected to occur, such as the metal-organic kagome-lattice magnet Cu(1,3-benzenedicarboxylate), the van der Waals honeycomb-lattice magnet CrI3, and the multiferroic kamiokite (Fe2Mo3O8). Our results demonstrate that interactions can play an important role in generating nontrivial topology.

Clock model and parafermions in Rashba nanowires

Flavio Ronetti, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 103, 235410 (2021); arXiv:2011.06367. We consider a semiconducting nanowire with Rashba spin-orbit interaction subjected to a magnetic field and in the presence of strong electron-electron interactions. When the ratio between Fermi and Rashba momenta is tuned to 1/2, two competing resonant multi-particle scattering processes are present simultaneously and the interplay between them brings the system into a gapless critical parafermion phase. This critical phase is described by a self-dual sine-Gordon model, which we are able to map explicitly onto the low-energy sector of the ℤ4 parafermion clock chain model. Finally, we show that by alternating regions in which only one of these two processes is present one can generate localized zero-energy parafermion bound states.

Isotropic and Anisotropic g-factor Corrections in GaAs Quantum Dots

Leon C. Camenzind, Simon Svab, Peter Stano, Liuqi Yu, Jeramy D. Zimmerman, Arthur C. Gossard, Daniel Loss, and Dominik M. Zumbühl Phys. Rev. Lett. 127, 057701 (2021); arXiv:2010.11185. We experimentally determine isotropic and anisotropic g-factor corrections in lateral GaAs single-electron quantum dots. We extract the Zeeman splitting by measuring the tunnel rates into the individual spin states of an empty quantum dot for an in-plane magnetic field with various strengths and directions. We quantify the Zeeman energy and find a linear dependence on the magnetic field strength which allows us to extract the g-factor. The measured g-factor is understood in terms of spin-orbit interaction induced isotropic and anisotropic corrections to the GaAs bulk g-factor. Because this implies a dependence of the spin splitting on the magnetic field direction, these findings are of significance for spin qubits in GaAs quantum dots.

Chiral Hinge Magnons in Second-Order Topological Magnon Insulators

Alexander Mook, Sebastián A. Díaz, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 104, 024406 (2021); arXiv:2010.04142. When interacting spins in condensed matter order ferromagnetically, their ground state wave function is topologically trivial. Nonetheless, in two dimensions, the ferromagnetic state can support spin excitations with nontrivial topology, an exotic state known as topological magnon insulator (TMI). Here, we theoretically unveil and numerically confirm a novel ferromagnetic state in three dimensions dubbed second-order TMI, whose hallmarks are excitations at its hinges, where facets intersect. Since ferromagnetism naturally comes with broken time-reversal symmetry, the hinge magnons are chiral, rendering backscattering impossible. Hence, they trace out a three-dimensional path about the sample unimpeded by defects and are topologically protected by the spectral gap. They are remarkably robust against disorder and simultaneously highly tunable by atomic-level engineering of the sample termination. Our findings empower magnonics with the tools of higher-order topology, a promising route to combine low-energy information transfer free of Joule heating with three-dimensional vertical integration.

Universality of Boundary Charge Fluctuations

Clara S. Weber, Kiryl Piasotski, Mikhail Pletyukhov, Jelena Klinovaja, Daniel Loss, Herbert Schoeller, and Dante M. Kennes Phys. Rev. Lett. 126, 016803 (2021); Editor's suggestion; arXiv:2008.08431. We establish the quantum fluctuations ΔQ2B of the charge QB accumulated at the boundary of an insulator as an integral tool to characterize phase transitions where a direct gap closes (and reopens), typically occurring for insulators with topological properties. The power of this characterization lies in its capability to treat different kinds of insulators on equal footing; being applicable to transitions between topological and non-topological band, Anderson, and Mott insulators alike. In the vicinity of the phase transition we find a universal scaling ΔQ2B(Eg) as function of the gap size Eg and determine its generic form in various dimensions. For prototypical phase transitions with a massive Dirac-like bulk spectrum we demonstrate a scaling with the inverse gap in one dimension and a logarithmic one in two dimensions.

Magnetic phase transitions in two-dimensional two-valley semiconductors with in-plane magnetic field

Dmitry Miserev, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 103, 024401 (2021); arXiv:2008.07518. A two-dimensional electron gas (2DEG) in two-valley semiconductors has two discrete degrees of freedom given by the spin and valley quantum numbers. We analyze the zero-temperature magnetic instabilities of two-valley semiconductors with SOI, in-plane magnetic field, and electron-electron interaction. The interplay of an applied in-plane magnetic field and the SOI results in non-collinear spin quantization in different valleys. Together with the exchange intervalley interaction this results in a rich phase diagram containing four non-trivial magnetic phases. The negative non-analytic cubic correction to the free energy, which is always present in an interacting 2DEG, is responsible for first order phase transitions. Here, we show that non-zero ground state values of the order parameters can cut this cubic non-analyticity and drive certain magnetic phase transitions second order. We also find two tri-critical points at zero temperature which together with the line of second order phase transitions constitute the quantum critical sector of the phase diagram. The phase transitions can be tuned externally by electrostatic gates or by the in-plane magnetic field.

Quadrupole spin polarization as signature of second-order topological superconductors

Kirill Plekhanov, Niclas Müller, Yanick Volpez, Dante M. Kennes, Herbert Schoeller, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 103, L041401 (2021); arXiv:2008.03611. We study theoretically second-order topological superconductors characterized by the presence of pairs of zero-energy Majorana corner states. We uncover a quadrupole spin polarization at the system edges that provides a striking signature to identify topological phases, thereby complementing standard approaches based on zero-bias conductance peaks due to Majorana corner states. We consider two different classes of second-order topological superconductors with broken time-reversal symmetry and show that both classes are characterized by a quadrupolar structure of the spin polarization that disappears as the system passes through the topological phase transition. This feature can be accessed experimentally using spin-polarized scanning tunneling microscopes. We study different models hosting second-order topological phases, both analytically and numerically, and using Keldysh techniques we provide numerical simulations of the spin-polarized currents probed by scanning tips.

Pinning of Andreev bound states to zero energy in two-dimensional superconductor-semiconductor Rashba heterostructures

Olesia Dmytruk, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 102, 245431 (2020); arXiv:2007.14369. We consider a two-dimensional electron gas (2DEG) with Rashba spin-orbit interaction (SOI) partially covered by an s-wave superconductor, where the uncovered region remains normal but is exposed to a perpendicular Zeeman field. We find analytically and numerically Andreev bound states (ABSs) formed in the normal region and show that, by tuning the SOI to certain values, one can reach a regime where the energy of the lowest ABS becomes pinned close to zero as a function of Zeeman field. In addition, we also consider a configuration with a superconducting vortex and find again ABSs pinned close to zero energy in the topologically trivial phase for a wide range of parameters. Thus, ABSs and Majorana bound states show similar behavior in such structures.

Kramers pairs of Majorana corner states in a topological insulator bilayer

Katharina Laubscher, Danial Chughtai, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 102, 195401 (2020); arXiv:2007.13579. We consider a system consisting of two tunnel-coupled two-dimensional topological insulators proximitized by a top and bottom superconductor with a phase difference of π between them. We show that this system exhibits a time-reversal invariant second-order topological superconducting phase characterized by the presence of a Kramers pair of Majorana corner states at all four corners of a rectangular sample. We furthermore investigate the effect of a weak time-reversal symmetry breaking perturbation and show that an in-plane Zeeman field leads to an even richer phase diagram exhibiting two nonequivalent phases with two Majorana corner states per corner as well as an intermediate phase with only one Majorana corner state per corner. We derive our results analytically from continuum models describing our system. In addition, we also provide independent numerical confirmation of the resulting phases using discretized lattice representations of the models, which allows us to demonstrate the robustness of the topological phases and the Majorana corner states against parameter variations and potential disorder.

Fermi Surface Resonance and Quantum Criticality in Strongly Interacting Fermi Gases

Dmitry Miserev, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 103, 075104 (2021); arXiv:2007.04913. Fermions in the Fermi gas obey the Pauli exclusion principle restricting any two fermions from filling the same quantum state. Strong interaction between fermions can completely change the properties of the Fermi gas. In our theoretical study we find a new exotic quantum phase in strongly interacting Fermi gases constrained to a certain condition imposed on the Fermi surfaces which we call the Fermi surface resonance. The new phase is quantum critical which can be identified by the power-law frequency tail of the spectral density and divergent static susceptibilities. An especially striking feature of the new phase is the anomalous power-law temperature dependence of the dc resistivity that is similar to strange metals. The new quantum critical phase can be experimentally found in ordinary semiconductor heterostructures.

Strong spin-orbit interaction and g-factor renormalization of hole spins in Ge/Si nanowire quantum dots

F. N. M. Froning, M. J. Rančić, B. Hetényi, S. Bosco, M. K. Rehmann, A. Li, E. P. A. M. Bakkers, F. A. Zwanenburg, D. Loss, D. M. Zumbühl, and F. R. Braakman Phys. Rev. Res. 3, 013081 (2021); arXiv:2007.04308. The spin-orbit interaction lies at the heart of quantum computation with spin qubits, research on topologically non-trivial states, and various applications in spintronics. Hole spins in Ge/Si core/shell nanowires experience a spin-orbit interaction that has been predicted to be both strong and electrically tunable, making them a particularly promising platform for research in these fields. We experimentally determine the strength of spin-orbit interaction of hole spins confined to a double quantum dot in a Ge/Si nanowire by measuring spin-mixing transitions inside a regime of spin-blockaded transport. We find a remarkably short spin-orbit length of ∼65 nm, comparable to the quantum dot length and the interdot distance. We additionally observe a large orbital effect of the applied magnetic field on the hole states, resulting in a large magnetic field dependence of the spin-mixing transition energies. Strikingly, together with these orbital effects, the strong spin-orbit interaction causes a significant enhancement of the g-factor with magnetic field.The large spin-orbit interaction strength demonstrated is consistent with the predicted direct Rashba spin-orbit interaction in this material system and is expected to enable ultrafast Rabi oscillations of spin qubits and efficient qubit-qubit interactions, as well as provide a platform suitable for studying Majorana zero modes

Critical current for an insulating regime of an underdamped current-biased topological Josephson junction

Aleksandr E. Svetogorov, Daniel Loss, and Jelena Klinovaja Phys. Rev. Research 2, 033448 (2020); arXiv:2006.16643. We study analytically an underdamped current-biased topological Josephson junction. First, we consider a simplified model at zero temperature, where the parity of the non-local fermionic state formed by Majorana bound states (MBSs) localized on the junction is fixed, and show that a transition from insulating to conducting state in this case is governed by single-quasiparticle tunneling rather than by Cooper pair tunneling in contrast to a non-topological Josephson junction. This results in a significantly lower critical current for the transition from insulating to conducting state. We propose that, if the length of the system is finite, the transition from insulating to conducting state occurs at exponentially higher bias current due to hybridization of the states with different parities as a result of the overlap of MBSs localized on the junction and at the edges of the topological nanowire forming the junction. Finally, we discuss how the appearance of MBSs can be established experimentally by measuring the critical current for an insulating regime at different values of the applied magnetic field.

Majorana zero modes and their bosonization

Victor Chua, Katharina Laubscher, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 102, 155416 (2020); arXiv:2006.03344. The simplest continuum model of a one-dimensional non-interacting superconducting fermionic symmetry-protected topological (SPT) phase is analyzed in great detail using analytic methods. A full exact diagonalization of the mean-field Bogoliubov-de Gennes Hamiltonian is carried out with open boundaries and finite lengths. Majorana zero modes are derived and studied in great detail. Thereafter exact operator bosonization in both open and closed geometries is carried out. The complementary viewpoints provided by fermionic and bosonic formulations of the superconducting SPT phase are then reconciled. In particular, we provide a complete and exact account of how the topological Majorana zero modes manifest in a bosonized description of an SPT phase.

Magnonic Quadrupole Topological Insulator in Antiskyrmion Crystals

Tomoki Hirosawa, Sebastian A. Diaz, Jelena Klinovaja, and Daniel Loss Phys. Rev. Lett. 125, 207204 (2020); arXiv:2005.05884. We uncover that antiskyrmion crystals provide an experimentally accessible platform to realize a magnonic quadrupole topological insulator, whose hallmark signatures are robust magnonic corner states. Furthermore, we show that tuning an applied magnetic field can trigger the self-assembly of antiskyrmions carrying a fractional topological charge along the sample edges. Crucially, these fractional antiskyrmions restore the symmetries needed to enforce the emergence of the magnonic corner states. Using the machinery of nested Wilson loops, adapted to magnonic systems supported by noncollinear magnetic textures, we demonstrate the quantization of the bulk quadrupole moment, edge dipole moments, and corner charges.

Optimal frequency estimation and its application to quantum dots

Angel Gutierrez-Rubio, Peter Stano, and Daniel Loss arXiv:2004.12049 We address the interaction-time optimization for frequency estimation in a general two-level system. The goal is to track with maximum precision a stochastic perturbation with arbitrary dynamics. Our approach is valid for any figure of merit used to define optimality, and is illustrated for the variance and entropy. For the entropy, we clarify the connection to maximum-likelihood estimation. We devise novel estimation protocols with and without feedback. They outperform common protocols given in the literature. We design a probabilistic self-consistent protocol as a generically optimal estimation without feedback. It can improve current experimental techniques and boost coherence times in quantum computing.

The germanium quantum information route

Giordano Scappucci, Christoph Kloeffel, Floris A. Zwanenburg, Daniel Loss, Maksym Myronov, Jian-Jun Zhang, Silvano De Franceschi, Georgios Katsaros, and Menno Veldhorst Nat Rev Mater (2020); arXiv:2004.08133. In the worldwide endeavor for disruptive quantum technologies, germanium is emerging as a versatile material to realize devices capable of encoding, processing, or transmitting quantum information. These devices leverage special properties of the germanium valence-band states, commonly known as holes, such as their inherently strong spin-orbit coupling and the ability to host superconducting pairing correlations. In this Review, we initially introduce the physics of holes in low-dimensional germanium structures with key insights from a theoretical perspective. We then examine the material science progress underpinning germanium-based planar heterostructures and nanowires. We review the most significant experimental results demonstrating key building blocks for quantum technology, such as an electrically driven universal quantum gate set with spin qubits in quantum dots and superconductor-semiconductor devices for hybrid quantum systems. We conclude by identifying the most promising prospects toward scalable quantum information processing.

Majorana bound states in topological insulators with hidden Dirac points

Ferdinand Schulz, Kirill Plekhanov, Daniel Loss, and Jelena Klinovaja Phys. Rev. Research 2, 033215 (2020); arXiv:2004.10623. We address the issue whether it is possible to generate Majorana bound states at the magnetic-superconducting interface in two-dimensional topological insulators with hidden Dirac points in the spectrum. In this case, the Dirac point of edge states is located at the energies of the bulk states such that two types of states are strongly hybridized. Here, we show that well-defined Majorana bound states can be obtained even in materials with hidden Dirac point provided that the width of the magnetic strip is chosen to be comparable with the localization length of the edge states. The obtained topological phase diagram allows one to extract precisely the position of the Dirac point in the spectrum. In addition to standard zero-bias peak features caused by Majorana bound states in transport experiments, we propose to supplement future experiments with measurements of charge and spin polarization. In particular, we demonstrate that both observables flip their signs at the topological phase transition, thus, providing an independent signature of the presence of topological superconductivity. All features remain stable against substantially strong disorder.

Exchange interaction of hole-spin qubits in double quantum dots in highly anisotropic semiconductors

Bence Hetényi, Christoph Kloeffel, and Daniel Loss Phys. Rev. Research 2, 033036 (2020); arXiv:2004.07658. We study the exchange interaction between two hole-spin qubits in a double quantum dot setup in a silicon nanowire in the presence of magnetic and electric fields. Based on symmetry arguments we show that there exists an effective spin that is conserved even in highly anisotropic semiconductors, provided that the system has a twofold symmetry with respect to the direction of the applied magnetic field. This finding facilitates the definition of qubit basis states and simplifies the form of exchange interaction for two-qubit gates in coupled quantum dots. If the magnetic field is applied along a generic direction, cubic anisotropy terms act as an effective spin-orbit interaction introducing novel exchange couplings even for an inversion symmetric setup. Considering the example of a silicon nanowire double dot, we present the relative strength of these anisotropic exchange interaction terms and calculate the fidelity of the SQRT-of-SWAP gate. Furthermore, we show that the anisotropy-induced spin-orbit effects can be comparable to that of the direct Rashba spin-orbit interaction for experimentally feasible electric field strengths.

Superconducting Quantum Interference in Edge State Josephson Junctions

Tamás Haidekker Galambos, Silas Hoffman, Patrik Recher, Jelena Klinovaja, and Daniel Loss Phys. Rev. Lett. 125, 157701 (2020); arXiv:2004.01733. We study superconducting quantum interference in a Josephson junction linked via edge states in two-dimensional (2D) insulators. We consider two scenarios in which the 2D insulator is either a topological or a trivial insulator supporting one-dimensional (1D) helical or nonhelical edge states, respectively. In equilibrium, we find that the qualitative dependence of critical supercurrent on the flux through the junction is insensitive to the helical nature of the mediating states and can, therefore, not be used to verify the topological features of the underlying insulator. However, upon applying a finite voltage bias smaller than the superconducting gap to a relatively long junction, the finite-frequency interference pattern in the non-equilibrium transport current is qualitatively different for helical edge states as compared to nonhelical ones.

Rational boundary charge in one-dimensional systems with interaction and disorder

Mikhail Pletyukhov, Dante M. Kennes, Kiryl Piasotski, Jelena Klinovaja, Daniel Loss, and Herbert Schoeller Phys. Rev. Res. 2, 033345 (2020); arXiv:2004.00463. We study the boundary charge QB of generic semi-infinite one-dimensional insulators with translational invariance and show that non-local symmetries (i.e., including translations) lead to rational quantizations p/q of QB. In particular, we find that (up to an unknown integer) the quantization of QB is given in integer units of 12ρ¯ and 12(ρ¯−1), where ρ¯ is the average charge per site (which is a rational number for an insulator). This is a direct generalization of the known half-integer quantization of QB for systems with local inversion or local chiral symmetries to any rational value. Quite remarkably, this rational quantization remains valid even in the presence of short-ranged electron-electron interactions as well as static random disorder (breaking translational invariance). This striking stability can be traced back to the fact that local perturbations in insulators induce only local charge redistributions. We establish this result with complementary methods including density matrix renormalization group calculations, bosonization methods, and exact solutions for particular lattice models. Furthermore, for the special case of half-filling ρ¯=12, we present explicit results in single-channel and nearest-neighbor hopping models and identify Weyl semimetal physics at gap closing points. Our general framework also allows us to shed new light on the well-known rational quantization of soliton charges at domain walls.

Spin orbit field in a physically defined p type MOS silicon double quantum dot

Marian Marx, Jun Yoneda, Ángel Gutiérrez Rubio, Peter Stano, Tomohiro Otsuka, Kenta Takeda, Sen Li, Yu Yamaoka, Takashi Nakajima, Akito Noiri, Daniel Loss, Tetsuo Kodera, and Seigo Tarucha arXiv:2003.07079 We experimentally and theoretically investigate the spin orbit (SO) field in a physically defined, p type metal oxide semiconductor double quantum dot in silicon. We measure the magnetic field dependence of the leakage current through the double dot in the Pauli spin blockade. A finite magnetic field lifts the blockade, with the lifting least effective when the external and SO fields are parallel. In this way, we find that the spin flip of a tunneling hole is due to a SO field pointing perpendicular to the double dot axis and almost fully out of the quantum well plane. We augment the measurements by a derivation of SO terms using group symmetric representations theory. It predicts that without in plane electric fields (a quantum well case), the SO field would be mostly within the plane, dominated by a sum of a Rashba and a Dresselhaus like term. We, therefore, interpret the observed SO field as originated in the electric fields with substantial in plane components.

Quantum Damping of Skyrmion Crystal Eigenmodes due to Spontaneous Quasiparticle Decay

Alexander Mook, Jelena Klinovaja, and Daniel Loss Phys. Rev. Res. 2, 033491 (2020); arXiv:2002.12676. The elementary excitations of skyrmion crystals experience both emergent magnetic fields and anharmonic interactions brought about by the topologically nontrivial noncollinear texture. The resulting flat bands cause strong spontaneous quasiparticle decay, dressing the eigenmodes of skyrmion crystals with a finite zero-temperature quantum lifetime. Sweeping the flat bands through the spectrum by changing the magnetic field leads to an externally controllable energy-selective magnon breakdown. In particular, we uncover that the three fundamental modes, i.e., the anticlockwise, breathing, and clockwise mode, exhibit distinct decay behavior, with the clockwise (anticlockwise) mode being the least (most) stable mode out of the three.

Spin Wave Radiation by a Topological Charge Dipole

Sebastian A. Diaz, Tomoki Hirosawa, Daniel Loss, and Christina Psaroudaki Nano Lett. 20, 6556 (2020); arXiv:2002.12282. The use of spin waves (SWs) as data carriers in spintronic and magnonic logic devices offers operation at low power consumption, free of Joule heating. Nevertheless, the controlled emission and propagation of SWs in magnetic materials remains a significant challenge. Here, we propose that skyrmion-antiskyrmion bilayers form topological charge dipoles and act as efficient sub-100 nm SW emitters when excited by in-plane ac magnetic fields. The propagating SWs have a preferred radiation direction, with clear dipole signatures in their radiation pattern, suggesting that the bilayer forms a SW antenna. Bilayers with the same topological charge radiate SWs with spiral and antispiral spatial profiles, enlarging the class of SW patterns. We demonstrate that the characteristics of the emitted SWs are linked to the topology of the source, allowing for full control of the SW features, including their amplitude, preferred direction of propagation, and wavelength.

Site‐Controlled Uniform Ge/Si Hut Wires with Electrically Tunable Spin–Orbit Coupling

Fei Gao, Jian-Huan Wang, Hannes Watzinger, Hao Hu, Marko J. Rancic, Jie-Yin Zhang, Ting Wang, Yuan Yao, Gui-Lei Wang, Josip Kukucka, Lada Vukusic, Christoph Kloeffel, Daniel Loss, Feng Liu, Georgios Katsaros, and Jian-Jun Zhang Adv. Mater. 2020, 1906523; arXiv:2001.11305. Semiconductor nanowires have been playing a crucial role in the development of nanoscale devices for the realization of spin qubits, Majorana fermions, single photon emitters, nanoprocessors, etc. The monolithic growth of site‐controlled nanowires is a prerequisite toward the next generation of devices that will require addressability and scalability. Here, combining top‐down nanofabrication and bottom‐up self‐assembly, the growth of Ge wires on prepatterned Si (001) substrates with controllable position, distance, length, and structure is reported. This is achieved by a novel growth process that uses a SiGe strain‐relaxation template and can be potentially generalized to other material combinations. Transport measurements show an electrically tunable spin–orbit coupling, with a spin–orbit length similar to that of III–V materials. Also, charge sensing between quantum dots in closely spaced wires is observed, which underlines their potential for the realization of advanced quantum devices. The reported results open a path toward scalable qubit devices using nanowires on silicon.

Magnetic field independent sub-gap states in hybrid Rashba nanowires

Christian Juenger, Raphaelle Delagrange, Denis Chevallier, Sebastian Lehmann, Kimberly A. Dick, Claes Thelander, Jelena Klinovaja, Daniel Loss, Andreas Baumgartner, and Christian Schoenenberger Phys. Rev. Lett. 125, 017701 (2020); arXiv:2001.07666. Sub-gap states in semiconducting-superconducting nanowire hybrid devices are controversially discussed as potential topologically non-trivial quantum states. One source of ambiguity is the lack of an energetically and spatially well defined tunnel spectrometer. Here, we use quantum dots directly integrated into the nanowire during the growth process to perform tunnel spectroscopy of discrete sub-gap states in a long nanowire segment. In addition to sub-gap states with a standard magnetic field dependence, we find topologically trivial sub-gap states that are independent of the external magnetic field, i.e. that are pinned to a constant energy as a function of field. We explain this effect qualitatively and quantitatively by taking into account the strong spin-orbit interaction in the nanowire, which can lead to a decoupling of Andreev bound states from the field due to a spatial spin texture of the confined eigenstates.

Transport signatures of topological phases in double nanowires probed by spin-polarized STM

Manisha Thakurathi, Denis Chevallier, Daniel Loss, and Jelena Klinovaja Phys. Rev. Research 2, 023197 (2020); arXiv:2001.05470. We study a double-nanowire setup proximity coupled to an s-wave superconductor and search for the bulk signatures of the topological phase transition that can be observed experimentally, for example, with an STM tip. Three bulk quantities, namely, the charge, the spin polarization, and the pairing amplitude of intrawire superconductivity are studied in this work. The spin polarization and the pairing amplitude flip sign as the system undergoes a phase transition from the trivial to the topological phase. In order to identify promising ways to observe bulk signatures of the phase transition in transport experiments, we compute the spin current flowing between a local spin-polarized probe, such as an STM tip, and the double-nanowire system in the Keldysh formalism. We find that the spin current contains information about the sign flip of the bulk spin polarization and can be used to determine the topological phase transition point.

Coherence of a driven electron spin qubit actively decoupled from quasi-static noise

Takashi Nakajima, Akito Noiri, Kento Kawasaki, Jun Yoneda, Peter Stano, Shinichi Amaha, Tomohiro Otsuka, Kenta Takeda, Matthieu R. Delbecq, Giles Allison, Arne Ludwig, Andreas D. Wieck, Daniel Loss, and Seigo Tarucha Phys. Rev. X 10, 011060 (2020); arXiv:2001.02884. The coherence of electron spin qubits in semiconductor quantum dots suffers mostly from low-frequency noise. During the last decade, efforts have been devoted to mitigate such noise by material engineering, leading to substantial enhancement of the spin dephasing time for an idling qubit. However, the role of the environmental noise during spin manipulation, which determines the control fidelity, is less understood. We demonstrate an electron spin qubit whose coherence in the driven evolution is limited by high-frequency charge noise rather than the quasi-static noise inherent to any semiconductor device. We employed a feedback control technique to actively suppress the latter, demonstrating a π-flip gate fidelity as high as 99.04±0.23% in a gallium arsenide quantum dot. We show that the driven-evolution coherence is limited by the longitudinal noise at the Rabi frequency, whose spectrum resembles the 1/f noise observed in isotopically purified silicon qubits.

Universal conductance dips and fractional excitations in a two-subband quantum wire

Chen-Hsuan Hsu, Flavio Ronetti, Peter Stano, Jelena Klinovaja, and Daniel Loss Phys. Rev. Research 2, 043208 (2020); arXiv:1912.11592. We theoretically investigate a quantum wire based on a quasi-one-dimensional Kondo lattice formed by localized spins and itinerant electrons, where the lowest two subbands of the quantum wire are populated. We uncover a backscattering mechanism involving helically ordered spins and Coulomb interaction between the electrons. The combination of these ingredients results in scattering resonances and partial gaps which give rise to non-standard plateaus and conductance dips at certain electron densities. The positions and values of these dips are independent of material parameters, serving as direct transport signatures of this mechanism. While our theory describes a generic Kondo lattice, an experimentally relevant realization is provided by quantum wires made out of III-V semiconductors hosting nuclear spins such as InAs. Observation of the universal conductance dips would not only confirm the presence of a nuclear spin helix but also identify a strongly correlated fermion system hosting fractional excitations, resembling the fractional quantum Hall states even without external magnetic fields.

Majorana and parafermion corner states from two coupled sheets of bilayer graphene

Katharina Laubscher, Daniel Loss, and Jelena Klinovaja Phys. Rev. Res. 2, 013330 (2020); arXiv:1912.10931. We consider a setup consisting of two coupled sheets of bilayer graphene in the regime of strong spin-orbit interaction, where electrostatic confinement is used to create an array of effective quantum wires. We show that for suitable interwire couplings the system supports a topological insulator phase exhibiting Kramers partners of gapless helical edge states, while the additional presence of a small in-plane magnetic field and weak proximity-induced superconductivity leads to the emergence of zero-energy Majorana corner states at all four corners of a rectangular sample, indicating the transition to a second-order topological superconducting phase. The presence of strong electron-electron interactions is shown to promote the above phases to their exotic fractional counterparts. In particular, we find that the system supports a fractional topological insulator phase exhibiting fractionally charged gapless edge states and a fractional second-order topological superconducting phase exhibiting zero-energy Z2m parafermion corner states, where m is an odd integer determined by the position of the chemical potential.

First-order magnetic phase-transition of mobile electrons in monolayer MoS2

Jonas Gaël Roch, Dmitry Miserev, Guillaume Froehlicher, Nadine Leisgang, Lukas Sponfeldner, Kenji Watanabe, Takashi Taniguchi, Jelena Klinovaja, Daniel Loss, and Richard John Warburton Phys. Rev. Lett. 124, 187602 (2020); arXiv:1911.10238. Evidence is presented for a first-order magnetic phase transition in a gated two-dimensional semiconductor, monolayer-MoS2. The phase boundary separates a spin-polarised (ferromagnetic) phase at low electron density and a paramagnetic phase at high electron density. Abrupt changes in the optical response signal an abrupt change in the magnetism. The magnetic order is thereby controlled via the voltage applied to the gate electrode of the device. Accompanying the change in magnetism is a large change in the electron effective mass.

Topological invariants to characterize universality of boundary charge in one-dimensional insulators beyond symmetry constraints

Mikhail Pletyukhov, Dante M. Kennes, Jelena Klinovaja, Daniel Loss, and Herbert Schoeller Phys. Rev. B 101, 161106 (2020); arXiv:1911.06890. In the absence of any symmetry constraints we address universal properties of the boundary charge QB for a wide class of tight-binding models with non-degenerate bands in one dimension. We provide a precise formulation of the bulk-boundary correspondence by splitting QB via a gauge invariant decomposition in a Friedel, polarisation, and edge part. We reveal the topological nature of QB by proving the quantization of a topological index I=ΔQB−ρ, where ΔQB is the change of QB when shifting the lattice by one site towards a boundary and ρ is the average charge per site. For a single band we find this index to be given by the winding number of the fundamental phase difference of the Bloch wave function between two adjacent sites. For a given chemical potential we establish a central topological constraint I in {−1,0} related to charge conservation and particle-hole duality. Our results are shown to be stable against disorder and we propose generalizations to multi-channel and interacting systems.

Surface charge theorem and topological constraints for edge states: Analytical study of one-dimensional nearest-neighbor tight-binding models

Mikhail Pletyukhov, Dante M. Kennes, Jelena Klinovaja, Daniel Loss, and Herbert Schoeller Phys. Rev. B 101, 165304 (2020); arXiv:1911.06886. For a wide class of noninteracting tight-binding models in one dimension, we present an analytical solution for all scattering and edge states on a half-infinite system. Without assuming any symmetry constraints, we consider models with nearest-neighbor hoppings and one orbital per site but the arbitrary size of the unit cell and generic modulations of on-site potentials and hoppings. The solutions are parametrized by determinants that can be straightforwardly calculated from recursion relations. We show that this representation allows for an elegant analytic continuation to complex quasimomentum consistent with previous treatments for continuum models. Two important analytical results are obtained based on the explicit knowledge of all eigenstates. (1) An explicit proof of the surface charge theorem is presented including a unique relationship between the boundary charge Q_B(a) of a single band a and the bulk polarization in terms of the Zak-Berry phase. In particular, the Zak-Berry phase is determined within a special gauge of the Bloch states such that no unknown integer is left. This establishes a precise form of a bulk-boundary correspondence relating the boundary charge of a single band to bulk properties. (2) We derive a topological constraint for the phase dependence of the edge state energies, where the phase variable describes a continuous shift of the lattice towards the boundary. The topological constraint is shown to be equivalent to the quantization of a topological index I = Q_B − ρ= −1, 0, introduced in an accompanying paper [M. Pletyukhov et al., Phys. Rev. B 101, 161106 (2020)]. Here, Q_B is the change of the boundary charge Q_B for a given chemical potential in the insulating regime when the lattice is shifted by one site towards the boundary, and ρ is the average charge per site (both in units of the elementary charge e = 1). This establishes an interesting link between universal properties of the boundary charge and edge state physics discussed within the field of topological insulators. In accordance with previous results for continuum systems, we also establish the localization of the boundary charge and determine the explicit form of the density given by an exponential decay and a pre-exponential function following a power law with generic exponent −1/2 at large distances

From Andreev to Majorana bound states in hybrid superconductor-semiconductor nanowires

Elsa Prada, Pablo San-Jose, Michiel W. A. de Moor, Attila Geresdi, Eduardo J. H. Lee, Jelena Klinovaja, Daniel Loss, Jesper Nygard, Ramon Aguado, and Leo P. Kouwenhoven Nat. Rev. Phys. 2, 575 (2020); arXiv:1911.04512. Electronic excitations above the ground state must overcome an energy gap in superconductors with spatially-homogeneous pairing. In contrast, inhomogeneous superconductors such as those with magnetic impurities, weak links or heterojunctions containing normal metals can host subgap electronic excitations that are generically known as Andreev bound states (ABSs). With the advent of topological superconductivity, a new kind of ABS with exotic qualities, known as Majorana bound state (MBS), has been discovered. We review the main properties of all such subgap states and the state-of-the-art techniques for their detection. We focus on hybrid superconductor-semiconductor nanowires, possibly coupled to quantum dots, as one of the most flexible and promising experimental platforms. We discuss how the combined effect of spin-orbit coupling and Zeeman energy in these wires triggers the transition from ABSs into MBSs and show theoretical progress beyond minimal models in understanding experiments, including the possibility of a new type of robust Majorana zero mode without the need of a band topological transition. We examine the role of spatial non-locality, a special property of MBS wavefunctions that, together with non-Abelian braiding, is the key ingredient for realizing topological quantum computing.

Magnetically-Confined Bound States in Rashba Systems

Flavio Ronetti, Kirill Plekhanov, Daniel Loss, and Jelena Klinovaja Phys. Rev. Research 2, 022052(R) (2020); arXiv:1911.03133. A Rashba nanowire is subjected to a magnetic field that assumes opposite signs in two sections of the nanowire, and, thus, creates a magnetic domain wall. The direction of magnetic field is chosen to be perpendicular to the Rashba spin-orbit vector such that there is only a partial gap in the spectrum. Nevertheless, we prove analytically and numerically that such a domain wall hosts a bound state whose energy is at bottom of the spectrum below the energy of all bulk states. Thus, this magnetically confined bound state is well-isolated and can be accessed experimentally. We further show that the same type of magnetic confinement can be implemented in two-dimensional systems with strong spin-orbit interaction. A quantum channel along the magnetic domain wall emerges with a nondegenerate dispersive band that lies energetically below the bulk states. We show that this magnetic confinement is robust against disorder and various parameter variations.

Electronic transport in one-dimensional Floquet topological insulators via topological- and non-topological edge states

Niclas Müller, Dante M. Kennes, Jelena Klinovaja, Daniel Loss, and Herbert Schoeller Phys. Rev. B 101, 155417 (2020); arXiv:1911.02295. Based on probing electronic transport properties, we propose an experimental test for the recently discovered rich topological phase diagram of one-dimensional Floquet topological insulators with Rashba spin-orbit interaction [Kennes et al., Phys. Rev. B 100, 041104(R) (2019)]. Using the Keldysh-Floquet formalism, we compute electronic transport properties of these nanowires, where we propose to couple the leads in such a way, as to primarily address electronic states with a large relative weight at one edge of the system. By tuning the Fermi energy of the leads to the center of the topological gap, we are able to directly address the topological edge states, granting experimental access to the topological phase diagram. Surprisingly, we find conductance values similar or even larger in magnitude to those corresponding to topological edge states, when tuning the lead Fermi energy to special values in the bulk, which coincide with bifurcation points of the dispersion relation in complex quasimomentum space. These peaks reveal the presence of narrow bands of states whose wave functions are linear combinations of delocalized bulk states and exponentially localized edge states, where the amplitude of the edge-state component is sharply peaked at the aforementioned bifurcation point, resulting in an unusually large relative edge-weight. We discuss the transport properties of these \emph{non-topological edge states} and explain their emergence in terms of an intuitive yet quantitative physical picture. The mechanism giving rise to these states is not specific to the model we consider here, suggesting that they may be present in a wide class of systems.

Quantum Depinning of a Magnetic Skyrmion

Christina Psaroudaki and Daniel Loss Phys. Rev. Lett. 124, 097202 (2020); Editor's suggestion; arXiv:1910.09585. We investigate the quantum depinning of a weakly driven skyrmion out of an impurity potential in a mesoscopic magnetic insulator. For small barrier height, the Magnus force dynamics dominates over the inertial one, and the problem is reduced to a massless charged particle in a strong magnetic field. The universal form of the WKB exponent, the rate of tunneling, and the crossover temperature between thermal and quantum tunneling is provided, independently of the detailed form of the pinning potential. The results are discussed in terms of macroscopic parameters of the insulator Cu2OSeO3 and various skyrmion radii. We demonstrate that small enough magnetic skyrmions, with a radius of ~ 10 lattice sites, consisting of some thousands of spins, can behave as quantum objects at low temperatures in the mK regime.

Time-Reversal Invariant Topological Superconductivity in Planar Josephson Bijunction

Yanick Volpez, Daniel Loss, and Jelena Klinovaja Phys. Rev. Research 2, 023415 (2020); arXiv:1910.06921. We consider a Josephson bijunction consisting of a thin SIS π-Josephson junction sandwiched between two-dimensional semiconducting layers with strong Rashba spin-orbit interaction. Each of these layers forms an SNS junction due to proximity-induced superconductivity. The SIS junction is assumed to be thin enough such that the two Rashba layers are tunnel-coupled. We show that, by tuning external gates, this system can be controllably brought into a time-reversal invariant topological superconducting phase with a Kramers pair of Majorana bound states being localized at the end of the normal region for a large parameter phase space. In particular, in the strong spin-orbit interaction limit, the topological phase can be accessed already in the regime of small tunneling amplitudes.

Chiral Magnonic Edge States in Ferromagnetic Skyrmion Crystals Controlled by Magnetic Fields

Sebastian A. Diaz, Tomoki Hirosawa, Jelena Klinovaja, and Daniel Loss Phys. Rev. Res. 2, 013231 (2020); arXiv:1910.05214. Achieving control over magnon spin currents in insulating magnets - where dissipation due to Joule heating is highly suppressed - is an active area of research that could lead to energy-efficient spintronics applications. However, magnon spin currents supported by conventional systems with uniform magnetic order have proven hard to control. An alternative approach that relies on topologically protected magnonic edge states of spatially periodic magnetic textures has recently emerged. A prime example of such textures is the ferromagnetic skyrmion crystal which hosts chiral edge states providing a platform for magnon spin currents. Here, we show, for the first time, an external magnetic field can drive a topological phase transition in the spin wave spectrum of a ferromagnetic skyrmion crystal. The topological phase transition is signaled by the closing of a low-energy bulk magnon gap at a critical field. In the topological phase, below the critical field, two topologically protected chiral magnonic edge states lie within this gap, but they unravel in the trivial phase, above the critical field. Remarkably, the topological phase transition involves an inversion of two magnon bands that at the Γ point correspond to the breathing and anticlockwise modes of the skyrmions in the crystal. Our findings suggest that an external magnetic field could be used as a knob to switch on and off magnon spin currents carried by topologically protected chiral magnonic edge states.

Hinge Modes and Surface States in Second-Order Topological Three-Dimensional Quantum Hall Systems induced by Charge Density

Pawel Szumniak, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 102, 125126 (2020); arXiv:1910.05090. We consider a system of weakly coupled one-dimensional wires forming a three-dimensional stack in the presence of a spatially periodic modulation of the chemical potential along the wires, equivalent to a charge density wave (CDW). An external static magnetic field is applied parallel to the wire axes. We show that, for a certain parameter regime, due to interplay between the CDW and magnetic field, the system can support a second-order topological phase characterized by the presence of chiral quasi-1D Quantum Hall Effect (QHE) hinge modes. Interestingly, we demonstrate that direction of propagation of the hinge modes depends on the phase of the CDW and can be reversed only by electrical means without the need of changing the orientation of the magnetic field. Furthermore, we show that the system can also support 2D chiral surface QHE states, which can coexist with one-dimensional hinge modes, realizing a scenario of a hybrid high-order topology. We show that the hinge modes are robust against static disorder.

Interaction Driven Floquet Engineering of Topological Superconductivity in Rashba Nanowires

Manisha Thakurathi, Pavel P. Aseev, Daniel Loss, and Jelena Klinovaja Phys. Rev. Res. 2, 013292 (2020); arXiv:1910.03730. We analyze, analytically and numerically, a periodically driven Rashba nanowire proximity coupled to an s-wave superconductor using bosonization and renormalization group analysis in the regime of strong electron-electron interactions. Due to the repulsive interactions, the superconducting gap is suppressed, whereas the Floquet Zeeman gap is enhanced, resulting in a higher effective value of g-factor compared to the non-interacting case. The flow equations for different coupling constants, velocities, and Luttinger-liquid parameters explicitly establish that even for small initial values of the Floquet Zeeman gap compared to the superconducting proximity gap, the interactions drive the system into the topological phase and the interband interaction term helps to achieve larger regions of the topological phase in parameter space.

Hinge states in a system of coupled Rashba layers

Kirill Plekhanov, Flavio Ronetti, Daniel Loss, and Jelena Klinovaja Phys. Rev. Research 2, 013083 (2020); arXiv:1910.01655. We consider a system of stacked tunnel-coupled two-dimensional electron- and hole-gas layers with Rashba spin-orbit interactions subjected to a staggered Zeeman field. The interplay of different intra-layer tunnel couplings results in a phase transition to a topological insulator phase in three dimensions hosting gapless surface states. The staggered Zeeman field further enriches the topological phase diagram by generating a second-order topological insulator phase hosting gapless hinge states. The emergence of the topological phases is proven analytically in the regime of small Zeeman field and confirmed by numerical simulations in the non-perturbative region of the phase diagram. The topological phases are stable against external perturbations and disorder.

Low-symmetry nanowire cross-sections for enhanced Dresselhaus spin-orbit interaction

Miguel J. Carballido, Christoph Kloeffel, Dominik M. Zumbuehl, and Daniel Loss Phys. Rev. B 103, 195444 (2021); arXiv:1910.00562. We study theoretically the spin-orbit interaction of low-energy electrons in semiconducting nanowires with a zinc-blende lattice. The effective Dresselhaus term is derived for various growth directions, including <11(-2)>-oriented nanowires. While a specific configuration exists where the Dresselhaus spin-orbit coupling is suppressed even at confinement potentials of low symmetry, many configurations allow for a strong Dresselhaus coupling. In particular, we discuss qualitative and quantitative results for nanowire cross-sections modeled after sectors of rings or circles. The parameter dependence is analyzed in detail, enabling predictions for a large variety of setups. For example, we gain insight into the spin-orbit coupling in recently fabricated GaAs-InAs nanomembrane-nanowire structures. By combining the effective Dresselhaus and Rashba terms, we find that such structures are promising platforms for applications where an electrically controllable spin-orbit interaction is needed. If the nanowire cross-section is scaled down and InAs replaced by InSb, remarkably high Dresselhaus-based spin-orbit energies of the order of millielectronvolt are expected. A Rashba term that is similar to the effective Dresselhaus term can be induced via electric gates, providing means to switch the spin-orbit interaction on and off. By varying the central angle of the circular sector, we find, among other things, that particularly strong Dresselhaus couplings are possible when nanowire cross-sections resemble half-disks.

Majorana Fermions in Magnetic Chains

Rémy Pawlak, Silas Hoffman, Jelena Klinovaja, Daniel Loss, and Ernst Meyer Progress in Particle and Nuclear Physics 107, 1-19 (2019); arXiv:1909.10778. Majorana fermions have recently garnered a great attention outside the field of particle physics, in condensed matter physics. In contrast to their particle physics counterparts, Majorana fermions are zero energy, chargeless, spinless, composite quasiparticles, residing at the boundaries of so-called topological superconductors. Furthermore, in opposition to any particles in the standard model, Majorana fermions in solid-state systems obey non-Abelian exchange statistics that make them attractive candidates for decoherence-free implementations of quantum computers. In this review, we report on the recent advances to realize synthetic topological superconductors supporting Majorana fermions with an emphasis on chains of magnetic impurities on the surface of superconductors. After outlining the theoretical underpinning responsible for the formation of Majorana fermions, we report on the subsequent experimental efforts to build topological superconductors and the resulting evidence in favor of Majorana fermions, focusing on scanning tunneling microscopy and the hunt for zero-bias peaks in the measured current. We conclude by summarizing the open questions in the field and propose possible experimental measurements to answer them.

Coherent backaction between spins and an electronic bath: Non-Markovian dynamics and low temperature quantum thermodynamic electron cooling

Stephanie Matern, Daniel Loss, Jelena Klinovaja, and Bernd Braunecker Phys. Rev. B 100, 134308 (2019); arXiv:1905.11422. We provide a general analytical framework for calculating the dynamics of a spin system in contact with a bath beyond the Markov approximation. The approach is based on a systematic expansion of the Nakashima-Zwanzig master equation in the weak-coupling limit but makes no assumption on the time dynamics and includes all quantum coherent memory effects leading to non-Markovian dynamics. Our results describe, for the free induction decay, the full time range from the non-Markovian dynamics at short times, to the well-known exponential thermal decay at long times. We provide full analytic results for the entire time range using a bath of itinerant electrons as an archetype for universal quantum fluctuations. Furthermore, we propose a quantum thermodynamic scheme to employ the temperature insensitivity of the non-Markovian decay to transport heat out of the electron system and thus, by repeated re-initialisation of a cluster of spins, to efficiently cool the electrons at very low temperatures.

Floquet Second-Order Topological Superconductor Driven via Ferromagnetic Resonance

Kirill Plekhanov, Manisha Thakurathi, Daniel Loss, and Jelena Klinovaja Phys. Rev. Research 1, 032013 (2019); arXiv:1905.09241. We consider a Floquet triple-layer setup composed of a two-dimensional electron gas with spin-orbit interactions, proximity coupled to an s-wave superconductor and to a ferromagnet driven at resonance. The ferromagnetic layer generates a time-oscillating Zeeman field which competes with the induced superconducting gap and leads to a topological phase transition. The resulting Floquet states support a second-order topological superconducting phase with a pair of localized zero-energy Floquet Majorana corner states. Moreover, the phase diagram comprises a Floquet helical topological superconductor, hosting a Kramers pair of Majorana edge modes protected by an effective time-reversal symmetry, as well as a gapless Floquet Weyl phase. The topological phases are stable against disorder and parameter variations and are within experimental reach.

Fractional Topological Superconductivity and Parafermion Corner States

Katharina Laubscher, Daniel Loss, and Jelena Klinovaja Phys. Rev. Research 1, 032017(R) (2019); arXiv:1905.00885. We consider a system of weakly coupled Rashba nanowires in the strong spin-orbit interaction (SOI) regime. The nanowires are arranged into two tunnel-coupled layers proximitized by a top and bottom superconductor such that the superconducting phase difference between them is pi. We show that in such a system strong electron-electron interactions can stabilize a helical topological superconducting phase hosting Kramers partners of Z_2m parafermion edge modes, where m is an odd integer determined by the position of the chemical potential. Furthermore, upon turning on a weak in-plane magnetic field, the system is driven into a second-order topological superconducting phase hosting zero-energy Z_2m parafermion bound states localized at two opposite corners of a rectangular sample. As a special case, zero-energy Majorana corner states emerge in the non-interacting limit m=1, where the chemical potential is tuned to the SOI energy of the single nanowires.

Quantum non-demolition measurement of an electron spin qubit

Takashi Nakajima, Akito Noiri, Jun Yoneda, Matthieu R. Delbecq, Peter Stano, Tomohiro Otsuka, Kenta Takeda, Shinichi Amaha, Giles Allison, Kento Kawasaki, Arne Ludwig, Andreas D. Wieck, Daniel Loss, and Seigo Tarucha Nature Nanotechnology, 14, 555 (2019); arXiv:1904.11220. Measurement of quantum systems inevitably involves disturbance in various forms. Within the limits imposed by quantum mechanics, however, one can design an "ideal" projective measurement that does not introduce a back action on the measured observable, known as a quantum nondemolition (QND) measurement. Here we demonstrate an all-electrical QND measurement of a single electron spin in a gate-defined quantum dot via an exchange-coupled ancilla qubit. The ancilla qubit, encoded in the singlet-triplet two-electron subspace, is entangled with the single spin and subsequently read out in a single shot projective measurement at a rate two orders of magnitude faster than the spin relaxation. The QND nature of the measurement protocol is evidenced by observing a monotonic increase of the readout fidelity over one hundred repetitive measurements against arbitrary input states. We extract information from the measurement record using the method of optimal inference, which is tolerant to the presence of the relaxation and dephasing. The QND measurement allows us to observe spontaneous spin flips (quantum jumps) in an isolated system with small disturbance. Combined with the high-fidelity control of spin qubits, these results pave the way for various measurement-based quantum state manipulations including quantum error correction protocols.

Quantum Brownian Motion of a Magnetic Skyrmion

Christina Psaroudaki, Pavel Aseev, and Daniel Loss Phys. Rev. B 100, 134404 (2019); arXiv:1904.09215. Within a microscopic theory, we study the quantum Brownian motion of a skyrmion in a magnetic insulator coupled to a bath of magnon-like quantum excitations. The intrinsic skyrmion-bath coupling gives rise to damping terms for the skyrmion center-of-mass, which remain finite down to zero temperature due to the quantum nature of the magnon bath. We show that the quantum version of the fluctuation-dissipation theorem acquires a non-trivial temperature dependence. As a consequence, the skyrmion mean square displacement is finite at zero temperature and has a fast thermal activation that scales quadratically with temperature, contrary to the linear increase predicted by the classical phenomenological theory. The effects of an external oscillating drive which couples directly on the magnon bath are investigated. We generalize the standard quantum theory of dissipation and we show explicitly that additional time-dependent dissipation terms are generated by the external drive. From these we emphasize a friction and a topological charge renormalization term, which are absent in the static limit. The skyrmion response function inherits the time periodicity of the driving field and it is thus enhanced and lowered over a driving cycle. Finally, we provide a generalized version of the nonequilibrium fluctuation-dissipation theorem valid for weakly driven baths.

Charge transport of a spin-orbit-coupled Tomonaga-Luttinger liquid

Chen-Hsuan Hsu, Peter Stano, Yosuke Sato, Sadashige Matsuo, Seigo Tarucha, and Daniel Loss Phys. Rev. B 100, 195423 (2019); arXiv:1904.06869. The charge transport of a (Tomonaga-) Luttinger liquid with tunnel barriers exhibits universal scaling: the current-voltage curves measured at various temperatures collapse into a single curve upon rescaling. The exponent characterizing this single curve can be used to extract the strength of electron-electron interaction. Motivated by a recent experiment on InAs nanowires [Sato et al., Phys. Rev. B 99, 155304 (2019)], we theoretically investigate the analogous behavior of a spin-orbit-coupled Luttinger liquid. We find that the scaling exponent differs for different impurity strengths, being weak (disorder potential) or strong (tunnel barriers), and their positions, either in the bulk or near the edge of the wire. For each case we quantify the exponent of the universal scaling and its modification due to the spin-orbit coupling. Our findings serve as a guide in the determination of the interaction strength of quasi-one-dimensional spin-orbit-coupled quantum wires from transport measurements.

Degeneracy lifting of Majorana bound states due to electron-phonon interactions

Pavel P. Aseev, Pasquale Marra, Peter Stano, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 99, 205435 (2019); arXiv:1903.12066. We study theoretically how electron-phonon interaction affects the energies and level broadening (inverse lifetime) of Majorana bound states (MBSs) in a clean topological nanowire at low temperatures. At zero temperature, the energy splitting between the right and left MBSs remains exponentially small with increasing nanowire length L. At finite temperatures, however, the absorption of thermal phonons leads to the broadening of energy levels of the MBSs that does not decay with system length, and the coherent absorption/emission of phonons at opposite ends of the nanowire results in MBSs energy splitting that decays only as an inverse power-law in L. Both effects remain exponential in temperature. In the case of quantized transverse motion of phonons, the presence of Van Hove singularities in the phonon density of states causes additional resonant enhancement of both the energy splitting and the level broadening of the MBSs. This is the most favorable case to observe the phonon-induced energy splitting of MBSs as it becomes much larger than the broadening even if the topological nanowire is much longer than the coherence length. We also calculate the charge and spin associated with the energy splitting of the MBSs induced by phonons. We consider both a spinless low-energy continuum model, which we evaluate analytically, as well as a spinful lattice model for a Rashba nanowire, which we evaluate numerically.

Majorana Bound States in Double Nanowires with Reduced Zeeman Thresholds due to Supercurrents

Olesia Dmytruk, Manisha Thakurathi, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 99, 245416 (2019); arXiv:1902.11232. We study the topological phase diagram of a setup composed of two nanowires with strong Rashba spin-orbit interaction subjected to an external magnetic field and brought into the proximity to a bulk s-wave superconductor in the presence of a supercurrent flowing through it. The supercurrent reduces the critical values of the Zeeman energy and crossed Andreev superconducting pairing required to reach the topological phase characterized by the presence of one Majorana bound state localized at each system end. We demonstrate that, even in the regime of the crossed Andreev pairing being smaller than the direct proximity pairing, a relatively weak magnetic field drives the system into the topological phase due to the presence of the supercurrent.

Entangling Spins in Double Quantum Dots and Majorana Bound States

Marko J. Rancic, Silas Hoffman, Constantin Schrade, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 99, 165306 (2019); arXiv:1902.10251. We study the coupling between a singlet-triplet qubit realized in a double quantum dot to a topological qubit realized by spatially well-separated Majorana bound states. We demonstrate that the singlet-triplet qubit can be leveraged for readout of the topological qubit and for supplementing the gate operations that cannot be performed by braiding of Majorana bound states. Furthermore, we extend our setup to a network of singlet-triplet and topological hybrid qubits that paves the way to scalable fault-tolerant quantum computing.

Spontaneous Symmetry Breaking in Monolayers of Transition Metal Dichalcogenides

Dmitry Miserev, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 100, 014428 (2019); arXiv:1902.07961. We analyze magnetic phases of monolayers of transition metal dichalcogenides that are two-valley materials with electron-electron interactions. The exchange intervalley scattering makes two-valley systems less stable to the spin fluctuations but more stable to the valley fluctuations. We predict a first-order ferromagnetic phase transition governed by the nonanalytic and negative cubic term in the free energy that results in a large spontaneous spin magnetization. Finite spin-orbit interaction leads to the out-of-plane Ising order of the ferromagnetic phase. Our theoretical prediction is consistent with the recent experiment on the electron-doped monolayers of MoS2 reported by Roch et al. [Nat. Nanotechnol. 14, 432 (2019)]. The proposed first-order phase transition can also be tested by measuring the linear magnetic field dependence of the spin susceptibility in the paramagnetic phase which is a direct consequence of the nonanalyticity of the free energy.

Topological Magnons and Edge States in Antiferromagnetic Skyrmion Crystals

Sebastian A. Diaz, Jelena Klinovaja, and Daniel Loss Phys. Rev. Lett. 122, 187203 (2019); arXiv:1812.11125. Antiferromagnetic skyrmion crystals are magnetic phases predicted to exist in antiferromagnets with Dzyaloshinskii-Moriya interactions. Their spatially periodic noncollinear magnetic texture gives rise to topological bulk magnon bands characterized by nonzero Chern numbers. We find topologically-protected chiral magnonic edge states over a wide range of magnetic fields and Dzyaloshinskii-Moriya interaction values. Moreover, and of particular importance for experimental realizations, edge states appear at the lowest possible energies, namely, within the first bulk magnon gap. Thus, antiferromagnetic skyrmion crystals show great promise as novel platforms for topological magnonics.

Chiral 1D Floquet topological insulators beyond rotating wave approximation

Dante M. Kennes, Niclas Mueller, Mikhail Pletyukhov, Clara Weber, Christoph Bruder, Fabian Hassler, Jelena Klinovaja, Daniel Loss, and Herbert Schoeller Phys. Rev. B 100, 041103(R) (2019); arXiv:1811.12062. We study one-dimensional (1D) Floquet topological insulators with chiral symmetry going beyond the standard rotating wave approximation. The occurrence of many anticrossings between Floquet replicas leads to a dramatic extension of phase diagram regions with stable topological edge states (TESs). We present an explicit construction of all TESs in terms of a truncated Floquet Hamiltonian in frequency space, prove the bulk-boundary correspondence, and analyze the stability of the TESs in terms of their localization lengths. We propose experimental tests of our predictions in curved bilayer graphene.

Universal quantum computation in the surface code using non-Abelian islands

Katharina Laubscher, Daniel Loss, and James R. Wootton Phys. Rev. A 100, 012338 (2019); arXiv:1811.06738. The surface code is currently the primary proposed method for performing quantum error correction. However, despite its many advantages, it has no native method to fault-tolerantly apply non-Clifford gates. Additional techniques are therefore required to achieve universal quantum computation. Here we propose a new method, using small islands of a qudit variant of the surface code. This allows the non-trivial action of the non-Abelian anyons in the latter to process information stored in the former. Specifically, we show that a non-stabilizer state can be prepared, which allows universality to be achieved.

Second Order Topological Superconductivity in pi-Junction Rashba Layers

Yanick Volpez, Daniel Loss, and Jelena Klinovaja Phys. Rev. Lett. 122, 126402 (2019); arXiv:1811.01827. We consider a Josephson junction bilayer consisting of two tunnel-coupled two-dimensional electron gas layers with Rashba spin-orbit interaction, proximitized by a top and bottom s-wave superconductor with phase difference phi close to pi. We show that, in the presence of a finite weak in-plane Zeeman field, the bilayer can be driven into a second order topological superconducting phase, hosting two Majorana corner states (MCSs). If phi=pi, in a rectangular geometry, these zero-energy bound states are located at two opposite corners determined by the direction of the Zeeman field. If the phase difference phi deviates from pi by a critical value, one of the two MCSs gets relocated to an adjacent corner. As the phase difference phi increases further, the system becomes trivially gapped. The obtained MCSs are robust against static and magnetic disorder. We propose two setups that could realize such a model: one is based on controlling phi by magnetic flux, the other involves an additional layer of randomly-oriented magnetic impurities responsible for the phase shift of pi in the proximity-induced superconducting pairing.

Superfluid Transport in Quantum Spin Chains

Silas Hoffman, Daniel Loss, and Yaroslav Tserkovnyak Phys. Rev. B 107, 085403 (2023); arXiv:1810.11470. Spin superfluids enable long-distance spin transport through classical ferromagnets by developing topologically stable magnetic textures. For small spins at low dimensions, however, the topological protection suffers from strong quantum fluctuations. We study the remanence of spin superfluidity inherited from the classical magnet by considering the two-terminal spin transport through a finite spin-1/2 magnetic chain with planar exchange. By fermionizing the system, we recast the spin-transport problem in terms of quasiparticle transmission through a superconducting region. We show that the topological underpinnings of a semiclassical spin superfluid relate to the topological superconductivity in the fermionic representation. In particular, we find an efficient spin transmission through the magnetic region of a characteristic resonant length, which can be related to the properties of the boundary Majorana zero modes.

Zero-energy Andreev bound states from quantum dots in proximitized Rashba nanowires

Christopher Reeg, Olesia Dmytruk, Denis Chevallier, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 98, 245407 (2018); arXiv:1810.09840. We study an analytical model of a Rashba nanowire that is partially covered by and coupled to a thin superconducting layer, where the uncovered region of the nanowire forms a quantum dot. We find that, even if there is no topological superconducting phase possible, there is a trivial Andreev bound state that becomes pinned exponentially close to zero energy as a function of magnetic field strength when the length of the quantum dot is tuned with respect to its spin-orbit length such that a resonance condition of Fabry-Perot type is satisfied. In this case, we find that the Andreev bound state remains pinned near zero energy for Zeeman energies that exceed the characteristic spacing between Andreev bound state levels but that are smaller than the spin-orbit energy of the quantum dot. Importantly, as the pinning of the Andreev bound state depends only on properties of the quantum dot, we conclude that this behavior is unrelated to topological superconductivity. To support our analytical model, we also perform a numerical simulation of a hybrid system while explicitly incorporating a thin superconducting layer, showing that all qualitative features of our analytical model are also present in the numerical results.

Strong Electron-Electron Interactions of a Tomonaga--Luttinger Liquid Observed in InAs Quantum Wires

Yosuke Sato, Sadashige Matsuo, Chen-Hsuan Hsu, Peter Stano, Kento Ueda, Yuusuke Takeshige, Hiroshi Kamata, Joon Sue Lee, Borzoyeh Shojaei, Kaushini Wickramasinghe, Javad Shabani, Chris Palmstroem, Yasuhiro Tokura, Daniel Loss, and Seigo Tarucha Phys. Rev. B 99, 155304 (2019); arXiv:1810.06259. We report strong electron-electron interactions in quantum wires etched from an InAs quantum well, a material known to have strong spin-orbit interactions. We find that the current through the wires as a function of the bias voltage and temperature follows the universal scaling behavior of a Tomonaga--Luttinger liquid. Using a universal scaling formula, we extract the interaction parameter and find strong electron-electron interactions, increasing as the wires become more depleted. We establish theoretically that spin-orbit interactions cause only minor modifications of the interaction parameter in this regime, indicating that genuinely strong electron-electron interactions are indeed achieved in the device. Our results suggest that etched InAs wires provide a platform with both strong electron-electron and strong spin-orbit interactions.

From fractional boundary charges to quantized Hall conductance

Manisha Thakurathi, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 98, 245404 (2018); arXiv:1809.00538. We study the fractional boundary charges (FBCs) occurring in nanowires in the presence of periodically modulated chemical potentials and connect them to the FBCs occurring in a two-dimensional electron gas in the presence of a perpendicular magnetic field in the integer quantum Hall effect (QHE) regime. First, we show that in nanowires the FBCs take fractional values and change linearly as a function of phase offset of the modulated chemical potential. This linear slope takes quantized values determined by the period of the modulation and depends only on the number of the filled bands. Next, we establish a mapping from the one-dimensional system to the QHE setup, where we again focus on the properties of the FBCs. By considering a cylinder topology with an external flux similar to the Laughlin construction, we find that the slope of the FBCs as function of flux is linear and assumes universal quantized values, also in the presence of arbitrary disorder. We establish that the quantized slopes give rise to the quantization of the Hall conductance. Importantly, the approach via FBCs is valid for arbitrary flux values and disorder. The slope of the FBCs plays the role of a topological invariant for clean and disordered QHE systems. Our predictions for the FBCs can be tested experimentally in nanowires and in Corbino disk geometries in the integer QHE regime.

Tunable Magnonic Thermal Hall Effect in Skyrmion Crystal Phases of Ferrimagnets

Se Kwon Kim, Kouki Nakata, Daniel Loss, and Yaroslav Tserkovnyak Phys. Rev. Lett. 122, 057204 (2019); arXiv:1808.06690. We theoretically study the thermal Hall effect by magnons in skyrmion crystal phases of ferrimagnets in the vicinity of the angular momentum compensation point (CP). To this end, we start by deriving the equation of motion for magnons in the background of an arbitrary equilibrium spin texture, which gives rise to the fictitious electromagnetic field for magnons. As the net spin density varies, the resultant equation of motion interpolates between the relativistic Klein-Gordon equation at CP and the nonrelativistic Schrodinger-like equation away from it. In skyrmion crystal phases, the right- and the left-circularly polarized magnons with respect to the order parameter are shown to form the Landau levels separately within the uniform skyrmion-density approximation. For an experimental proposal, we predict that the magnonic thermal Hall conductivity changes its sign when the ferrimagnet is tuned across CP, providing a way to control heat flux in spin-caloritronic devices on the one hand and a feasible way to detect CP of ferrimagnets on the other hand.

Difference in charge and spin dynamics in a quantum dot-lead coupled system

Tomohiro Otsuka, Takashi Nakajima, Matthieu R. Delbecq, Peter Stano, Shinichi Amaha, Jun Yoneda, Kenta Takeda, Giles Allison, Sen Li, Akito Noiri, Takumi Ito, Daniel Loss, Arne Ludwig, Andreas D. Wieck, and Seigo Tarucha Phys. Rev. B 99, 085402 (2019); arXiv:1808.05303. We analyze time evolution of charge and spin states in a quantum dot coupled to an electric reservoir. Utilizing high-speed single-electron detection, we focus on dynamics induced by the first-order tunneling. We find that there is a difference between the spin and the charge relaxation: The former appears slower than the latter. The difference depends on the Fermi occupation factor and the spin relaxation becomes slower when the energy level of the quantum dot is lowered. We explain this behavior by a theory including the first-order tunneling processes and find a good agreement between the experiment and the theory.

Gate-defined quantum dot in a strong in-plane magnetic field: spin-orbit and g-factor effects

Peter Stano, Chen-Hsuan Hsu, Marcel Serina, Leon C. Camenzind, Dominik M. Zumbuhl, and Daniel Loss Phys. Rev. B 98, 195314 (2018); arXiv:1808.03963. We analyze orbital effects of an in-plane magnetic field on the spin structure of states of a gated quantum dot based in a two-dimensional electron gas. Starting with a k⋅p Hamiltonian, we perturbatively calculate these effects for the conduction band of GaAs, up to the third power of the magnetic field. We quantify several corrections to the g-tensor and reveal their relative importance. We find that for typical parameters, the Rashba spin-orbit term and the isotropic term, H43∝P2B⋅σ, give the largest contributions in magnitude. The in-plane anisotropy of the g-factor is, on the other hand, dominated by the Dresselhaus spin-orbit term. At zero magnetic field, the total correction to the g-factor is typically 5-10% of its bulk value. In strong in-plane magnetic fields, the corrections are modified appreciably.

Benchmarks for approximate CNOTs based on a 17-Qubit Surface Code

Andreas Peter, Daniel Loss, and James R. Wootton arXiv:1808.03927 Scalable and fault-tolerant quantum computation will require error correction. This will demand constant measurement of many-qubit observables, implemented using a vast number of CNOT gates. Indeed, practically all operations performed by a fault-tolerant device will be these CNOTs, or equivalent two-qubit controlled operations. It is therefore important to devise benchmarks for these gates that explicitly quantify their effectiveness at this task. Here we develop such benchmarks, and demonstrate their use by applying them to a range of differently implemented controlled gates and a particular quantum error correcting code. Specifically, we consider spin qubits confined to quantum dots that are coupled either directly or via floating gates to implement the minimal 17-qubit instance of the surface code. Our results show that small differences in the gate fidelity can lead to large differences in the performance of the surface code. This shows that gate fidelity is not, in general, a good predictor of code performance.

Lifetime of Majorana qubits in Rashba nanowires with non-uniform chemical potential

Pavel P. Aseev, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 98, 155414 (2018); arXiv:1807.07997. We study the lifetime of topological qubits based on Majorana bound states hosted in a one-dimensional Rashba nanowire (NW) with proximity-induced superconductivity and non-uniform chemical potential needed for manipulation and read-out. If nearby gates tune the chemical potential locally so that part of the NW is in the trivial phase, Andreev bound states (ABSs) can emerge which are localized at the interface between topological and trivial phases with energies significantly less than the gap. The emergence of such subgap states strongly decreases the Majorana qubit lifetime at finite temperatures due to local perturbations that can excite the system into these ABSs. Using Keldysh formalism, we study such excitations caused by fluctuating charges in capacitively coupled gates and calculate the corresponding Majorana lifetimes due to thermal noise, which are shown to be much shorter than those in NWs with uniform chemical potential.

Renormalization of quantum dot g-factor in superconducting Rashba nanowires

Olesia Dmytruk, Denis Chevallier, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 98, 165403 (2018); arXiv:1806.06842. We study analytically and numerically the renormalization of the g-factor in semiconducting Rashba nanowires (NWs), consisting of a normal and superconducting section. If the potential barrier between the sections is high, a quantum dot (QD) is formed in the normal section. For harmonic (hard-wall) confinement, the effective g-factor of all QD levels is suppressed exponentially (power-law) in the product of the spin-orbit interaction (SOI) wavevector and the QD length. If the barrier between the two sections is removed, the g-factor of the emerging Andreev bound states is suppressed less strongly. In the strong SOI regime and if the chemical potential is tuned to the SOI energy in both sections, the g-factor saturates to a universal constant. Remarkably, the effective g-factor shows a pronounced peak at the SOI energy as function of the chemical potentials. In addition, if the SOI is uniform, the g-factor renormalization as a function of the chemical potential is given by a universal dependence which is independent of the QD size. This prediction provides a powerful tool to determine experimentally whether the SOI in the whole NW is uniform and, moreover, gives direct access to the SOI strengths of the NW via g-factor measurements. In addition, it allows one to find the optimum position of the chemical potential for bringing the NW into the topological phase at large magnetic fields.

Majorana Kramers pairs in higher-order topological insulators

Chen-Hsuan Hsu, Peter Stano, Jelena Klinovaja, and Daniel Loss Phys. Rev. Lett. 121, 196801 (2018); arXiv:1805.12146. We propose a tune-free scheme to realize Kramers pairs of Majorana bound states in recently discovered higher-order topological insulators (HOTIs). We show that, by bringing two hinges of a HOTI into the proximity of an s-wave superconductor, the competition between local and crossed-Andreev pairing leads to formation of Majorana Kramers pairs, when the latter pairing dominates over the former. We demonstrate that such a topological superconductivity is stabilized by moderate electron-electron interactions. The proposed setup avoids the application of a magnetic field or local voltage gates, and requires weaker interactions comparing to nonhelical nanowires.

Proximity effect in a two-dimensional electron gas coupled to a thin superconducting layer

Christopher Reeg, Daniel Loss, and Jelena Klinovaja Beilstein Journal of Nanotechnology 9, 1263 (2018); arXiv:1804.08337. There have recently been several experiments studying induced superconductivity in semiconducting two-dimensional electron gases that are strongly coupled to thin superconducting layers, as well as probing possible topological phases supporting Majorana bound states in such setups. We show that a large band shift is induced in the semiconductor by the superconductor in this geometry, thus making it challenging to realize a topological phase. Additionally, we show that while increasing the thickness of the superconducting layer reduces the magnitude of the band shift, it also leads to a more significant renormalization of the semiconducting material parameters and does not reduce the challenge of tuning into a topological phase.

A fast quantum interface between different spin qubit encodings

A. Noiri, T. Nakajima, J. Yoneda, M. R. Delbecq, P. Stano, T. Otsuka, K. Takeda, S. Amaha, G. Allison, K. Kawasaki, A. Ludwig, A. D. Wieck, D. Loss, and S. Tarucha Nature Communications 9, 5066 (2018); arXiv:1804.04764. Single-spin qubits in semiconductor quantum dots hold promise for universal quantum computation with demonstrations of a high single-qubit gate fidelity above 99.9% and two-qubit gates in conjunction with a long coherence time. However, initialization and readout of a qubit is orders of magnitude slower than control, which is detrimental for implementing measurement-based protocols such as error-correcting codes. In contrast, a singlet-triplet qubit, encoded in a two-spin subspace, has the virtue of fast readout with high fidelity. Here, we present a hybrid system which benefits from the different advantages of these two distinct spin-qubit implementations. A quantum interface between the two codes is realized by electrically tunable inter-qubit exchange coupling. We demonstrate a controlled-phase gate that acts within 5.5 ns, much faster than the measured dephasing time of 211 ns. The presented hybrid architecture will be useful to settle remaining key problems with building scalable spin-based quantum computers.

Spectroscopy of Quantum-Dot Orbitals with In-Plane Magnetic Fields

Leon C. Camenzind, Liuqi Yu, Peter Stano, Jeramy Zimmerman, Arthur C. Gossard, Daniel Loss, and Dominik M. Zumbuhl Phys. Rev. Lett. 122, 207701 (2019); Viewpoint; arXiv:1804.00162. We show that in-plane magnetic-field-assisted spectroscopy allows extraction of the in-plane orientation and full 3D size parameters of the quantum mechanical orbitals of a single electron GaAs lateral quantum dot with subnanometer precision. The method is based on measuring the orbital energies in a magnetic field with various strengths and orientations in the plane of the 2D electron gas. From such data, we deduce the microscopic confinement potential landscape and quantify the degree by which it differs from a harmonic oscillator potential. The spectroscopy is used to validate shape manipulation with gate voltages, agreeing with expectations from the gate layout. Our measurements demonstrate a versatile tool for quantum dots with one dominant axis of strong confinement.

Orbital effects of a strong in-plane magnetic field on a gate-defined quantum dot

Peter Stano, Chen-Hsuan Hsu, Leon Camenzind, Liuqi Yu, Dominik Zumbuhl, and Daniel Loss Phys. Rev. B 99, 085308 (2019); arXiv:1804.00128. We theoretically investigate the orbital effects of an in-plane magnetic field on the spectrum of a quantum dot embedded in a two-dimensional electron gas (2DEG). We derive an effective two-dimensional Hamiltonian where these effects enter in proportion to the flux penetrating the 2DEG. We quantify the latter in detail for harmonic, triangular, and square potential of the heterostructure. We show how the orbital effects allow one to extract a wealth of information, for example, on the heterostructure interface, the quantum dot size and orientation, and the spin-orbit fields. We illustrate the formalism by extracting this information from recent measured data [L.~C.~Camenzind, et al., Nat. Commun. 9, 3454 (2018)].

Conductance of fractional Luttinger liquids at finite temperatures

Pavel P. Aseev, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 98, 045416 (2018); arXiv:1803.07359. We study the electrical conductance in single-mode quantum wires with Rashba spin-orbit interaction subjected to externally applied magnetic fields in the regime in which the ratio of spin-orbit momentum to the Fermi momentum is close to an odd integer, so that a combined effect of multi-electron interaction and applied magnetic field leads to a partial gap in the spectrum. We study how this partial gap manifests itself in the temperature dependence of the fractional conductance of the quantum wire. We use two complementing techniques based on bosonization: refermionization of the model at a particular value of the interaction parameter and a semiclassical approach within a dilute soliton gas approximation of the functional integral. We show how the low-temperature fractional conductance can be affected by the finite length of the wire, by the properties of the contacts, and by a shift of the chemical potential, which takes the system away from the resonance condition. We also predict an internal resistivity caused by a dissipative coupling between gapped and gapless modes.

Skyrmions Driven by Intrinsic Magnons

Christina Psaroudaki and Daniel Loss Phys. Rev. Lett. 120, 237203 (2018); arXiv:1803.04001. We study the dynamics of a skyrmion in a magnetic insulating nanowire in the presence of time-dependent oscillating magnetic field gradients. These ac fields act as a net driving force on the skyrmion via its own intrinsic magnetic excitations. In a microscopic quantum field theory approach we include the unavoidable coupling of the external field to the magnons, which gives rise to time-dependent dissipation for the skyrmion. We demonstrate that the magnetic ac field induces a super-Ohmic to Ohmic crossover behavior for the skyrmion dissipation kernels with time-dependent Ohmic terms. The ac driving of the magnon bath at resonance results in a unidirectional helical propagation of the skyrmion in addition to the otherwise periodic bounded motion.

Rashba Sandwiches with Topological Superconducting Phases

Yanick Volpez, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 97, 195421 (2018); arXiv:1803.00987. We introduce a versatile heterostructure harboring various topological superconducting phases characterized by the presence of helical, chiral, or unidirectional edge states. Changing parameters, such as an effective Zeeman field or chemical potential, one can tune between these three topological phases in the same setup. Our model relies only on conventional non-topological ingredients. The bilayer setup consists of an s-wave superconductor sandwiched between two two-dimensional electron gas layers with strong Rashba spin-orbit interaction. The interplay between two different pairing mechanisms, proximity induced direct and crossed Andreev superconducting pairings, gives rise to multiple topological phases. In particular, helical edge states occur if crossed Andreev superconducting pairing is dominant. In addition, an in-plane Zeeman field leads to a 2D gapless topological phase with unidirectional edge states, which were previously predicted to exist only in non-centrosymmetric superconductors. If the Zeeman field is tilted out of the plane, the system is in a topological phase hosting chiral edge states.

Boundary spin polarization as robust signature of topological phase transition in Majorana nanowires

Marcel Serina, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 98, 035419 (2018); arXiv:1803.00544. We show that the boundary charge and spin can be used as alternative signatures of the topological phase transition in topological models such as semiconducting nanowires with strong Rashba spin-orbit interaction in the presence of a magnetic field and in proximity to an s-wave superconductor. We identify signatures of the topological phase transition that do not rely on the presence of Majorana zero-energy modes and, thus, can serve as independent probes of topological properties. The boundary spin component along the magnetic field, obtained by summing contributions from all states below the Fermi level, has a pronounced peak at the topological phase transition point. Generally, such signatures can be observed at boundaries between topological and trivial sections in nanowires and are stable against disorder.

Metallization of Rashba wire by superconducting layer in the strong-proximity regime

Christopher Reeg, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 97, 165425 (2018); arXiv:1801.06509. Semiconducting quantum wires defined within two-dimensional electron gases and strongly coupled to thin superconducting layers have been extensively explored in recent experiments as promising platforms to host Majorana bound states. We study numerically such a geometry, consisting of a quasi-one-dimensional wire coupled to a disordered three-dimensional superconducting layer. We find that, in the strong-coupling limit of a sizable proximity-induced superconducting gap, all transverse subbands of the wire are significantly shifted in energy relative to the chemical potential of the wire. For the lowest subband, this band shift is comparable in magnitude to the spacing between quantized levels that arise due to the finite thickness of the superconductor (which typically is ca. 500 meV for a 10-nm-thick layer of Aluminum); in higher subbands, the band shift is much larger. Additionally, we show that the width of the system, which is usually much larger than the thickness, and moderate disorder within the superconductor have almost no impact on the induced gap or band shift. We provide a detailed discussion of the ramifications of our results, arguing that a huge band shift and significant renormalization of semiconducting material parameters in the strong-coupling limit make it challenging to realize a topological phase in such a setup, as the strong coupling to the superconductor essentially metallizes the semiconductor. This metallization of the semiconductor can be tested experimentally through the measurement of the band shift.

Effects of nuclear spins on the transport properties of the edge of two-dimensional topological insulators

Chen-Hsuan Hsu, Peter Stano, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 97, 125432 (2018); arXiv:1712.09040. The electrons in the edge channels of two-dimensional topological insulators can be described as a helical Tomonaga-Luttinger liquid. They couple to nuclear spins embedded in the host materials through the hyperfine interaction, and are therefore subject to elastic spin-flip backscattering on the nuclear spins. We investigate the nuclear-spin-induced edge resistance due to such backscattering by performing a renormalization-group analysis. Remarkably, the effect of this backscattering mechanism is stronger in a helical edge than in nonhelical channels, which are believed to be present in the trivial regime of InAs/GaSb quantum wells. In a system with sufficiently long edges, the disordered nuclear spins lead to an edge resistance which grows exponentially upon lowering the temperature. On the other hand, electrons from the edge states mediate an anisotropic Ruderman-Kittel-Kasuya-Yosida nuclear spin-spin interaction, which induces a spiral nuclear spin order below the transition temperature. We discuss the features of the spiral order, as well as its experimental signatures. In the ordered phase, we identify two backscattering mechanisms, due to charge impurities and magnons. The backscattering on charge impurities is allowed by the internally generated magnetic field, and leads to an Anderson-type localization of the edge states. The magnon-mediated backscattering results in a power-law resistance, which is suppressed at zero temperature. Overall, we find that in a sufficiently long edge the nuclear spins, whether ordered or not, suppress the edge conductance to zero as the temperature approaches zero.

Direct Rashba spin-orbit interaction in Si and Ge nanowires with different growth directions

Christoph Kloeffel, Marko J. Rancic, and Daniel Loss Phys. Rev. B 97, 235422 (2018); Editor's suggestion; arXiv:1712.03476. We study theoretically the low-energy hole states in Si, Ge, and Ge/Si core/shell nanowires (NWs). The NW core in our model has a rectangular cross section, the results for a square cross section are presented in detail. In the case of Ge and Ge/Si core/shell NWs, we obtain very good agreement with previous theoretical results for cylindrically symmetric NWs. In particular, the NWs allow for an unusually strong and electrically controllable spin-orbit interaction (SOI) of Rashba type. We find that the dominant contribution to the SOI is the "direct Rashba spin-orbit interaction" (DRSOI), which is an important mechanism for systems with heavy-hole-light-hole mixing. Our results for Si NWs depend significantly on the orientation of the crystallographic axes. The numerically observed dependence on the growth direction is consistent with analytical results from a simple model, and we identify a setup where the DRSOI enables spin-orbit energies of the order of millielectronvolts in Si NWs. Furthermore, we analyze the dependence of the SOI on the electric field and the cross section of the Ge or Si core. A helical gap in the spectrum can be opened with a magnetic field. We obtain the largest g factors with magnetic fields applied perpendicularly to the NWs.

Majorana Kramers pairs in Rashba double nanowires with interactions and disorder

Manisha Thakurathi, Pascal Simon, Ipsita Mandal, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 97, 045415 (2018); arXiv:1711.04682. We analyze the effects of electron-electron interactions and disorder on a Rashba double-nanowire setup coupled to an s-wave superconductor, which has been recently proposed as a versatile platform to generate Kramers pairs of Majorana bound states in the absence of magnetic fields. We identify the regime of parameters for which these Kramers pairs are stable against interaction and disorder effects. We use bosonization, perturbative renormalization group, and replica techniques to derive the flow equations for various parameters of the model and evaluate the corresponding phase diagram with topological and disorder-dominated phases. We confirm aforementioned results by considering a more microscopic approach which starts from the tunneling Hamiltonian between the three-dimensional s-wave superconductor and the nanowires. We find again that the interaction drives the system into the topological phase and, as the strength of the source term coming from the tunneling Hamiltonian increases, strong electron-electron interactions are required to reach the topological phase.

Hyperfine-phonon spin relaxation in a single-electron GaAs quantum dot

Leon C. Camenzind, Liuqi Yu, Peter Stano, Jeramy Zimmerman, Arthur C. Gossard, Daniel Loss, and Dominik M. Zumbuhl Nature Communications 9, 3454 (2018); arXiv:1711.01474. Understanding and control of the spin relaxation time T1 is among the key challenges for spin based qubits. A larger T1 is generally favored, setting the fundamental upper limit to the qubit coherence and spin readout fidelity. In GaAs quantum dots at low temperatures and high in-plane magnetic fields B, the spin relaxation relies on phonon emission and spin-orbit coupling. The characteristic dependence T1 ~1/B^5and pronounced B-field anisotropy were already confirmed experimentally. However, it has also been predicted 15 years ago that at low enough fields, the spin-orbit interaction is replaced by the coupling to the nuclear spins, where the relaxation becomes isotropic, and the scaling changes to T1~1/B^3. We establish these predictions experimentally, by measuring T1 over an unprecedented range of magnetic fields -- made possible by lower temperature -- and report a maximum T1=57 s at the lowest fields, setting a new record for the electron spin lifetime in a nanostructure.

A repetition code of 15 qubits

James R. Wootton and Daniel Loss Phys. Rev. A 97, 052313 (2018); arXiv:1709.00990. The repetition code is an important primitive for the techniques of quantum error correction. Here we implement repetition codes of at most 15 qubits on the 16 qubit \emph{ibmqx3} device. Each experiment is run for a single round of syndrome measurements, achieved using the standard quantum technique of using ancilla qubits and controlled operations. The size of the final syndrome is small enough to allow for lookup table decoding using experimentally obtained data. The results show strong evidence that the logical error rate decays exponentially with code distance, as is expected and required for the development of fault-tolerant quantum computers. The results also give insight into the nature of noise in the device.

Topological Phase Detection in Rashba Nanowires with a Quantum Dot

Denis Chevallier, Pawel Szumniak, Silas Hoffman, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 97, 045404 (2018); arXiv:1710.05576. We study theoretically the detection of the topological phase transition occurring in Rashba nanowires with proximity-induced superconductivity using a quantum dot. The bulk states lowest in energy of such a nanowire have a spin polarization parallel or antiparallel to the applied magnetic field in the topological or trivial phase, respectively. We show that this property can be probed by the quantum dot created at the end of the nanowire by external gates. By tuning one of the two spin-split levels of the quantum dot to be in resonance with nanowire bulk states, one can detect the spin polarization of the lowest band via transport measurement. This allows one to determine the topological phase of the Rashba nanowire independently of the presence of Majorana bound states.

DIII Topological Superconductivity with Emergent Time-Reversal Symmetry

Christopher Reeg, Constantin Schrade, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 96, 161407(R) (2017); arXiv:1708.06755. We find a new class of topological superconductors which possess an emergent time-reversal symmetry that is present only after projecting to an effective low-dimensional model. We show that a topological phase in symmetry class DIII can be realized in a noninteracting system coupled to an s-wave superconductor only if the physical time-reversal symmetry of the system is broken, and we provide three general criteria that must be satisfied in order to have such a phase. We also provide an explicit model which realizes the class DIII topological superconductor in 1D. We show that, just as in time-reversal invariant topological superconductors, the topological phase is characterized by a Kramers pair of Majorana fermions that are protected by the emergent time-reversal symmetry.

Finite-size effects in a nanowire strongly coupled to a thin superconducting shell

Christopher Reeg, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 96, 125426 (2017); arXiv:1707.08417. We study the proximity effect in a one-dimensional nanowire strongly coupled to a finite superconductor with a characteristic size which is much shorter than its coherence length. Such geometries have become increasingly relevant in recent years in the experimental search for Majorana fermions with the development of thin epitaxial Al shells which form a very strong contact with either InAs or InSb nanowires. So far, however, no theoretical treatment of the proximity effect in these systems has accounted for the finite size of the superconducting film. We show that the finite-size effects become very detrimental when the level spacing of the superconductor greatly exceeds its energy gap. Without any fine-tuning of the size of the superconductor (on the scale of the Fermi wavelength), the tunneling energy scale must be larger than the level spacing in order to reach the hard gap regime which is seen ubiquitously in the experiments. However, in this regime, the large tunneling energy scale induces a large shift in the effective chemical potential of the nanowire and pushes the topological phase transition to magnetic field strengths which exceed the critical field of Al.

Magnonic topological insulators in antiferromagnets

Kouki Nakata, Se Kwon Kim (UCLA), Jelena Klinovaja, and Daniel Loss Phys. Rev. B 96, 224414 (2017); arXiv:1707.07427. Extending the notion of symmetry protected topological phases to insulating antiferromagnets (AFs) described in terms of opposite magnetic dipole moments associated with the magnetic Né el order, we establish a bosonic counterpart of topological insulators in semiconductors. Making use of the Aharonov-Casher effect, induced by electric field gradients, we propose a magnonic analog of the quantum spin Hall effect (magnonic QSHE) for edge states that carry helical magnons. We show that such up and down magnons form the same Landau levels and perform cyclotron motion with the same frequency but propagate in opposite direction. The insulating AF becomes characterized by a topological ℤ2 number consisting of the Chern integer associated with each helical magnon edge state. Focusing on the topological Hall phase for magnons, we study bulk magnon effects such as magnonic spin, thermal, Nernst, and Ettinghausen effects, as well as the thermomagnetic properties of helical magnon transport both in topologically trivial and nontrivial bulk AFs and establish the magnonic Wiedemann-Franz law. We show that our predictions are within experimental reach with current device and measurement techniques.

Three-Dimensional Fractional Topological Insulators in Coupled Rashba Layers

Yanick Volpez, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 96, 085422 (2017); arXiv:1706.09863. We propose a model of three-dimensional topological insulators consisting of weakly coupled electron- and hole-gas layers with Rashba spin-orbit interaction stacked along a given axis. We show that in the presence of strong electron-electron interactions the system realizes a fractional strong topological insulator, where the rotational symmetry and condensation energy arguments still allow us to treat the problem as quasi-one-dimensional with bosonization techniques. We also show that if Rashba and Dresselhaus spin-orbit interaction terms are equally strong, by doping the system with magnetic impurities, one can bring it into the Weyl semimetal phase.

Low-field Topological Threshold in Majorana Double Nanowires

Constantin Schrade, Manisha Thakurathi, Christopher Reeg, Silas Hoffman, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 96, 035306 (2017); arXiv:1705.09364. A hard proximity-induced superconducting gap has recently been observed in semiconductor nanowire systems at low magnetic fields. However, in the topological regime at high magnetic fields a soft gap re-emerges and represents a fundamental obstacle to topologically protected quantum information processing with Majorana bound states. Here we show that this obstacle can be overcome in a setup of double Rashba nanowires which are coupled to an $s$-wave superconductor and subjected to an external magnetic field along the wires. Specifically, we demonstrate that the required field strength for the topological threshold can be significantly reduced by the destructive interference of direct and crossed-Andreev pairing in this setup; precisely down to the regime in which current experimental technology allows for a hard superconducting gap. We also show that the resulting Majorana bound states exhibit sufficiently short localization lengths which makes them ideal candidates for future braiding experiments.

Spin-dependent coupling between quantum dots and topological quantum wires

Silas Hoffman, Denis Chevallier, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 96, 045440 (2017); arXiv:1705.03002. Considering Rashba quantum wires with a proximity-induced superconducting gap as physical realizations of Majorana fermions and quantum dots, we calculate the overlap of the Majorana wave functions with the local wave functions on the dot. We determine the spin-dependent tunneling amplitudes between these two localized states and show that we can tune into a fully spin polarized tunneling regime by changing the distance between dot and Majorana fermion. Upon directly applying this to the tunneling model Hamiltonian, we calculate the effective magnetic field on the quantum dot flanked by two Majorana fermions. The direction of the induced magnetic field on the dot depends on the occupation of the nonlocal fermion formed from the two Majorana end states which can be used as a readout for such a Majorana qubit.

Nuclear spin-induced localization of the edge states in two-dimensional topological insulators

Chen-Hsuan Hsu (RIKEN), Peter Stano (RIKEN), Jelena Klinovaja, and Daniel Loss Phys. Rev. B 96, 081405(R) (2017); arXiv:1703.03421. We investigate the influence of nuclear spins on the resistance of helical edge states of two-dimensional topological insulators (2DTIs). Via the hyperfine interaction, nuclear spins allow electron backscattering, otherwise forbidden by time-reversal symmetry. We identify two backscattering mechanisms, depending on whether the nuclear spins are ordered or not. Their temperature dependence is distinct but both give resistance, which increases with the edge length, decreasing temperature, and increasing strength of the electron-electron interaction. Overall, we find that the nuclear spins will typically shut down the conductance of the 2DTI edges at zero temperature.

Spin and Charge Signatures of Topological Superconductivity in Rashba Nanowires

Pawel Szumniak, Denis Chevallier, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 96, 041401(R) (2017); arXiv:1703.00265. We consider a Rashba nanowire with proximity gap which can be brought into the topological phase by tuning external magnetic field or chemical potential. We study spin and charge of the bulk quasiparticle states when passing through the topological transition for open and closed systems. We show, analytically and numerically, that the spin of bulk states around the topological gap reverses its sign when crossing the transition due to band inversion, independent of the presence of Majorana fermions in the system. This spin reversal can be considered as a bulk signature of topological superconductivity that can be accessed experimentally. We find a similar behaviour for the charge of the bulk quasiparticle states, also exhibiting a sign reversal at the transition. We show that these signatures are robust against random static disorder.

Destructive interference of direct and crossed Andreev pairing in a system of two nanowires coupled via an s-wave superconductor

Christopher R. Reeg, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 96, 081301(R) (2017); arXiv:1701.07107. We consider a system of two one-dimensional nanowires coupled via an s-wave superconducting strip, a geometry that is capable of supporting Kramers pairs of Majorana fermions. By performing an exact analytical diagonalization of a tunneling Hamiltonian describing the proximity effect (via a Bogoliubov transformation), we show that the excitation gap of the system varies periodically on the scale of the Fermi wavelength in the limit where the interwire separation is shorter than the superconducting coherence length. Comparing with the excitation gaps in similar geometries containing only direct pairing, where one wire is decoupled from the superconductor, or only crossed Andreev pairing, where each nanowire is considered as a spin-polarized edge of a quantum Hall state, we find that the gap is always reduced, by orders of magnitude in certain cases, when both types of pairing are present. Our analytical results are further supported by numerical calculations on a tight-binding lattice. Finally, we show that treating the proximity effect by integrating out the superconductor cannot reproduce the results of our exact diagonalization.

Robust Single-Shot Spin Measurement with 99.5% Fidelity in a Quantum Dot Array

Takashi Nakajima, Matthieu R. Delbecq, Tomohiro Otsuka, Peter Stano, Shinichi Amaha, Jun Yoneda, Akito Noiri, Kento Kawasaki, Kenta Takeda, Giles Allison, Arne Ludwig, Andreas D. Wieck, Daniel Loss, Seigo Tarucha Phys. Rev. Lett. 119, 017701 (2017); arXiv:1701.03622. We demonstrate a new method for projective single-shot measurement of two electron spin states (singlet versus triplet) in an array of gate-defined lateral quantum dots in GaAs. The measurement has very high fidelity and is robust with respect to electric and magnetic fluctuations in the environment. It exploits a long-lived metastable charge state, which increases both the contrast and the duration of the charge signal distinguishing the two measurement outcomes. This method allows us to evaluate the charge measurement error and the spin-to-charge conversion error separately. We specify conditions under which this method can be used, and project its general applicability to scalable quantum dot arrays in GaAs or silicon.

Superconducting grid-bus surface code architecture for hole-spin qubits

Simon E. Nigg, Andreas Fuhrer (IBM Zurich), and Daniel Loss Phys. Rev. Lett. 118, 147701 (2017); arXiv:1612.07292. We present a scalable hybrid architecture for the 2D surface code combining superconducting resonators and hole-spin qubits in nanowires with tunable direct Rashba spin-orbit coupling. The back-bone of this architecture is a square lattice of capacitively coupled coplanar waveguide resonators each of which hosts a nanowire hole-spin qubit. Both the frequency of the qubits and their coupling to the microwave field are tunable by a static electric field applied via the resonator center pin. In the dispersive regime, an entangling two-qubit gate can be realized via a third order process, whereby a virtual photon in one resonator is created by a first qubit, coherently transferred to a neighboring resonator, and absorbed by a second qubit in that resonator. Numerical simulations with state-of-the-art coherence times yield gate fidelities approaching the 99% fault tolerance threshold.

Quantum dynamics of skyrmions in chiral magnets

Christina Psaroudaki, Silas Hoffman, Jelena Klinovaja, and Daniel Loss Phys. Rev. X 7, 041045 (2017); arXiv:1612.01885. We study the quantum propagation of a Skyrmion in chiral magnetic insulators by generalizing the micromagnetic equations of motion to a finite-temperature path integral formalism, using field theoretic tools. Promoting the center of the Skyrmion to a dynamic quantity, the fluctuations around the Skyrmionic configuration give rise to a time-dependent damping of the Skyrmion motion. From the frequency dependence of the damping kernel, we are able to identify the Skyrmion mass, thus providing a microscopic description of the kinematic properties of Skyrmions. When defects are present or a magnetic trap is applied, the Skyrmion mass acquires a finite value proportional to the effective spin, even at vanishingly small temperature. We demonstrate that a Skyrmion in a confined geometry provided by a magnetic trap behaves as a massive particle owing to its quasi-one-dimensional confinement. An additional quantum mass term is predicted, independent of the effective spin, with an explicit temperature dependence which remains finite even at zero temperature.

Finite-temperature conductance of strongly interacting quantum wire with a nuclear spin order

Pavel Aseev, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 95, 125440 (2017); arXiv:1611.10238. We study the temperature dependence of the electrical conductance of a clean strongly interacting quantum wire in the presence of a helical nuclear spin order. The nuclear spin helix opens a temperature-dependent partial gap in the electron spectrum. Using a bosonization framework we describe the gapped electron modes by sine-Gordon-like kinks. We predict an internal resistivity caused by an Ohmic-like friction these kinks experience via interacting with gapless excitations. As a result, the conductance rises from G=e2/h at temperatures below the critical temperature when nuclear spins are fully polarized to G=2e2/h at higher temperatures when the order is destroyed, featuring a relatively wide plateau in the intermediate regime. The theoretical results are compared with the experimental data for GaAs quantum wires obtained recently by Scheller et al. [Phys. Rev. Lett. 112, 066801 (2014)].

Magnonic quantum Hall effect and Wiedemann-Franz law

Kouki Nakata, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 95, 125429 (2017); arXiv:1611.09752. We present a quantum Hall effect of magnons in two-dimensional clean insulating magnets at finite temperature. Through the Aharonov-Casher effect, a magnon moving in an electric field acquires a geometric phase and forms Landau levels in an electric field gradient of sawtooth form. At low temperatures, the lowest energy band being almost flat carries a Chern number associated with a Berry curvature. Appropriately defining the thermal conductance for bosons, we find that the magnon Hall conductances get quantized and show a universal thermomagnetic behavior, i.e., are independent of materials, and obey a Wiedemann-Franz law for magnon transport. We consider magnons with quadratic and linear (Dirac-like) dispersions. Finally, we show that our predictions are within experimental reach for ferromagnets and skyrmion lattices with current device and measurement techniques.

Spin Currents and Magnon Dynamics in Insulating Magnets

Kouki Nakata, Pascal Simon (Orsay), and Daniel Loss J. Phys. D: Appl. Phys. 50 114004 (2017); arXiv:1610.08901. Nambu-Goldstone theorem provides gapless modes to both relativistic and nonrelativistic systems. The Nambu-Goldstone bosons in insulating magnets are called magnons or spin-waves and play a key role in magnetization transport. We review here our past works on magnetization transport in insulating magnets and also add new insights, with a particular focus on magnon transport. We summarize in detail the magnon counterparts of electron transport, such as the Wiedemann-Franz law, the Onsager reciprocal relation between the Seebeck and Peltier coefficients, the Hall effects, the superconducting state, the Josephson effects, and the persistent quantized current in a ring to list a few. Focusing on the electromagnetism of moving magnons, i.e., magnetic dipoles, we theoretically propose a way to directly measure magnon currents. As a consequence of the Mermin-Wagner-Hohenberg theorem, spin transport is drastically altered in one-dimensional antiferromagnetic (AF) spin-1/2 chains; where the N\'eel order is destroyed by quantum fluctuations and a quasiparticle magnon-like picture breaks down. Instead, the the low-energy collective excitations of the AF spin chain are described by a Tomonaga-Luttinger liquid (TLL) which provides to the spin transport properties in such antiferromagnets some universal features at low enough temperature. Finally, we enumerate open issues and provide a platform to discuss the future directions of magnonics.

Floquet Majorana and Para-Fermions in Driven Rashba Nanowires

Manisha Thakurathi, Daniel Loss, and Jelena Klinovaja Phys. Rev. B 95, 155407 (2017); arXiv:1608.08143. We study a periodically driven nanowire with Rashba-like conduction and valence bands in the presence of a magnetic field. We identify topological regimes in which the system hosts zero-energy Majorana fermions. We further investigate the effect of strong electron-electron interactions that give rise to parafermion zero energy modes hosted at the nanowire ends. The first setup we consider allows for topological phases by applying only static magnetic fields without the need of superconductivity. The second setup involves both superconductivity and time-dependent magnetic fields and allows one to generate topological phases without fine-tuning of the chemical potential. Promising candidate materials are graphene nanoribbons due to their intrinsic particle-hole symmetry.

Higher-order spin and charge dynamics in a quantum dot-lead hybrid system

Tomohiro Otsuka, Takashi Nakajima, Matthieu R. Delbecq, Shinichi Amaha, Jun Yoneda, Kenta Takeda, Giles Allison, Peter Stano, Akito Noiri, Takumi Ito, Daniel Loss, Arne Ludwig, Andreas D. Wieck, and Seigo Tarucha Scientific Reports 7, 12201 (2017); arXiv:1608.07646. Understanding the dynamics of open quantum systems is important and challenging in basic physics and applications for quantum devices and quantum computing. Semiconductor quantum dots offer a good platform to explore the physics of open quantum systems because we can tune parameters including the coupling to the environment or leads. Here, we apply the fast single-shot measurement techniques from spin qubit experiments to explore the spin and charge dynamics due to tunnel coupling to a lead in a quantum dot-lead hybrid system. We experimentally observe both spin and charge time evolution via first- and second-order tunneling processes, and reveal the dynamics of the spin-flip through the intermediate state. These results enable and stimulate the exploration of spin dynamics in dot-lead hybrid systems, and may offer useful resources for spin manipulation and simulation of open quantum systems.

Proposal for a minimal surface code experiment

James R. Wootton, Andreas Peter, Janos R. Winkler, and Daniel Loss Phys. Rev. A 96, 032338 (2017); arXiv:1608.05053. Current quantum technology is approaching the system sizes and fidelities required for quantum error correction. It is therefore important to determine exactly what is needed for proof-of-principle experiments, which will be a major step towards fault-tolerant quantum computation. Here we propose a surface code based experiment that is the smallest, both in terms of code size and circuit depth, that would allow errors to be detected and corrected for both the X and Z bases of a qubit. This requires 17 physical qubits initially prepared in a product state, on which 16 two-qubit entangling gates are applied before a final measurement of all qubits. A platform agnostic error model is applied to give some idea of the noise levels required for success. It is found that a true demonstration of quantum error correction will require fidelities for the preparation and measurement of qubits and the entangling gates to be above 99%.

Detecting Topological Superconductivity with phi_0 Josephson Junctions

Constantin Schrade, Silas Hoffman, and Daniel Loss Phys. Rev. B 95, 195421 (2017); arXiv:1607.07794. The interplay of superconductivity, magnetic fields, and spin-orbit interaction lies at the heart of topological superconductivity. Remarkably, the recent experimental discovery of phi_0 Josephson junctions by Szombati et al., Nat. Phys. 12, 568 (2016), characterized by a finite phase offset in the supercurrent, require the same ingredients as topological superconductors, which suggests a profound connection between these two distinct phenomena. Here, we theoretically show that a quantum dot phi_0 Josephson junction can serve as a new qualitative indicator for topological superconductivity: Microscopically, we find that the phase shift in a junction of s-wave superconductors is due to the spin-orbit induced mixing of singly occupied states on the quantum dot, while for a topological superconductor junction it is due to singlet-triplet mixing. Because of this important difference, when the spin-orbit vector of the quantum dot and the external Zeeman field are orthogonal, the s-wave superconductors form a pi Josephson junction while the topological superconductors have a finite offset phi_0 by which topological superconductivity can be distinguished from conventional superconductivity. Our prediction can be immediately tested in nanowire systems currently used for Majorana fermion experiments and thus offers a new and realistic approach for detecting topological bound states.

Heavy hole states in Germanium hut wires

Hannes Watzinger, Christoph Kloeffel, Lada Vukusic, Marta D. Rossell, Violetta Sessi, Josip Kukucka, Raimund Kirchschlager, Elisabeth Lausecker, Alisha Truhlar, Martin Glaser, Armando Rastelli, Andreas Fuhrer, Daniel Loss, and Georgios Katsaros Nano Lett. 16, 6879 (2016); arXiv:1607.02977. Hole spins have gained considerable interest in the past few years due to their potential for fast electrically controlled qubits. Here, we study holes confined in Ge hut wires, a so far unexplored type of nanostructure. Low temperature magnetotransport measurements reveal a large anisotropy between the in-plane and out-of-plane g-factors of up to 18. Numerical simulations verify that this large anisotropy originates from a confined wave function which is of heavy hole character. A light hole admixture of less than 1% is estimated for the states of lowest energy, leading to a surprisingly large reduction of the out-of-plane g-factors. However, this tiny light hole contribution does not influence the spin lifetimes, which are expected to be very long, even in non isotopically purified samples.

Role of the electron spin in determining the coherence of the nuclear spins in a quantum dot

Gunter Wuest, Mathieu Munsch, Franziska Maier, Andreas V. Kuhlmann, Arne Ludwig, Andreas D. Wieck, Daniel Loss, Martino Poggio, and Richard J. Warburton Nature Nanotechnology 11, 885 (2016) A huge effort is underway to develop semiconductor nanostructures as low-noise qubits. A key source of dephasing for an electron spin qubit in GaAs and in naturally occurring Si is the nuclear spin bath. The electron spin is coupled to each nuclear spin by the hyperfine interaction. The same interaction also couples two remote nuclear spins via a common coupling to the delocalized electron. It has been suggested that this interaction limits both electron and nuclear spin coherence, but experimental proof is lacking. We show that the nuclear spin decoherence time decreases by two orders of magnitude on occupying an empty quantum dot with a single electron, recovering to its original value for two electrons. In the case of one electron, agreement with a model calculation verifies the hypothesis of an electron-mediated nuclear spin–nuclear spin coupling. The results establish a framework to understand the main features of this complex interaction in semiconductor nanostructures.

Fractional boundary charges in quantum dot arrays with density modulation

Jin-Hong Park, Guang Yang (Riken), Jelena Klinovaja, Peter Stano (Riken), and Daniel Loss. Phys. Rev. B 94, 075416 (2016); arXiv:1604.05437. We show that fractional charges can be realized at the boundaries of a linear array of tunnel coupled quantum dots in the presence of a periodically modulated onsite potential. While the charge fractionalization mechanism is similar to the one in polyacetylene, here the values of fractional charges can be tuned to arbitrary values by varying the phase of the onsite potential or the total number of dots in the array. We also find that the fractional boundary charges, unlike the in-gap bound states, are stable against static random disorder. We discuss the minimum array size where fractional boundary charges can be observed.

Universal Quantum Computation with Hybrid Spin-Majorana Qubits

Silas Hoffman, Constantin Schrade, Jelena Klinovaja, and Daniel Loss. Phys. Rev. B 94, 045316 (2016); arXiv:1602.06923. We theoretically propose a set of universal quantum gates acting on a hybrid qubit formed by coupling a quantum dot spin qubit and Majorana fermion qubit. First, we consider a quantum dot tunnel-coupled to two topological superconductors. The effective spin-Majorana exchange facilitates a hybrid CNOT gate for which either qubit can be the control or target. The second setup is a modular scalable network of topological superconductors and quantum dots. As a result of the exchange interaction between adjacent spin qubits, a CNOT gate is implemented that acts on neighboring Majorana qubits, and eliminates the necessity of inter-qubit braiding. In both setups the spin-Majorana exchange interaction allows for a phase gate, acting on either the spin or the Majorana qubit, and for a SWAP or hybrid SWAP gate which is sufficient for universal quantum computation without projective measurements.

Majorana bound states in magnetic skyrmions

Guang Yang (Riken), Peter Stano (Riken), Jelena Klinovaja, and Daniel Loss. Phys. Rev. B 93, 224505 (2016); arXiv:1602.00968. Magnetic skyrmions are highly mobile nanoscale topological spin textures. We show, both analytically and numerically, that a magnetic skyrmion of an even azimuthal winding number placed in proximity to an s-wave superconductor hosts a zero-energy Majorana bound state in its core, when the exchange coupling between the itinerant electrons and the skyrmion is strong. This Majorana bound state is stabilized by the presence of a spin-orbit interaction. We propose the use of a superconducting trijunction to realize non-Abelian statistics of such Majorana bound states.

Optimal geometry of lateral GaAs and Si/SiGe quantum dots for electrical control of spin qubits

Ognjen Malkoc (Riken/Lund), Peter Stano (Riken), and Daniel Loss. Phys. Rev. B 93, 235413 (2016); arXiv:1601.05881. We investigate the effects of the orientation of the magnetic field and the orientation of a quantum dot, with respect to crystallographic coordinates, on the quality of an electrically controlled qubit realized in a gated semiconductor quantum dot. We find that, due to the anisotropy of the spin-orbit interactions, varying the two orientations it is possible to tune the qubit in the sense of optimizing the ratio of its couplings to phonons and to a control electric field. We find conditions under which such optimal setup can be reached by solely reorienting the magnetic field, and when a specific positioning of the dot is required. We also find that the knowledge of the relative sign of the spin-orbit interactions strengths allows to choose a robust optimal dot geometry, with the dot main axis along [110], or [1-10], where the qubit can be always optimized by reorienting the magnetic field.

Topological Phases of Inhomogeneous Superconductivity

Silas Hoffman, Jelena Klinovaja, and Daniel Loss Phys. Rev. B 93, 165418 (2016); arXiv:1601.04270. We theoretically consider the effect of a spatially periodic modulation of the superconducting order parameter on the formation of Majorana fermions induced by a one-dimensional system with magnetic impurities brought into close proximity to an s-wave superconductor. When the magnetic exchange energy is larger than the inter-impurity electron hopping we model the effective system as a chain of coupled Shiba states. While in the opposite regime, the effective system is accurately described by a quantum wire model. Upon including a spatially modulated superconducting pairing, we find, for sufficiently large magnetic exchange energy, the system is able to support a single pair of Majorana fermions with one Majorana fermion on the left end of the system and one on the right end. When the modulation of superconductivity is large compared to the magnetic exchange energy, the Shiba chain returns to a trivially gapped regime while the quantum wire enters a new topological phase capable of supporting two pairs of Majorana fermions.

Long-Range Interaction between Charge and Spin Qubits in Quantum Dots

Marcel Serina, Christoph Kloeffel, and Daniel Loss Phys. Rev. B 95, 245422 (2017); arXiv:1601.03564. We analyze and give estimates for the long-distance coupling via floating metallic gates between different types of spin qubits in quantum dots made of different commonly used materials. In particular, we consider the hybrid, the singlet-triplet, and the spin-1/2 qubits, and the pairwise coupling between each type of these qubits with another hybrid qubit in GaAs, InAs, Si, and Si0.9Ge0.1. We show that hybrid qubits can be capacitively coupled strongly enough to implement two-qubit gates, as long as the dimensions of the dots and their distance from the metallic gates are small enough. Thus, hybrid qubits are good candidates for scalable implementations of quantum computing in semiconducting nanostructures.

Chiral and Non-Chiral Edge States in Quantum Hall Systems with Charge Density Modulation

Pawel Szumniak, Jelena Klinovaja, and Daniel Loss. Phys. Rev. B 93, 245308 (2016); arXiv:1512.05971. We consider a system of weakly coupled wires with quantum Hall effect (QHE) and in the presence of a spatially periodic modulation of the chemical potential along the wire, equivalent to a charge density wave (CDW). We investigate the competition between the two effects which both open a gap. We show that by changing the ratio between the amplitudes of the CDW modulation and the tunneling between wires, one can switch between non-topological CDW-dominated phase to topological QHE-dominated phase. Both phases host edge states of chiral and non-chiral nature robust to on-site disorder. However, only in the topological phase, the edge states are immune to disorder in the phase shifts of the CDWs. We provide analytical solutions for filling factor nu=1 and study numerically effects of disorder as well as present numerical results for higher filling factors.

Phonon-assisted relaxation and decoherence of singlet-triplet qubits in Si/SiGe quantum dots

Viktoriia Kornich, Christoph Kloeffel, and Daniel Loss. Quantum 2, 70 (2018); arXiv:1511.07369. We study theoretically the phonon-induced relaxation and decoherence of spin states of two electrons in a lateral double quantum dot in a SiGe/Si/SiGe heterostructure. We consider two types of singlet-triplet spin qubits and calculate their relaxation and decoherence times, in particular as a function of level hybridization, temperature, magnetic field, spin orbit interaction, and detuning between the quantum dots, using Bloch-Redfield theory. We show that the magnetic field gradient, which is usually applied to operate the spin qubit, may suppress the relaxation time by more than an order of magnitude. Using this insight, we identify an optimal regime where the magnetic field gradient does not affect the relaxation time significantly, and we propose regimes of longest decay times. We take into account the effects of one-phonon and two-phonon processes and suggest how our theory can be tested experimentally. The spin lifetimes we find here for Si-based quantum dots are significantly longer than the ones reported for their GaAs counterparts.

Quantum Computing with Parafermions

Adrian Hutter and Daniel Loss. Phys. Rev. B 93, 125105 (2016); arXiv:1511.02704. ℤd Parafermions are exotic non-Abelian quasiparticles generalizing Majorana fermions, which correspond to the case d=2. In contrast to Majorana fermions, braiding of parafermions with d>2 allows to perform an entangling gate. This has spurred interest in parafermions and a variety of condensed matter systems have been proposed as potential hosts for them. In this work, we study the computational power of braiding parafermions more systematically. We make no assumptions on the underlying physical model but derive all our results from the algebraical relations that define parafermions. We find a familiy of 2d representations of the braid group that are compatible with these relations. The braiding operators derived this way reproduce those derived previously from physical grounds as special cases. We show that if a d-level qudit is encoded in the fusion space of four parafermions, braiding of these four parafermions allows to generate the entire single-qudit Clifford group (up to phases), for any d. If d is odd, then we show that in fact the entire many-qudit Clifford group can be generated.

From Coupled Rashba Electron and Hole Gas Layers to 3D Topological Insulators

Luka Trifunovic, Daniel Loss, and Jelena Klinovaja. Phys. Rev. B 93, 205406 (2016); arXiv:1511.01742. We introduce a system of stacked two-dimensional electron and hole gas layers with Rashba spin orbit interaction and show that the tunnel coupling between the layers induces a strong three- dimensional (3D) topological insulator phase. At each of the two-dimensional bulk boundaries we find the spectrum consisting of a single anistropic Dirac cone, which we show by analytical and numerical calculations. Our setup has a unit-cell consisting of four tunnel coupled Rashba layers and presents a synthetic strong 3D topological insulator and is distinguished by its rather high experimental feasibility.

Josephson Junction through a 3D Topological Insulator with Helical Magnetization

Alexander Zyuzin, Mohammad Alidoust, and Daniel Loss. Phys. Rev. B 93, 214502 (2016); arXiv:1511.01486. We study supercurrent and proximity vortices in a Josephson junction made of disordered surface states of a three-dimensional topological insulator with a proximity induced in-plane helical magnetization. In a regime where the rotation period of helical magnetization is larger than the junction width, we find supercurrent 0-{\pi} crossovers as a function of junction thickness, magnetization strength, and parameters inherent to the helical modulation and surface states. The supercurrent reversals are associated with proximity induced vortices, nucleated along the junction width, where the number of vortices and their locations can be manipulated by means of the superconducting phase difference and the parameters mentioned above.

Topological Floquet Phases in Driven Coupled Rashba Nanowires

Jelena Klinovaja, Peter Stano (Riken), and Daniel Loss. Phys. Rev. Lett. 116, 176401 (2016); arXiv:1510.03640. We consider periodically-driven arrays of weakly coupled wires with conduction and valence bands of Rashba type and study the resulting Floquet states. This non-equilibrium system can be tuned into non-trivial phases such as of topological insulators, Weyl semimetals, and dispersionless zero-energy edge mode regimes. In the presence of strong electron-electron interactions, we generalize these regimes to the fractional case, where elementary excitations have fractional charges e/m with m being an odd integer.

Long-Distance Entanglement of Spin Qubits via Quantum Hall Edge States

Guang Yang (Riken), Chen-Hsuan Hsu (Riken), Peter Stano (Riken), Jelena Klinovaja, and Daniel Loss. Phys. Rev. B 93, 075301 (2016); arXiv:1509.09006. The implementation of a functional quantum computer involves entangling and coherent manipulation of a large number of qubits. For qubits based on electron spins confined in quantum dots, which are among the most investigated solid-state qubits at present, architectural challenges are often encountered in the design of quantum circuits attempting to assemble the qubits within the very limited space available. Here, we provide a solution to such challenges based on an approach to realizing entanglement of spin qubits over long distances. We show that long-range Ruderman-Kittel-Kasuya-Yosida interaction of confined electron spins can be established by quantum Hall edge states, leading to an exchange coupling of spin qubits. The coupling is anisotropic and can be either Ising-type or XY-type, depending on the spin polarization of the edge state. Such a property, combined with the dependence of the electron spin susceptibility on the chirality of the edge state, can be utilized to gain valuable insights into the topological nature of various quantum Hall states.

Anti-ferromagnetic nuclear spin helix and topological superconductivity in 13C nanotubes

Chen-Hsuan Hsu (RIKEN), Peter Stano (RIKEN), Jelena Klinovaja, and Daniel Loss. Phys. Rev. B 92, 235435 (2015); arXiv:1509.01685. We investigate the RKKY interaction arising from the hyperfine coupling between localized nuclear spins and conduction electrons in interacting 13C carbon nanotubes. Using the Luttinger liquid formalism, we show that the RKKY interaction is sublattice dependent, consistent with the spin susceptibility calculation in non-interacting carbon nanotubes, and it leads to an anti-ferromagnetic nuclear spin helix in finite-size systems. The transition temperature reaches up to tens of millikelvins, due to a strong boost by a positive feedback through the Overhauser field from ordered nuclear spins. Similar to GaAs nanowires, the formation of the helical nuclear spin order gaps out half of the conduction electrons, and is therefore observable as a reduction of conductance by a factor of two in a transport experiment. The nuclear spin helix leads to a density wave combining spin and charge degrees of freedom in the electron subsystem, resulting in synthetic spin-orbit interaction, which induces non-trivial topological phases. As a result, topological superconductivity with Majorana fermion bound states can be realized in the system in the presence of proximity-induced superconductivity without the need of fine tuning the chemical potential. We present the phase diagram as function of system parameters, including the pairing gaps, the gap due to the nuclear spin helix, and the Zeeman field perpendicular to the helical plane.

Dephasing due to nuclear spins in large-amplitude electric dipole spin resonance

Li-Ping Yang (Beijing), Stefano Chesi (Beijing), and Daniel Loss. Phys. Rev. Lett. 116, 066806 (2016); arXiv:1508.06894. We analyze effects of the hyperfine interaction on electric dipole spin resonance when the amplitude of the quantum-dot motion becomes comparable or larger than the quantum dot’s size. Away from the wellknown small-drive regime, the important role played by transverse nuclear fluctuations leads to a Gaussian decay with characteristic dependence on drive strength and detuning. A characterization of spin-flip gate fidelity, in the presence of such additional drive-dependent dephasing, shows that vanishingly small errors can still be achieved at sufficiently large amplitudes. Based on our theory, we analyze recent electric dipole spin resonance experiments relying on spin-orbit interactions or the slanting field of a micromagnet. We find that such experiments are already in a regime with significant effects of transverse nuclear fluctuations and the form of decay of the Rabi oscillations can be reproduced well by our theory.

Supercurrent Reversal in Two-Dimensional Topological Insulators

Alexander Zyuzin, Mohammad Alidoust, Jelena Klinovaja, and Daniel Loss. Phys. Rev. B 92, 174515 (2015); arXiv:1507.05089. We theoretically demonstrate that a supercurrent across a two-dimensional topological insulator subjected to an external magnetic field unambiguously reveals the existence of edge-mode superconductivity. When the edge states of a narrow two-dimensional topological insulator are hybridized, an external magnetic field can close the hybridization gap, thus driving a quantum phase transition from insulator to semimetal states of the topological insulator. Importantly, we find a sign reversal of the supercurrent at the quantum phase transition which offers a simple and experimentally feasible way to observe intrinsic properties of topological insulators including edge-mode superconductivity.

Wiedemann-Franz Law for Magnon Transport

Kouki Nakata, Pascal Simon (Orsay), and Daniel Loss Phys. Rev. B 92, 134425 (2015); arXiv:1507.03807. One of the main goals of spintronics is to improve transport of information carriers and to achieve new functionalities with ultra-low dissipation. A most promising strategy for this holy grail is to use pure magnon currents created and transported in insulating magnets, in the complete absence of any conducting metallic elements. Here we propose a realistic solution to this fundamental challenge by analyzing magnon and heat transport in insulating ferromagnetic junctions. We calculate all transport coefficients for magnon transport and establish Onsager relations between them. We theoretically discover that magnon transport in junctions has a universal behavior, i.e. is independent of material parameters, and establish a magnon analog of the celebrated Wiedemann-Franz law which governs charge transport at low temperatures. We calculate the Seebeck and Peltier coefficients which are crucial quantities for spin caloritronics and demonstrate that they assume universal values in the low temperature limit. Finally, we show that our predictions are within experimental reach with current device and measurement technologies.

Persistent Skyrmion Lattice of Non-Interacting Electrons in Spin-Orbit Coupled Double Wells

Jiyong Fu, Poliana H. Penteado, Marco O. Hachiya, Daniel Loss, and J. Carlos Egues. Phys. Rev. Lett. 117, 226401 (2016); arXiv:1507.00811. A persistent spin helix (PSH) is a robust helical spin-density pattern arising in disordered 2D electron gases with Rashba a and Dresselhaus b spin-orbit (SO) tuned couplings, i.e., |a|=|b|. Here, we investigate the emergence of a persistent Skyrmion lattice (PSL) resulting from the coherent superposition of PSHs along orthogonal directions -crossed PSHs- in wells with two occupied subbands n=1, 2. For realistic GaAs wells, we show that the Rashba a_n and Dresselhaus b_n couplings can be simultaneously tuned to equal strengths but opposite signs, e.g., a_1=b_1 and a_2=-b_2. In this regime, and away from band anticrossings, our noninteracting electron gas sustains a topologically nontrivial Skyrmion-lattice spin-density excitation, which inherits the robustness against spin-independent disorder and interactions from its underlying crossed PSHs. We find that the spin relaxation rate due to the interband SO coupling is comparable to that of the cubic Dresselhaus term as a mechanism of the PSL decay. Near anticrossings, the interband-induced spin mixing leads to unusual spin textures along the energy contours beyond those of the Rahsba-Dresselhaus bands. Our PSL opens up the unique possibility of observing topological phenomena, e.g., topological and Skyrmion Hall effects, in ordinary GaAs wells with noninteracting electrons.

Proximity-Induced pi Josephson Junctions in Topological Insulators and Kramers Pairs of Majorana Fermions

Constantin Schrade, A.A. Zyuzin, Jelena Klinovaja,, and Daniel Loss. Phys. Rev. Lett. 115, 237001 (2015); arXiv:1506.09120. We study two microscopic models of topological insulators in contact with an s-wave superconductor. In the first model the superconductor and the topological insulator are tunnel coupled via a layer of randomly distributed scalar and of randomly oriented spin impurities. Here, we demonstrate that spin-flip tunneling dominates over the spin-conserving one. In the second model the tunnel coupling is realized by a spatially nonuniform array of single-level quantum dots with randomly oriented spins. We find that the tunnel region forms a \pi junction where the effective order parameter changes sign. Because of the random spin orientation, effectively both models exhibit time-reversal symmetry. The proposed \pi junctions support topological superconductivity without magnetic fields and can be used to generate and manipulate Kramers pairs of Majorana fermions by gates.

Probing Atomic Structure and Majorana Wavefunctions in Mono-Atomic Fe-chains on Superconducting Pb-Surface

Remy Pawlak, Marcin Kisiel, Jelena Klinovaja, Tobias Meier, Shigeki Kawai, Thilo Glatzel, Daniel Loss, and Ernst Meyer. npj Quantum Information 2, 16035 (2016); arXiv:1505.06078. Motivated by the striking promise of quantum computation, Majorana bound states (MBSs) in solid-state systems have attracted wide attention in recent years. In particular, the wavefunction localization of MBSs is a key feature and crucial for their future implementation as qubits. Here, we investigate the spatial and electronic characteristics of topological superconducting chains of iron atoms on the surface of Pb(110) by combining scanning tunneling microscopy (STM) and atomic force microscopy (AFM). We demonstrate that the Fe chains are mono-atomic, structured in a linear fashion, and exhibit zero-bias conductance peaks at their ends which we interprete as signature for a Majorana bound state. Spatially resolved conductance maps of the atomic chains reveal that the MBSs are well localized at the chain ends (below 25 nm), with two localization lengths as predicted by theory. Our observation lends strong support to use MBSs in Fe chains as qubits for quantum computing devices.

Fractional Charge and Spin States in Topological Insulator Constrictions

Jelena Klinovaja and Daniel Loss. Phys. Rev. B 92, 121410(R) (2015); arXiv:1505.02682. We investigate theoretically properties of two-dimensional topological insulator constrictions both in the integer and fractional regimes. In the presence of a perpedicular magnetic field, the constriction functions as a spin filter with near-perfect efficiency and can be switched by electric fields only. Domain walls between different topological phases can be created in the constriction as an interface between tunneling, magnetic fields, charge density wave, or electron-electron interactions dominated regions. These domain walls host non-Abelian bound states with fractional charge and spin and result in degenerate ground states with parafermions. If a proximity gap is induced bound states give rise to an exotic Josephson current with 8π-peridiodicity.

Parafermions in a Kagome lattice of qubits for topological quantum computation

Adrian Hutter, James R. Wootton, and Daniel Loss. Phys. Rev. X 5, 041040 (2015); arXiv:1505.01412. Engineering complex non-Abelian anyon models with simple physical systems is crucial for topological quantum computation. Unfortunately, the simplest systems are typically restricted to Majorana zero modes (Ising anyons). Here, we go beyond this barrier, showing that the ℤ4 parafermion model of non-Abelian anyons can be realized on a qubit lattice. Our system additionally contains the Abelian D(ℤ4) anyons as low-energetic excitations. We show that braiding of these parafermions with each other and with the D(ℤ4) anyons allows the entire d=4 Clifford group to be generated. The error-correction problem for our model is also studied in detail, guaranteeing fault tolerance of the topological operations. Crucially, since the non-Abelian anyons are engineered through defect lines rather than as excitations, non-Abelian error correction is not required. Instead, the error-correction problem is performed on the underlying Abelian model, allowing high noise thresholds to be realized.

Impurity Induced Quantum Phase Transitions and Magnetic Order in Conventional Superconductors: Competition between Bound and Quasiparticle states

Silas Hoffman, Jelena Klinovaja, Tobias Meng (TU Dresden), and Daniel Loss. Phys. Rev. B 92, 125422 (2015); arXiv:1503.08762. We theoretically study bound states generated by magnetic impurities within conventional s-wave superconductors, both analytically and numerically. In determining the effect of the hybridization of two such bound states on the energy spectrum as a function of magnetic exchange coupling, relative angle of magnetization, and distance between impurities, we find that quantum phase transitions can be modulated by each of these parameters. Accompanying such transitions, there is a change in the preferred spin configuration of the impurities. Although the interaction between the impurity spins is overwhelmingly dominated by the quasiparticle contribution, the ground state of the system is determined by the bound state energies. Self-consistently calculating the superconducting order parameter, we find a discontinuity when the system undergoes a quantum phase transition as indicated by the bound state energies.

Voltage induced conversion of helical to uniform nuclear spin polarization in a quantum wire

Viktoriia Kornich, Peter Stano (Tokyo), Alexander A. Zyuzin, and Daniel Loss. Phys. Rev. B 91, 195423 (2015); arXiv:1503.06950. We study the effect of bias voltage on the nuclear spin polarization of a ballistic wire, which contains electrons and nuclei interacting via hyperfine interaction. In equilibrium, the localized nuclear spins are helically polarized due to the electron-mediated Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. Focusing here on non-equilibrium, we find that an applied bias voltage induces a uniform polarization, from both helically polarized and unpolarized spins available for spin flips. Once a macroscopic uniform polarization in the nuclei is established, the nuclear spin helix rotates with frequency proportional to the uniform polarization. The uniform nuclear spin polarization monotonically increases as a function of both voltage and temperature, reflecting a thermal activation behavior. Our predictions offer specific ways to test experimentally the presence of a nuclear spin helix polarization in semiconducting quantum wires.

Field-dependent superradiant quantum phase transition of molecular magnets in microwave cavities

Dimitrije Stepanenko (Belgrade), Mircea Trif (Paris/Beijing), Oleksandr Tsyplyatyev (Frankfurt), and Daniel Loss. Semicond. Sci. Technol. 31, 094003 (2016); arXiv:1502.04075. We study a superradiant quantum phase transition in the model of triangular molecular magnets coupled to the electric component of a microwave cavity field. The transition occurs when the coupling strength exceeds a critical value, dc, which, in sharp contrast to the standard two-level emitters, can be tuned by an external magnetic field. In addition to emitted radiation, the molecules develop an in-plane electric dipole moment at the transition. We estimate that the transition can be detected in state-of-the-art microwave cavities if their electric field couples to a crystal containing a sufficient number of oriented molecules.

Magnon transport through microwave pumping

Kouki Nakata, Pascal Simon (Paris), and Daniel Loss. Phys. Rev. B 92, 014422 (2015); arXiv:1502.03865. We present a microscopic theory ofmagnon transport in ferromagnetic insulators (FIs).Using magnon injection through microwave pumping, we propose a way to generate magnon dc currents and show how to enhance their amplitudes in hybrid ferromagnetic insulating junctions. To this end, focusing on a single FI, we first revisit microwave pumping at finite (room) temperature from the microscopic viewpoint of magnon injection. Next, we apply it to two kinds of hybrid ferromagnetic insulating junctions. The first is the junction between a quasiequilibrium magnon condensate and magnons being pumped by microwave, while the second is the junction between such pumped magnons and noncondensed magnons. We show that quasiequilibrium magnon condensates generate ac and dc magnon currents, while noncondensed magnons produce essentially a dc magnon current. The ferromagnetic resonance (FMR) drastically increases the density of the pumped magnons and enhances such magnon currents. Lastly, using microwave pumping in a single FI, we discuss the possibility that a magnon current through an Aharonov-Casher phase flows persistently even at finite temperature.We show that such a magnon current arises even at finite temperature in the presence of magnon-magnon interactions. Due to FMR, its amplitude becomes much larger than the condensed magnon current.

Superconducting Gap Renomalization around two Magnetic Impurities: From Shiba to Andreev Bound States

Tobias Meng (Dresden), Jelena Klinovaja, Silas Hoffman, Pascal Simon (Paris), and Daniel Loss. Phys. Rev. B 92, 064503 (2015); arXiv:1501.07901. We study the renormalization of the gap of an s-wave superconductor in the presence of two magnetic impurities. For weakly bound Shiba states, we analytically calculate the part of the gap renormalization that is sensitive to the relative orientation of the two impurity spins. For impurities with a strong exchange coupling to the conduction electrons, we solve the gap equation self-consistently by numerics and find that the subgap Shiba state turns into a supragap Andreev state when the local gap parameter changes sign under the impurities.

Long-Distance Entanglement of Soliton Spin Qubits in Gated Nanowires

Pawel Szumniak, Jaroslaw Pawlowski (Krakow), Stanislaw Bednarek (Krakow), and Daniel Loss. Phys. Rev. B 92, 035403 (2015); arXiv:1501.01932. We investigate numerically charge, spin, and entanglement dynamics of two electrons confined in a gated semiconductor nanowire. The electrostatic coupling between electrons in the nanowire and the charges in the metal gates leads to a self-trapping of the electrons which results in soliton-like properties. We show that the interplay of an all-electrically controlled coherent transport of the electron solitons and of the exchange interaction can be used to realize ultrafast SWAP and entangling SQRT-of-SWAP gates for distant spin qubits. We demonstrate that the latter gate can be used to generate a maximally entangled spin state of spatially separated electrons. The results are obtained by quantum mechanical time-dependent calculations with exact inclusion of electron-electron correlations.

Electrically-tunable hole g-factor of an optically-active quantum dot for fast spin rotations

Jonathan H. Prechtel, Franziska Maier, Julien Houel, Andreas V. Kuhlmann, Arne Ludwig, Andreas D. Wieck, Daniel Loss, and Richard J. Warburton. Phys. Rev. B 91, 165304 (2015); arXiv:1412.4238. We report a large g-factor tunability of a single hole spin in an InGaAs quantum dot via an electric field. The magnetic field lies in the in-plane direction x, the direction required for a coherent hole spin. The electrical field lies along the growth direction z and is changed over a large range, 100 kV/cm. Both electron and hole g-factors are determined by high resolution laser spectroscopy with resonance fluorescence detection. This, along with the low electrical-noise environment, gives very high quality experimental results. The hole g-factor g_xh depends linearly on the electric field Fz, dg_xh/dFz = (8.3 +/- 1.2)* 10^-4 cm/kV, whereas the electron g-factor g_xe is independent of electric field, dg_xe/dFz = (0.1 +/- 0.3)* 10^-4 cm/kV (results averaged over a number of quantum dots). The dependence of g_xh on Fz is well reproduced by a 4x4 k.p model demonstrating that the electric field sensitivity arises from a combination of soft hole confining potential, an In concentration gradient and a strong dependence of material parameters on In concentration. The electric field sensitivity of the hole spin can be exploited for electrically-driven hole spin rotations via the g-tensor modulation technique and based on these results, a hole spin coupling as large as ~ 1 GHz is expected to be envisaged.

Integer and Fractional Quantum Anomalous Hall Effect in a Strip of Stripes Model

Jelena Klinovaja, Yaroslav Tserkovnyak (UCLA), and Daniel Loss. Phys. Rev. B 91, 085426 (2015); arXiv:1412.0548. We study the quantum anomalous Hall effect in a strip of stripes model coupled to a magnetic texture with zero total magnetization and in the presence of strong electron-electron interactions. A helical magnetization along the stripes and a spin-selective coupling between the stripes gives rise to a bulk gap and chiral edge modes. Depending on the ratio between the period of the magnetic structure and the Fermi wavelength, the system can exhibit the integer or fractional quantum anomalous Hall effect. In the fractional regime, the quasiparticles have fractional charges and non-trivial Abelian braid statistics.

Quantum Memories at Finite Temperature

Benjamin J. Brown (Imperial London), Daniel Loss, Jiannis K. Pachos (Leeds), Chris N. Self (Leeds), and James R. Wootton. Rev. Mod. Phys. 88, 045005 (2016); arXiv:1411.6643. To use quantum systems for technological applications we first need to preserve their coherence for macroscopic timescales, even at finite temperature. Quantum error correction has made it possible to actively correct errors that affect a quantum memory. An attractive scenario is the construction of passive storage of quantum information with minimal active support. Indeed, passive protection is the basis of robust and scalable classical technology, physically realized in the form of the transistor and the ferromagnetic hard disk. The discovery of an analogous quantum system is a challenging open problem, plagued with a variety of no-go theorems. Several approaches have been devised to overcome these theorems by taking advantage of their loopholes. Here we review the state-of-the-art developments in this field in an informative and pedagogical way. We give the main principles of self-correcting quantum memories and we analyze several milestone examples from the literature of two-, three- and higher-dimensional quantum memories.

Spin and Orbital Magnetic Response on the Surface of a Topological Insulator

Yaroslav Tserkovnyak (UCLA), D. A. Pesin (Univ. of Utah), and Daniel Loss. Phys. Rev. B 91, 041121(R) (2015); arXiv:1411.2070. Coupling of the spin and orbital degrees of freedom on the surface of a strong three-dimensional insulator, on the one hand, and textured magnetic configuration in an adjacent ferromagnetic film, on the other, is studied using a combination of transport and thermodynamic considerations. Expressing exchange coupling between the localized magnetic moments and Dirac electrons in terms of the electrons' out-of-plane orbital and spin magnetizations, we relate the thermodynamic properties of a general ferromagnetic spin texture to the physics in the zeroth Landau level. Persistent currents carried by Dirac electrons endow the magnetic texture with a Dzyaloshinski-Moriya interaction, which exhibits a universal scaling form as a function of electron temperature, chemical potential, and the time-reversal symmetry breaking gap. In addition, the orbital motion of electrons establishes a direct magnetoelectric coupling between the unscreened electric field and local magnetic order, which furnishes complex long-ranged interactions within the magnetic film.

Improved HDRG decoders for qudit and non-Abelian quantum error correction

Adrian Hutter, Daniel Loss, and James R. Wootton. New J. Phys. 17, 035017 (2015); arXiv:1410.4478. Hard-decision renormalization group (HDRG) decoders are an important class of decoding algorithms for topological quantum error correction. Due to their versatility, they have been used to decode systems with fractal logical operators, color codes, qudit topological codes, and non-Abelian systems. In this work, we develop a method of performing HDRG decoding which combines strenghts of existing decoders and further improves upon them. In particular, we increase the minimal number of errors necessary for a logical error in a system of linear size L from Θ(L2/3) to Ω(L1−ϵ) for any ϵ>0. We apply our algorithm to decoding D(ℤd) quantum double models and a non-Abelian anyon model with Fibonacci-like fusion rules, and show that it indeed significantly outperforms previous HDRG decoders. Furthermore, we provide the first study of continuous error correction with imperfect syndrome measurements for the D(ℤd) quantum double models. The parallelized runtime of our algorithm is poly(logL) for the perfect measurement case. In the continuous case with imperfect syndrome measurements, the averaged runtime is O(1) for Abelian systems, while continuous error correction for non-Abelian anyons stays an open problem.

Majorana Fermions in Ge/Si Hole Nanowires

Franziska Maier, Jelena Klinovaja (Harvard), and Daniel Loss. Phys. Rev. B 90, 195421 (2014); arXiv:1409.8645. We consider Ge/Si core/shell nanowires with hole states coupled to an s-wave superconductor in the presence of electric and magnetic fields. We employ a microscopic model that takes into account material-specific details of the band structure such as strong and electrically tunable Rashba-type spin-orbit interaction and g factor anisotropy for the holes. In addition, the proximity-induced superconductivity Hamiltonian is derived starting from a microscopic model. In the topological phase, the nanowires host Majorana fermions with localization lengths that depend strongly on both the magnetic and electric fields. We identify the optimal regime in terms of the directions and magnitudes of the fields in which the Majorana fermions are the most localized at the nanowire ends. In short nanowires, the Majorana fermions hybridize and form a subgap fermion whose energy is split away from zero and oscillates as a function of the applied fields. The period of these oscillations could be used to measure the dependence of the spin-orbit interaction on the applied electric field and the g factor anisotropy.

Fast Long-Distance Control of Spin Qubits by Photon Assisted Cotunneling

Peter Stano (Riken), Jelena Klinovaja (Harvard), Floris R. Braakman, Lieven M. K. Vandersypen (Delft), and Daniel Loss. Phys. Rev. B 92, 075302 (2015); arXiv:1409.4852. We investigate theoretically the long-distance coupling and spin exchange in an array of quantum dot spin qubits in the presence of microwaves.We find that photon-assisted cotunneling is boosted at resonances between photon and energies of virtually occupied excited states and showhowtomake it spin selective.We identify configurations that enable fast switching and spin echo sequences for efficient and nonlocal manipulation of spin qubits.We devise configurations in which the near-resonantly boosted cotunneling provides nonlocal coupling which, up to certain limit, does not diminish with distance between themanipulated dots before it decays weakly with inverse distance.

High-efficiency resonant amplification of weak magnetic fields for single spin magnetometry

Luka Trifunovic, Fabio L. Pedrocchi, Silas Hoffman, Patrick Maletinsky, Amir Yacoby (Harvard), and Daniel Loss. Nature Nanotechnology 10, 541 (2015); arXiv:1409.1497. Magnetic resonance techniques not only provide powerful imaging tools that have revolutionized medicine, but they have a wide spectrum of applications in other fields of science such as biology, chemistry, neuroscience and physics. However, current state-of-the-art magnetometers are unable to detect a single nuclear spin unless the tip-to-sample separation is made sufficiently small. Here, we demonstrate theoretically that by placing a ferromagnetic particle between a nitrogen– vacancy magnetometer and a target spin, the magnetometer sensitivity is improved dramatically. Using materials and techniques that are already experimentally available, our proposed set-up is sensitive enough to detect a single nuclear spin within ten milliseconds of data acquisition at room temperature. The sensitivity is practically unchanged when the ferromagnet surface to the target spin separation is smaller than the ferromagnet lateral dimensions; typically about a tenth of a micrometre. This scheme further benefits when used for nitrogen–vacancy ensemble measurements, enhancing sensitivity by an additional three orders of magnitude.

Fermionic and Majorana Bound States in Hybrid Nanowires with Non-Uniform Spin-Orbit Interaction

Jelena Klinovaja (Harvard) and Daniel Loss. Eur. Phys. J. B 88, 62 (2015); arXiv:1408.3366. We study intragap bound states in the topological phase of a Rashba nanowire in the presence of a magnetic field and with non-uniform spin orbit interaction (SOI) and proximity-induced superconductivity gap. We show that fermionic bound states (FBS) can emerge inside the proximity gap. They are localized at the junction between two wire sections characterized by different directions of the SOI vectors, and they coexist with Majorana bound states (MBS) localized at the nanowire ends. The energy of the FBS is determined by the angle between the SOI vectors and the lengthscale over which the SOI changes compared to the Fermi wavelength and the localization length. We also consider double-junctions and show that the two emerging FBSs can hybridize and form a double quantum dot-like structure inside the gap. We find explicit analytical solutions of the bound states and their energies for certain parameter regimes such as weak and strong SOI. The analytical results are confirmed and complemented by an independent numerical tight-binding model approach. Such FBS can act as quasiparticle traps and thus can have implications for topological quantum computing schemes based on braiding MBSs.

Spontaneous Helical Order of Electron and Nuclear Spins in a Luttinger Liquid

Christian P Scheller, Bernd Braunecker, Daniel Loss, and Dominik M Zumbuhl Progress in Physics (44), COMMUNICATIONS of the Swiss Physical Society (SPS) 9, 23 (2014) In a one-dimensional (1D) conductor, electrons are confined to move along a single direction, occupying only the quantum mechanical ground state orbital of the transverse dimensions of the wire. What is the electrical conductance of such a quantum wire? This fundamental question was answered by Rolf Landauer many years ago for non-interacting electrons in a clean, ballistic conductor: each spin species carries the quantum of conductance, e2/h [1], with e the electron charge and h the Planck constant. For a spin degenerate 1D conductor with a single subband, the conductance is therefore 2e2/h. If the spin degeneracy is broken and transport of one spin direction is blocked, the conductance is thus reduced to 1e2/h. Similar to spin, other degeneracies such as valley degeneracies or multiple 1D subbands due to weaker confinement can also open additional conductance …

NMR Response of Nuclear Spin Helix in Quantum Wires with Hyperfine and Spin-Orbit Interaction

Peter Stano (RIKEN) and Daniel Loss. Phys. Rev. B 90, 195312 (2014); arXiv:1408.2353. We calculate the nuclear magnetic resonance (NMR) response of a quantum wire where at low temperature a self-sustained electron-nuclear spin order is created. Our model includes the electron mediated Ruderman-Kittel-Kasuya-Yosida (RKKY) exchange, electron spin-orbit interactions, nuclear dipolar interactions, and the static and oscillating NMR fields, all of which play an essential role. The paramagnet to helimagnet transition in the nuclear system is reflected in an unusual response: it absorbs at a frequency given by the internal RKKY exchange field, rather than the external static field, whereas the latter leads to a splitting of the resonance peak.

Strongly Interacting Holes in Ge/Si Nanowires

Franziska Maier, Tobias Meng, and Daniel Loss. Phys. Rev. B 90, 155437 (2014); arXiv:1408.0631. We consider holes confined to Ge/Si core/shell nanowires subject to strong Rashba spin-orbit interaction and screened Coulomb interaction. Such wires can, for instance, serve as host systems for Majorana bound states. Starting from a microscopic model, we find that the Coulomb interaction strongly influences the properties of experimentally realistic wires. To show this, a Luttinger liquid description is derived based on a renormalization group analysis. This description in turn allows to calculate the scaling exponents of various correlation functions as a function of the microscopic system parameters. It furthermore permits to investigate the effect of Coulomb interaction on a small magnetic field, which opens a strongly anisotropic partial gap.

Conductance behavior in nanowires with spin-orbit interaction -- A numerical study

Diego Rainis and Daniel Loss. Phys. Rev. B 90, 235415 (2014); arXiv:1407.8239. We consider electronic transport through semiconducting nanowires (W) with spin-orbit interaction (SOI), in a hybrid N-W-N setup where the wire is contacted by normal-metal leads (N). We investigate the conductance behavior of the system as a function of gate and bias voltage, magnetic field, wire length, temperature, and disorder. The transport calculations are performed numerically and are based on standard recursive Green's function techniques. In particular, we are interested in understanding if and how it is possible to deduce the strength of the SOI from the transport behavior. This is a very relevant question since so far no clear experimental observation in that direction has been produced. We find that the smoothness of the electrostatic potential profile between the contacts and the wire plays a crucial role, and we show that in realistic regimes the N-W-N setup may mask the effects of SOI, and a trivial behavior with apparent vanishing SOI is observed. We identify an optimal parameter regime, with neither too smooth nor too abrupt potentials, where the signature of SOI is best visible, with and without Fabry-Perot oscillations, and is most resilient to disorder and temperature effects.

RKKY Interaction On Surfaces of Topological Insulators With Superconducting Proximity Effect

Alexander A. Zyuzin and Daniel Loss. Phys. Rev. B 90, 125443 (2014); arXiv:1407.6632. We consider the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between magnetic impurities on the surface of a three-dimensional topological insulator with proximity induced superconductivity. A superconductor placed on the top of the topological insulator induces a gap in the surface electron states and gives rise to a long-ranged in-plane antiferromagnetic RKKY interaction. This interaction is frustrated due to strong spin-orbit coupling, decays as 1/r for r < xi, where r is the distance between two magnetic impurities and xi the superconducting coherence length, and dominates over the ferromagnetic and Dzyaloshinskii-Moriya type interactions for r > xi. We find the condition for the Yu-Shiba-Rusinov intragap states that are bound to the magnetic impurities.

Helical nuclear spin order in a strip of stripes in the Quantum Hall regime

Tobias Meng, Peter Stano (RIKEN), Jelena Klinovaja (Harvard), and Daniel Loss. Eur. Phys. J. B 87, 203 (2014); arXiv:1407.3726. We investigate nuclear spin effects in a two-dimensional electron gas in the quantum Hall regime modeled by a weakly coupled array of interacting quantum wires. We show that the presence of hyperfine interaction between electron and nuclear spins in such wires can induce a phase transition, ordering electrons and nuclear spins into a helix in each wire. Electron-electron interaction effects, pronounced within the one-dimensional stripes, boost the transition temperature up to tens to hundreds of millikelvins in GaAs. We predict specific experimental signatures of the existence of nuclear spin order, for instance for the resistivity of the system at transitions between different quantum Hall plateaus.

Nuclear Spin Relaxation in Rashba Nanowires

Alexander A. Zyuzin, Tobias Meng, Viktoriia Kornich, and Daniel Loss. Phys. Rev. B 90, 195125 (2014); arXiv:1407.2582. We study the nuclear spin relaxation in a ballistic nanowire with hyperfine and Rashba spin-orbit interactions (SOI) and in the presence of magnetic field and electron interactions. The relaxation rate shows pronounced peaks as function of magnetic field and chemical potential due to van Hove singularities in the Rashba bands. As a result, the regimes of weak and strong SOIs can be distinguished by the number of peaks in the rate. The relaxation rate increases with increasing magnetic field if both Rashba subbands are occupied, whereas it decreases if only the lowest one is occupied.

Josephson and Persistent Spin Currents in Bose-Einstein Condensates of Magnons

Kouki Nakata, Kevin A. van Hoogdalem, Pascal Simon (Paris), and Daniel Loss. Phys. Rev. B 90, 144419 (2014); arXiv:1406.7004. Using the Aharonov-Casher (A-C) phase, we present a microscopic theory of the Josephson and persistent spin currents in quasi-equilibrium Bose-Einstein condensates (BECs) of magnons in ferromagnetic insulators. Starting from a microscopic spin model that we map onto a Gross-Pitaevskii Hamiltonian, we derive a two-state model for the Josephson junction between the weakly coupled magnon-BECs. We then show how to obtain the alternating-current (ac) Josephson effect with magnons as well as macroscopic quantum self-trapping in a magnon-BEC. We next propose how to control the direct-current (dc) Josephson effect electrically using the A-C phase, which is the geometric phase acquired by magnons moving in an electric field. Finally, we introduce a magnon-BEC ring and show that persistent magnon-BEC currents flow due to the A-C phase. Focusing on the feature that the persistent magnon-BEC current is a steady flow of magnetic dipoles that produces an electric field, we propose a method to directly measure it experimentally.

Single-spin manipulation in a double quantum dot with micromagnet

Stefano Chesi (Beijing), Ying-Dan Wang (Beijing), Jun Yoneda (Tokyo), Tomohiro Otsuka (Tokyo), Seigo Tarucha (Tokyo), and Daniel Loss. Phys. Rev. B 90, 235311 (2014); arXiv:1405.7618. The manipulation of single spins in double quantum dots by making use of the exchange interaction and a highly inhomogeneous magnetic field was discussed in [W. A. Coish and D. Loss, Phys. Rev. B 75, 161302 (2007)]. However, such large inhomogeneity is difficult to achieve through the slanting field of a micromagnet in current designs of lateral double dots. Therefore, we examine an analogous spin manipulation scheme directly applicable to realistic GaAs double dot setups. We estimate that typical gate times, realized at the singlet-triplet anticrossing induced by the inhomogeneous micromagnet field, can be a few nanoseconds. We discuss the optimization of initialization, read-out, and single-spin gates through suitable choices of detuning pulses and an improved geometry. We also examine the effect of nuclear dephasing and charge noise. The latter induces fluctuations of both detuning and tunneling amplitude. Our results suggest that this scheme is a promising approach for the realization of fast single-spin operations.

Long-Distance Entanglement of Spin Qubits via Ferromagnet

Luka Trifunovic, Fabio L. Pedrocchi, and Daniel Loss. Phys. Rev. X 3, 041023 (2013); arXiv:1305.2451. We propose a mechanism of a long-range coherent interaction between two singlet-triplet qubits dipolarly coupled to a dogbone-shaped ferromagnet. An effective qubit-qubit interaction Hamiltonian is derived and the coupling strength is estimated. Furthermore we derive the effective coupling between two spin-1/2 qubits that are coupled via dipolar interaction to the ferromagnet and that lie at arbitrary positions and deduce the optimal positioning. We consider hybrid systems consisting of spin-1/2 and ST qubits and derive the effective Hamiltonian for this case. We then show that operation times vary between 1MHz and 100MHz and give explicit estimates for GaAs, Silicon, and NV-center based spin qubits. Finally, we explicitly construct the required sequences to implement a CNOT gate. The resulting quantum computing architecture retains all the single qubit gates and measurement aspects of earlier approaches, but allows qubit spacing at distances of order 1$\,\mu$m for two-qubit gates, achievable with current semiconductor technology.

Quantum charge pumping through fractional Fermions in charge density modulated quantum wires and Rashba nanowires

Arijit Saha, Diego Rainis, Rakesh P. Tiwari, and Daniel Loss. Phys. Rev. B 90, 035422 (2014); arXiv:1405.5719. We study the phenomenon of adiabatic quantum charge pumping in systems supporting fractionally charged fermionic bound states, in two different setups. The first quantum pump setup consists of a charge-density-modulated quantum wire, and the second one is based on a semiconducting nanowire with Rashba spin-orbit interaction, in the presence of a spatially oscillating magnetic field. In both these quantum pumps transport is investigated in a N-X-N geometry, with the system of interest (X) connected to two normal-metal leads (N), and the two pumping parameters are the strengths of the effective wire-lead barriers. Pumped charge is calculated within the scattering matrix formalism. We show that quantum pumping in both setups provides a unique signature of the presence of the fractional-fermion bound states, in terms of asymptotically quantized pumped charge. Furthermore, we investigate shot noise arising due to quantum pumping, verifying that quantized pumped charge corresponds to minimal shot noise.

Acoustic phonons and strain in core/shell nanowires

Christoph Kloeffel, Mircea Trif (Paris), and Daniel Loss. Phys. Rev. B 90, 115419 (2014); arXiv:1405.4834. We study theoretically the low-energy phonons and the static strain in cylindrical core/shell nanowires (NWs). Assuming pseudomorphic growth, isotropic media, and a force-free wire surface, we derive algebraic expressions for the dispersion relations, the displacement fields, and the stress and strain components from linear elasticity theory. Our results apply to NWs with arbitrary radii and arbitrary elastic constants for both core and shell. The expressions for the static strain are consistent with experiments, simulations, and previous analytical investigations; those for phonons are consistent with known results for homogeneous NWs. Among other things, we show that the dispersion relations of the torsional, longitudinal, and flexural modes change differently with the relative shell thickness, and we identify new terms in the corresponding strain tensors that are absent for uncapped NWs. We illustrate our results via the example of Ge/Si core/shell NWs and demonstrate that shell-induced strain has large effects on the hole spectrum of these systems.

Characterization of spin-orbit interactions of GaAs heavy holes using a quantum point contact

Fabrizio Nichele (ETHZ), Stefano Chesi (RIKEN), Szymon Hennel (ETHZ), Angela Wittmann (ETHZ), Christian Gerl (ETHZ), Werner Wegscheider (ETHZ), Daniel Loss, Thomas Ihn (ETHZ), and Klaus Ensslin (ETHZ). Phys. Rev. Lett. 113, 046801 (2014); arXiv:1405.2981. We present transport experiments performed in high quality quantum point contacts embedded in a GaAs two-dimensional hole gas. The strong spin-orbit interaction results in peculiar transport phenomena, including the previously observed anisotropic Zeeman splitting and level-dependent effective g-factors. Here we find additional effects, namely the crossing and the anti-crossing of spin-split levels depending on subband index and magnetic field direction. Our experimental observations are reconciled in an heavy hole effective spin-orbit Hamiltonian where cubic- and quadratic-in-momentum terms appear. The spin-orbit components, being of great importance for quantum computing applications, are characterized in terms of magnitude and spin structure. In the light of our results, we explain the level dependent effective g-factor in an in-plane field. Through a tilted magnetic field analysis, we show that the QPC out-of-plane g-factor saturates around the predicted 7.2 bulk value.

Kramers Pairs of Majorana Fermions and Parafermions in Fractional Topological Insulators

Jelena Klinovaja (Harvard), Amir Yacoby (Harvard), and Daniel Loss. Phys. Rev. B 90, 155447 (2014); arXiv:1403.4125. We propose a scheme based on topological insulators to generate Kramers pairs of Majorana fermions or parafermions in the complete absence of magnetic fields. Our setup consists of two topological insulators whose edge states are brought close to an s-wave superconductor. The resulting proximity effect leads to an interplay between a non-local crossed Andreev pairing, which is dominant in the strong electron-electron interaction regime, and usual superconducting pairing, which is dominant at large separation between the two topological insulator edges. As a result, there are zero-energy bound states localized at interfaces between spatial regions dominated by the two different types of pairing. Due to the preserved time-reversal symmetry, the bound states come in Kramers pairs. If the topological insulators carry fractional edge states, the zero-energy bound states are parafermions, otherwise, they are Majorana fermions.

Renormalization of anticrossings in interacting quantum wires with Rashba and Dresselhaus spin-orbit couplings

Tobias Meng, Jelena Klinovaja (Harvard), and Daniel Loss. Phys. Rev. B 89, 205133 (2014); arXiv:1403.2759. We discuss how electron-electron interactions renormalize the spin-orbit induced anticrossings between different subbands in ballistic quantum wires. Depending on the ratio of spin-orbit coupling and subband spacing, electron-electron interactions can either increase or decrease anticrossing gaps. When the anticrossings are closing due to a special combination of Rashba and Dresselhaus spin-orbit couplings, their gap approaches zero as an interaction dependent power law of the spin-orbit couplings, which is a consequence of Luttinger liquid physics. Monitoring the closing of the anticrossings allows to directly measure the related renormalization group scaling dimension in an experiment. If a magnetic field is applied parallel to the spin-orbit coupling direction, the anticrossings experience different renormalizations. Since this difference is entirely rooted in electron-electron interactions, unequally large anticrossings also serve as a direct signature of Luttinger liquid physics. Electron-electron interactions furthermore increase the sensitivity of conductance measurements to the presence of anticrossing.

Inhibition of dynamic nuclear polarization by heavy-hole noncollinear hyperfine interactions

Hugo Ribeiro (McGill), Franziska Maier, and Daniel Loss. Phys. Rev. B 92, 075421 (2015); arXiv:1403.0490. We show that the effective hyperfine interaction for heavy-hole states (or any particle described with a p-like Bloch function) can induce nontrivial dynamics of nuclear spins. Experimental evidence can be found, e.g., in self-assembled quantum dots by measuring the saturation of nuclear spin polarization with different orientations of an external magnetic field.

Breakdown of surface-code error correction due to coupling to a bosonic bath

Adrian Hutter and Daniel Loss. Phys. Rev. A 89, 042334 (2014); arXiv:1402.3108. We consider a surface code suffering decoherence due to coupling to a bath of bosonic modes at finite temperature and study the time available before the unavoidable breakdown of error correction occurs as a function of coupling and bath parameters. We derive an exact expression for the error rate on each individual qubit of the code, taking spatial and temporal correlations between the errors into account. We investigate numerically how different kinds of spatial correlations between errors in the surface code affect its threshold error rate. This allows us to derive the maximal duration of each quantum error-correction period by studying when the single-qubit error rate reaches the corresponding threshold. At the time when error correction breaks down, the error rate in the code can be dominated by the direct coupling of each qubit to the bath, by mediated subluminal interactions, or by mediated superluminal interactions. For a two-dimensional Ohmic bath, the time available per quantum error-correction period vanishes in the thermodynamic limit of a large code size L due to induced superluminal interactions, although it does so only like 1/√lnL. For all other bath types considered, this time remains finite as L→∞.

A quantum magnetic RC circuit

Kevin A. van Hoogdalem, Mathias Albert (Orsay), Pascal Simon (Orsay), and Daniel Loss. Phys. Rev. Lett. 113, 037201 (2014); arXiv:1401.5712. We propose a setup that is the spin analog of the charge-based quantum RC circuit. We define and compute the spin capacitance and the spin resistance of the circuit for both ferromagnetic (FM) and antiferromagnetic (AF) systems. We find that the antiferromagnetic setup has universal properties, but the ferromagnetic setup does not. We discuss how to use the proposed setup as a quantum source of spin excitations, and put forward a possible experimental realization using ultracold atoms in optical lattices.

Time-Reversal Invariant Parafermions in Interacting Rashba Nanowires

Jelena Klinovaja (Harvard) and Daniel Loss. Phys. Rev. B 90, 045118 (2014); arXiv:1312.1998. We propose a scheme to generate pairs of time-reversal invariant parafermions. Our setup consists of two quantum wires with Rashba spin-orbit interactions coupled to an s-wave superconductor, in the presence of electron-electron interactions. The zero-energy bound states localized at the wire ends arise from the interplay between two types of proximity-induced superconductivity: the usual intrawire superconductivity and the interwire superconductivity due to crossed Andreev reflections. If the latter dominates, which is the case for strong electron-electron interactions, the system supports Kramers pair of parafermions. Moreover, the scheme can be extended to a two-dimensional sea of time-reversal invariant parafermions.

Parafermions in Interacting Nanowire Bundle

Jelena Klinovaja (Harvard) and Daniel Loss. Phys. Rev. Lett. 112, 246403 (2014); arXiv:1311.3259. We propose a scheme to induce Z3 parafermion modes, exotic zero-energy bound states that possess non-Abelian statistics. We consider a minimal setup consisting of a bundle of four tunnel coupled nanowires hosting spinless electrons that interact strongly with each other. The hallmark of our setup is that it relies only on simple one-dimensional wires, uniform magnetic fields, and strong interactions, but does not require the presence of superconductivity or exotic quantum Hall phases.

Phonon-Mediated Decay of Singlet-Triplet Qubits in Double Quantum Dots

Viktoriia Kornich, Christoph Kloeffel, and Daniel Loss. Phys. Rev. B 89, 085410 (2014); arXiv:1311.2197. We study theoretically the phonon-induced decoherence and relaxation of singlet-triplet qubits in lateral GaAs double quantum dots. For typical setups, the decoherence and relaxation rates due to one-phonon processes are proportional to the temperature T, whereas the rates due to two-phonon processes reveal a transition from T^2 to higher powers as T is decreased. In contrast to previous calculations of the phonon-limited lifetimes T_1 (relaxation) and T_2 (decoherence), we find T_2 \neq 2 T_1 in this system. Remarkably, both T_1 and T_2 exhibit a maximum when the external magnetic field is applied along a certain axis within the plane of the two-dimensional electron gas. We compare our results with recent experiments and analyze the dependence of T_1 and T_2 on system properties such as the detuning, the spin-orbit parameters, and the orientation of the double quantum dot and the applied magnetic field with respect to the main crystallographic axes.

Anisotropic g factor in InAs self-assembled quantum dots

Robert Zielke, Franziska Maier, and Daniel Loss. Phys. Rev. B 89, 115438 (2014); arXiv:1311.0908. We investigate the wavefunctions, spectrum, and g factor anisotropy of low-energy electrons confined to self-assembled, pyramidal InAs quantum dots (QDs) subject to external magnetic and electric fields. We present the construction of trial wavefunctions for a pyramidal geometry with hard-wall confinement. We explicitly find the ground and first excited states and show the associated probability distributions and energies. Subsequently, we use these wavefunctions and 8-band k⋅p theory to derive a Hamiltonian describing the QD states close to the valence band edge. Using a perturbative approach, we find an effective conduction band Hamiltonian describing low-energy electronic states in the QD. From this, we further extract the magnetic field dependent eigenenergies and associated g factors. We examine the g factors regarding anisotropy and behavior under small electric fields. In particular, we find strong anisotropies, with the specific shape depending strongly on the considered subband. Our results are in good agreement with recent measurements [Takahashi et al., Phys. Rev. B 87, 161302 (2013)] and support the possibility to control a spin qubit by means of g tensor modulation.

Error Correction for Non-Abelian Topological Quantum Computation

James R. Wootton, Jan Burri, Sofyan Iblisdir, and Daniel Loss. Phys. Rev. X 4, 011051 (2014); Popular Summary; arXiv:1310.3846. The possibility of quantum computation using non-Abelian anyons has been considered for over a decade. However, the question of how to obtain and process information about what errors have occurred in order to negate their effects has not yet been considered. This is in stark contrast with quantum computation proposals for Abelian anyons, for which decoding algorithms have been tailor-made for many topological error-correcting codes and error models. Here, we address this issue by considering the properties of non-Abelian error correction, in general. We also choose a specific anyon model and error model to probe the problem in more detail. The anyon model is the charge submodel of D(S3). This shares many properties with important models such as the Fibonacci anyons, making our method more generally applicable. The error model is a straightforward generalization of those used in the case of Abelian anyons for initial benchmarking of error correction methods. It is found that error correction is possible under a threshold value of 7% for the total probability of an error on each physical spin. This is remarkably comparable with the thresholds for Abelian models.

Transport signature of fractional Fermions in Rashba nanowires

Diego Rainis, Arijit Saha, Jelena Klinovaja (Harvard), Luka Trifunovic, and Daniel Loss. Phys. Rev. Lett. 112, 196803 (2014); arXiv:1309.3738. We theoretically study transport through a semiconducting Rashba nanowire (NW) in the presence of uniform and spatially modulated magnetic fields. The system is fully gapped, and the interplay between the spin orbit interaction and the magnetic fields leads to fractionally charged fermion (FF) bound states of the Jackiw-Rebbi type at each end of the nanowire. We investigate the transport and noise behavior of a N=NW=N system, where the wire is contacted by two normal leads (N), and we look for possible signatures that could help in the experimental detection of such states. We find that the differential conductance and the shot noise exhibit a subgap structure which fully reveals the presence of the FF state. Alternatively, another confirmation of the presence of the FFs is provided by a conductance measurement in an Aharonov-Bohm setup, where the FFs are responsible for oscillations with double period. Our predictions can be tested in InSb or InAs nanowires and are within reach of the present technology.

Structure factor of interacting one-dimensional helical systems

Suhas Gangadharaiah, Thomas L. Schmidt, and Daniel Loss. Phys. Rev. B 89, 035131 (2014); arXiv:1308.5982. We calculate the dynamical structure factor S(q, {\omega}) of a weakly interacting helical edge state in the presence of a magnetic field B. The latter opens a gap of width 2B in the single-particle spectrum, which becomes strongly nonlinear near the Dirac point. For chemical potentials |{\mu}| > B, the system then behaves as a nonlinear helical Luttinger liquid, and a mobile-impurity analysis reveals interaction-dependent power-law singularities in S(q,{\omega}). For |{\mu}| < B, the low-energy excitations are gapped, and we determine S(q,{\omega}) by using an analogy to exciton physics.

Topological Superconductivity and Majorana Fermions in RKKY Systems

Jelena Klinovaja, Peter Stano (Riken), Ali Yazdani (Princeton), and Daniel Loss. Phys. Rev. Lett. 111, 186805 (2013); arXiv:1307.1442. We consider quasi one-dimensional RKKY systems in proximity to an s-wave superconductor. We show that a $2k_F$-peak in the spin susceptibility of the superconductor in the one-dimensional limit supports helical order of localized magnetic moments via RKKY interaction, where $k_F$ is the Fermi wavevector. The magnetic helix is equivalent to a uniform magnetic field and very strong spin-orbit interaction (SOI) with an effective SOI length $1/2k_F$. We find the conditions to establish such a magnetic state in atomic chains and semiconducting nanowires with magnetic atoms or nuclear spins. Generically, these systems are in a topological phase with Majorana fermions. The inherent self-tuning of the helix to $2k_F$ eliminates the need to tune the chemical potential.

Enhanced thermal stability of the toric code through coupling to a bosonic bath

Fabio L. Pedrocchi, Adrian Hutter, James R. Wootton, and Daniel Loss. Phys. Rev. A 88, 062313 (2013); arXiv:1309.0621. We propose and study a model of a quantum memory that features self-correcting properties and a lifetime growing arbitrarily with system size at non-zero temperature. This is achieved by locally coupling a 2D L x L toric code to a 3D bath of bosons hopping on a cubic lattice. When the stabilizer operators of the toric code are coupled to the displacement operator of the bosons, we solve the model exactly via a polaron transformation and show that the energy penalty to create anyons grows linearly with L. When the stabilizer operators of the toric code are coupled to the bosonic density operator, we use perturbation theory to show that the energy penalty for anyons scales with ln(L). For a given error model, these energy penalties lead to a lifetime of the stored quantum information growing respectively exponentially and polynomially with L. Furthermore, we show how to choose an appropriate coupling scheme in order to hinder the hopping of anyons (and not only their creation) with energy barriers that are of the same order as the anyon creation gaps. We argue that a toric code coupled to a 3D Heisenberg ferromagnet realizes our model in its low-energy sector. Finally, we discuss the delicate issue of the stability of topological order in the presence of perturbations. While we do not derive a rigorous proof of topological order, we present heuristic arguments suggesting that topological order remains intact when perturbative operators acting on the toric code spins are coupled to the bosonic environment.

Low-energy properties of fractional helical Luttinger liquids

Tobias Meng, Lars Fritz (Koeln), Dirk Schuricht (Aachen), and Daniel Loss. Phys. Rev. B 89, 045111 (2014); arXiv:1308.3169. We investigate the low-energy properties of (quasi) helical and fractional helical Luttinger liquids. In particular, we calculate the Drude peak of the optical conductivity, the density of states, as well as charge transport properties of the interacting system with and without attached Fermi liquid leads at small and large (compared to the gap) frequencies. For fractional wires, we find that the low energy tunneling density of states vanishes. The conductance of a fractional helical Luttinger liquid is non-integer. It is independent of the Luttinger parameters in the wire, despite the intricate mixing of charge and spin degrees of freedom, and only depends on the relative locking of charge and spin degrees of freedom.

Circuit QED with Hole-Spin Qubits in Ge/Si Nanowire Quantum Dots

Christoph Kloeffel, Mircea Trif (UCLA), Peter Stano (RIKEN), and Daniel Loss. Phys. Rev. B 88, 241405(R) (2013); arXiv:1306.3596. We propose a setup for universal and electrically controlled quantum information processing with hole spins in Ge/Si core/shell nanowire quantum dots (NW QDs). Single-qubit gates can be driven through electric-dipole-induced spin resonance, with spin-flip times shorter than 100 ps. Long-distance qubit-qubit coupling can be mediated by the cavity electric field of a superconducting transmission line resonator, where we show that operation times below 20 ns seem feasible for the entangling square-root-of-iSWAP gate. The absence of Dresselhaus spin-orbit interaction (SOI) and the presence of an unusually strong Rashba-type SOI enable precise control over the transverse qubit coupling via an externally applied, perpendicular electric field. The latter serves as an on-off switch for quantum gates and also provides control over the g factor, so that single- and two-qubit gates can be operated independently. Remarkably, we find that idle states are insensitive to charge noise and phonons, and we discuss strategies for enhancing noise-limited gate fidelities.

Vortex Loops and Majorana Fermions

Stefano Chesi (Riken), Arthur Jaffe (Harvard), Daniel Loss, and Fabio L. Pedrocchi. J. Math. Phys. 54, 112203 (2013); arXiv:1305.6270. We investigate the role that vortex loops play in characterizing eigenstates of certain systems of half-integer spins with nearest-neighbor interaction on a trivalent lattice. In particular we focus on ground states (and other low-lying states). We test our ideas on a "spin ladder" In certain cases we show how the vortex configuration of the ground state is determined by the relative signs of the coupling constants. Two methods yield exact results: i.) We utilize the equivalence of spin Hamiltonians with quartic interactions of Majorana fermions, and analyze that fermionic Hamiltonian. ii) We use reflection positivity for Majorana fermions to characterize vortices in ground states for reflection-symmetric couplings. Two additional methods suggest potential wider applicability of these results: iii.) Numerical evidence suggests similar behavior for certain systems without reflection symmetry. iv.) A perturbative analysis also suggests similar behavior without the assumption of reflection symmetry.

Integer and Fractional Quantum Hall Effect in a Strip of Stripes

Jelena Klinovaja and Daniel Loss. Eur. Phys. J. B (2014) 87: 171; arXiv:1305.1569. We study anisotropic stripe models of interacting electrons in the presence of magnetic fields in the quantum Hall regime with integer and fractional filling factors. The model consists of an infinite strip of finite width that contains periodically arranged stripes (forming supercells) to which the electrons are confined and between which they can hop with associated magnetic phases. The interacting electron system within the one-dimensional stripes are described by Luttinger liquids and shown to give rise to charge and spin density waves that lead to periodic structures within the stripe with a reciprocal wavevector 8k_F. This wavevector gives rise to Umklapp scattering and resonant scattering that results in gaps and chiral edge states at all known integer and fractional filling factors \nu. The integer and odd denominator filling factors arise for a uniform distribution of stripes, whereas the even denominator filling factors arise for a non-uniform stripe distribution. We calculate the Hall conductance via the Streda formula and show that it is given by \sigma_H=\nu e^2/h for all filling factors. We show that the composite fermion picture follows directly from the condition of the resonant Umklapp scattering.

Magnetically-Defined Qubits on 3D Topological Insulators

Gerson J. Ferreira and Daniel Loss. Phys. Rev. Lett. 111, 106802 (2013); arXiv:1305.5003. We explore time-reversal-symmetry-breaking potentials to confine the surface states of 3D topological insulators into quantum wires and quantum dots. A magnetic domain wall on a ferromagnet insulator cap layer provides interfacial states predicted to show the quantum anomalous Hall effect (QAHE). Here we show that confinement can also occur at magnetic domain heterostructures, with states extended in the inner domain, as well as interfacial QAHE states at the surrounding domain walls. The proposed geometry allows the isolation of the wire and dot from spurious circumventing surface states. For the quantum dots, we find that highly spin-polarized quantized QAHE states at the dot edge constitute a promising candidate for quantum computing qubits.

Correlations between Majorana fermions through a superconductor

A.A. Zyuzin, Diego Rainis, Jelena Klinovaja, and Daniel Loss. Phys. Rev. Lett. 111, 056802 (2013); arXiv:1305.4187. We consider a model of ballistic quasi-one dimensional semiconducting wire with intrinsic spin-orbit interaction placed on the surface of a bulk s-wave superconductor (SC), in the presence of an external magnetic field. This setup has been shown to give rise to a topological superconducting state in the wire, characterized by a pair of Majorana-fermion (MF) bound states formed at the two ends of the wire. Here we demonstrate that, besides the well-known direct overlap-induced energy splitting, the two MF bound states may hybridize via elastic correlated tunneling processes through virtual quasiparticles states in the SC, giving rise to an additional energy splitting between MF states from the same as well as from different wires.

Spintronics in MoS_2 monolayer quantum wires

Jelena Klinovaja and Daniel Loss. Phys. Rev. B 88, 075404 (2013); arXiv:1304.4542. We study analytically and numerically spin effects in MoS_2 monolayer armchair quantum wires and quantum dots. The interplay between intrinsic and Rashba spin orbit interactions induced by an electric field leads to helical modes, giving rise to spin filtering in time-reversal invariant systems. The Rashba spin orbit interaction can also be generated by spatially varying magnetic fields. In this case, the system can be in a helical regime with nearly perfect spin polarization. If such a quantum wire is brought into proximity to an s-wave superconductor, the system can be tuned into a topological phase, resulting in midgap Majorana fermions localized at the wire ends.

Strongly anisotropic spin response as a signature of the helical regime in Rashba nanowires

Tobias Meng and Daniel Loss. Phys. Rev. B 88, 035437 (2013); arXiv:1303.6994. Rashba nanowires in a magnetic field exhibit a helical regime when the spin-orbit momentum is close to the Fermi momentum, k_F \approx k_{SO}. We show that this regime is characterized by a strongly anisotropic electron spin susceptibility, with an exponentially suppressed signal along one direction in spin space, and that there are no low frequency spin fluctuations along this direction. Since the spin response in the gapless regime k_F \not \approx k_{SO} has a power law behavior in all three directions, spin measurements provide a signature of the helical regime that complements spin-insensitive conductance measurements.

Helical nuclear spin order in two-subband quantum wires

Tobias Meng and Daniel Loss. Phys. Rev. B 87, 235427 (2013); arXiv:1303.1542. In quantum wires, the hyperfine coupling between conduction electrons and nuclear spins can lead to a (partial) ordering of both of them at low temperatures. By an interaction-enhanced mechanism, the nuclear spin order, caused by RKKY exchange, acts back onto the electrons and gaps out part of their spectrum. In wires with two subbands characterized by distinct Fermi momenta kF1 and kF2, the nuclear spins form a superposition of two helices with pitches {\pi}/kF1 and {\pi}/kF2, thus exhibiting a beating pattern. This order results in a reduction of the electronic conductance in two steps upon lowering the temperature.

Local Spin Susceptibilities of Low-Dimensional Electron Systems

Peter Stano, Jelena Klinovaja, Amir Yacoby (Harvard), and Daniel Loss. Phys. Rev. B 88, 045441 (2013); arXiv:1303.1151. We investigate, assess, and suggest possibilities for a measurement of the local spin susceptibility of a conducting low-dimensional electron system. The basic setup of the experiment we envisage is a source-probe one. Locally induced spin density (e.g. by a magnetized atomic force microscope tip) extends in the medium according to its spin susceptibility. The induced magnetization can be detected as a dipolar magnetic field, for instance, by an ultra-sensitive nitrogen-vacancy center based detector, from which the spatial structure of the spin susceptibility can be deduced. We find that one-dimensional systems, such as semiconducting nanowires or carbon nanotubes, are expected to yield a measurable signal. The signal in a two-dimensional electron gas is weaker, though materials with high enough $g$-factor (such as InGaAs) seem promising for successful measurements.

Topological Edge States and Fractional Quantum Hall Effect from Umklapp Scattering

Jelena Klinovaja and Daniel Loss. Phys. Rev. Lett. 111, 196401 (2013); arXiv:1302.6132. We study anisotropic lattice strips in the presence of a magnetic field in the quantum Hall effect regime. At specific magnetic fields, causing resonant Umklapp scattering, the system is gapped in the bulk and supports chiral edge states in close analogy to topological insulators. These gaps result in plateaus for the Hall conductivity exactly at the known fillings n/m (both positive integers and m odd) for the integer and fractional quantum Hall effect. For double strips we find topological phase transitions with phases that support midgap edge states with flat dispersion. The topological effects predicted here could be tested directly in optical lattices.

Tunable g factor and phonon-mediated hole spin relaxation in Ge/Si nanowire quantum dots

Franziska Maier, Christoph Kloeffel, and Daniel Loss. Phys. Rev. B 87, 161305(R) (2013); arXiv:1302.5027. We theoretically consider g factor and spin lifetimes of holes in a longitudinal Ge/Si core/shell nanowire quantum dot that is exposed to external magnetic and electric fields. For the ground states, we find a large anisotropy of the g factor which is highly tunable by applying electric fields. This tunability depends strongly on the direction of the electric field with respect to the magnetic field. We calculate the single-phonon hole spin relaxation times T1 for zero and small electric fields and propose an optimal setup in which very large T1 of the order of tens of milliseconds can be reached. Increasing the relative shell thickness or the longitudinal confinement length prolongs T1 further. In the absence of electric fields, the dephasing vanishes and the decoherence time T2 is determined by T2 = 2 T1.

Dynamic Generation of Topologically Protected Self-Correcting Quantum Memory

Daniel Becker, Tetsufumi Tanamoto (Toshiba), Adrian Hutter, Fabio L. Pedrocchi, and Daniel Loss. Phys. Rev. A 87, 042340 (2013); arXiv:1302.3998. We propose a scheme to dynamically realize a thermally stable quantum memory based on the toric code. The code is generated from qubit systems with typical two-body interactions (Ising, XY, Heisenberg) using periodic, NMR-like, pulse sequences. It allows one to encode the logical qubits without measurements and to protect them dynamically against the time evolution of the physical qubits. Thermal stability is achieved by weakly coupling the qubits to additional cavity modes that mediate long-range attractive interactions between the stabilizer operators of the toric code. We investigate how the fidelity, with which the toric code is realized, depends on the period length T of the pulse sequence and the magnitude of possible pulse errors. We derive an optimal period T_opt that maximizes the fidelity.

An efficient decoding algorithm for stabilizer codes

Adrian Hutter, James R. Wootton, and Daniel Loss. Phys. Rev. A 89, 022326 (2014); arXiv:1302.2669. To date, the best classical algorithm for performing error correction in the surface code has been minimum-weight perfect matching. However, in this work we present a Markov chain Monte Carlo algorithm that achieves significantly lower logical error rates. It therefore allows any target logical error rate to be obtained using a significantly smaller code. This increase in performance does come at the cost of an increased runtime complexity, but only by a polynomial factor $O(L^\eps)$ for $\eps<2$. Our algorithm is based on an analytically exact rewriting of the probability of each logical equivalence class, which also suggests that for arbitrary stabilizer codes error correction can be performed to arbitrary accuracy in a runtime $O(\m{poly}(L))$. It is applicable to any stabilizer code, allows for parallelization, and can be used to correct in the case of imperfect stabilizer measurements.

Long-Range Interaction of Spin-Qubits via Ferromagnets

Luka Trifunovic, Fabio L. Pedrocchi, and Daniel Loss. Phys. Rev. X 3, 041023 (2013); arXiv:1302.4017. We propose a mechanism of long-range coherent coupling between spins coupled to a ferromagnet by exchnage or dipolar coupling. An effective two-spin interaction Hamiltonian is derived and the coupling strength is estimated. We also discuss mechanisms of decoherence and consider possibilities for gate control of the interaction between neighboring spin-qubits. The resulting quantum computing architecture retains all the single qubit gates and measurement aspects of earlier approaches, but allows qubit spacing at distances of order 1$\,\mu$m for two-qubit gates, achievable with current semiconductor technology. The clock speed depends strongly on the dimensionality of the ferromagnet and is between MHz and GHz.

Fractional Fermions with Non-Abelian Statistics

Jelena Klinovaja and Daniel Loss. Phys. Rev. Lett. 110, 126402 (2013); arXiv:1301.5822. We introduce a novel class of low-dimensional topological tight-binding models that allow for bound states that are fractionally charged fermions and exhibit non-Abelian braiding statistics. The proposed model consists of a double (single) ladder of spinless (spinful) fermions in the presence of magnetic fields. We study the system analytically in the continuum limit as well as numerically in the tight-binding representation. We find a topological phase transition with a topological gap that closes and reopens as a function of system parameters and chemical potential. The topological phase is of the type BDI and carries two degenerate mid-gap bound states that are localized at opposite ends of the ladders. We show numerically that these bound states are robust against a wide class of perturbations.

RKKY interaction in carbon nanotubes and graphene nanoribbons

Jelena Klinovaja and Daniel Loss. Phys. Rev. B 87, 045422 (2013); arXiv:1211.3067. We study Rudermann-Kittel-Kasuya-Yosida (RKKY) interaction in carbon nanotubes (CNTs) and graphene nanoribbons in the presence of spin-orbit interactions and magnetic fields. For this, we evaluate the static spin susceptibility tensor in real space in various regimes at zero temperature. In metallic CNTs, the RKKY interaction depends strongly on the sublattice and, at the Dirac point, is purely ferromagnetic (antiferromagnetic) for the localized spins on the same (different) sublattice, whereas in semiconducting CNTs, the spin susceptibility depends only weakly on the sublattice and is dominantly ferromagnetic. The spin-orbit interactions break the SU(2) spin symmetry of the system, leading to an anisotropic RKKY interaction of Ising and Dzyaloshinskii-Moriya form, aside from the usual isotropic Heisenberg interaction. All these RKKY terms can be made of comparable magnitude by tuning the Fermi level close to the gap induced by the spin-orbit interaction. We further calculate the spin susceptibility also at finite frequencies and thereby obtain the spin noise in real space via the fluctuation-dissipation theorem.

Giant spin orbit interaction due to rotating magnetic fields in graphene nanoribbons

Jelena Klinovaja and Daniel Loss. Phys. Rev. X 3, 011008 (2013); arXiv:1211.2739. We theoretically study graphene nanoribbons in the presence of spatially varying magnetic fields produced e.g. by nanomagnets. We show both analytically and numerically that an exceptionally large Rashba spin orbit interaction (SOI) of the order of 10 meV can be produced by the non-uniform magnetic field. As a consequence, helical modes exist in armchair nanoribbons that exhibit nearly perfect spin polarization and are robust against boundary defects. This paves the way to realizing spin filter devices in graphene nanoribbons in the temperature regime of a few Kelvins. If a nanoribbon in the helical regime is in proximity contact to an s-wave superconductor, the nanoribbon can be tuned into a topological phase sustaining Majorana fermions.

Effective quantum-memory Hamiltonian from local two-body interactions

Adrian Hutter, Fabio L. Pedrocchi, James R. Wootton, and Daniel Loss. Phys. Rev. A 90, 012321 (2014); arXiv:1209.5289. In Phys. Rev. A 88, 062313 (2013) we proposed and studied a model for a self-correcting quantum memory in which the energetic cost for introducing a defect in the memory grows without bounds as a function of system size. This positive behavior is due to attractive long-range interactions mediated by a bosonic field to which the memory is coupled. The crucial ingredients for the implementation of such a memory are the physical realization of the bosonic field as well as local five-body interactions between the stabilizer operators of the memory and the bosonic field. Here, we show that both of these ingredients appear in a low-energy effective theory of a Hamiltonian that involves only two-body interactions between neighboring spins. In particular, we consider the low-energy, long-wavelength excitations of an ordered Heisenberg ferromagnet (magnons) as a realization of the bosonic field. Furthermore, we present perturbative gadgets for generating the required five-spin operators. Our Hamiltonian involving only local two-body interactions is thus expected to exhibit self-correcting properties as long as the noise affecting it is in the regime where the effective low-energy description remains valid.

Ultrafast magnon-transistor at room temperature

Kevin A. van Hoogdalem and Daniel Loss. Phys. Rev. B 88, 024420 (2013); arXiv:1209.5594; See News and Views, SPINTRONICS: An insulator-based transistor, by Yaroslav Tserkovnyak; Nature Nanotechnology 8, 706 (2013). We study sequential tunneling of magnetic excitations in nonitinerant systems (either magnons or spinons) through triangular molecular magnets. It is known that the quantum state of such molecular magnets can be controlled by application of an electric- or a magnetic field. Here, we use this fact to control the flow of a spin current through the molecular magnet by electric- or magnetic means. This allows us to design a system that behaves as a magnon-transistor. We show how to combine three magnon-transistors to form a NAND-gate, and give several possible realizations of the latter, one of which could function at room temperature using transistors with a 11 ns switching time.

Magnetic texture-induced thermal Hall effects

Kevin A. van Hoogdalem, Yaroslav Tserkovnyak (UCLA), and Daniel Loss. Phys. Rev. B 87, 024402 (2013); arXiv:1208.1646. Magnetic excitations in ferromagnetic systems with a noncollinear ground state magnetization experience a fictitious magnetic field due to the equilibrium magnetic texture. Here, we investigate how such fictitious fields lead to thermal Hall effects in two-dimensional insulating magnets in which the magnetic texture is caused by spin-orbit interaction. We find that, besides the well-known geometric texture contribution to the fictitious magnetic field in such systems, there exists also an equally important but often neglected contribution due to the original spin-orbit term in the free energy. We consider the different possible ground states in the phase diagram of a two-dimensional ferromagnet with spin-orbit interaction: The spiral state and the skyrmion lattice, and find that thermal Hall effects can occur in certain domain walls as well as the skyrmion lattice.

Helical States in Curved Bilayer Graphene

Jelena Klinovaja, Gerson J. Ferreira, and Daniel Loss. Phys. Rev. B 86, 235416 (2012); arXiv:1208.2601. We study spin effects of quantum wires formed in bilayer graphene by electrostatic confinement. With a proper choice of the confinement direction, we show that in the presence of magnetic field, spin-orbit interaction induced by curvature, and intervalley scattering, bound states emerge that are helical. The localization length of these helical states can be modulated by the gate voltage which enables the control of the tunnel coupling between two parallel wires. Allowing for proximity effect via an s-wave superconductor, we show that the helical modes give rise to Majorana fermions in bilayer graphene.

Transition from fractional to Majorana fermions in Rashba nanowires

Jelena Klinovaja, Peter Stano, and Daniel Loss. Phys. Rev. Lett. 109, 236801 (2012); arXiv:1207.7322. We study hybrid superconducting-semiconducting nanowires in the presence of Rashba spin-orbit interaction as well as helical magnetic fields. We show that the interplay between them leads to a competition of phases with two topological gaps closing and reopening, resulting in unexpected reentrance behavior. Besides the topological phase with localized Majorana fermions (MFs) we find new phases characterized by fractionally charged fermion (FF) bound states of Jackiw-Rebbi type. The system can be fully gapped by the magnetic fields alone, giving rise to FFs that transmute into MFs upon turning on superconductivity. We find explicit analytical solutions for MF and FF bound states and determine the phase diagram numerically by determining the corresponding Wronskian null space. We show by renormalization group arguments that electron-electron interactions enhance the Zeeman gaps opened by the fields.

Realistic transport modeling for a superconducting nanowire with Majorana fermions

Diego Rainis, Luka Trifunovic, Jelena Klinovaja, and Daniel Loss. Phys. Rev. B 87, 024515 (2013); arXiv:1207.5907. Motivated by recent experiments searching for Majorana fermions (MFs) in hybrid semiconducting-superconducting nanostructures, we consider a realistic tight-binding model and analyze its transport behavior numerically. In particular, we take into account the presence of a superconducting contact, used in real experiments to extract the current, which is usually not included in theoretical calculations. We show that important features emerge that are absent in simpler models, such as the shift in energy of the proximity gap signal, and the enhanced visibility of the topological gap for increased spin-orbit interaction. We find oscillations of the zero bias peak as a function of the magnetic field and study them analytically. We argue that many of the experimentally observed features hint at an actual spin-orbit interaction larger than the one typically assumed. However, even taking into account all the known ingredients of the experiments and exploring many parameter regimes for MFs, we are not able to reach full agreement with the reported data. Thus, a different physical origin for the observed zero-bias peak cannot be excluded.

Exchange-based CNOT gates for singlet-triplet qubits with spin orbit interaction

Jelena Klinovaja, Dimitrije Stepanenko, Bertrand I. Halperin (Harvard), and Daniel Loss. Phys. Rev. B 86, 085423 (2012); arXiv:1206.2579. We propose a scheme for implementing the CNOT gate over qubits encoded in a pair of electron spins in a double quantum dot. The scheme is based on exchange and spin orbit interactions and on local gradients in Zeeman fields. We find that the optimal device geometry for this implementation involves effective magnetic fields that are parallel to the symmetry axis of the spin orbit interaction. We show that the switching times for the CNOT gate can be as fast as a few nanoseconds for realistic parameter values in GaAs semiconductors. Guided by recent advances in surface codes, we also consider the perpendicular geometry. In this case, leakage errors due to spin orbit interaction occur but can be suppressed in strong magnetic fields.

Self-correcting quantum memory with a boundary

Adrian Hutter, James R. Wootton, Beat Roethlisberger, and Daniel Loss. Phys. Rev. A 86, 052340 (2012); arXiv:1206.0991. We study the two-dimensional toric-code Hamiltonian with effective long-range interactions between its anyonic excitations induced by coupling the toric code to external fields. It has been shown that such interactions allow an arbitrary increase in the lifetime of the stored quantum information by making L, the linear size of the memory, larger [ Chesi et al. Phys. Rev. A 82 022305 (2010)]. We show that for these systems the choice of boundary conditions (open boundaries as opposed to periodic boundary conditions) is not a mere technicality; the influence of anyons produced at the boundaries becomes in fact dominant for large enough L. This influence can be either beneficial or detrimental. In particular, we study an effective Hamiltonian proposed by Pedrocchi et al. [ Phys. Rev. B 83 115415 (2011)] that describes repulsion between anyons and anyon holes. For this system, we find a lifetime of the stored quantum information that grows exponentially in L2 for both periodic and open boundary conditions, although the exponent in the latter case is found to be less favorable. However, L is upper bounded through the breakdown of the perturbative treatment of the underlying Hamiltonian.

Decoherence of Majorana qubits by noisy gates

Manuel J. Schmidt (Aachen), Diego Rainis, and Daniel Loss. Phys. Rev. B 86, 085414 (2012); arXiv:1206.0743. We propose and study a realistic model for the decoherence of topological qubits, based on Majorana fermions in one-dimensional topological superconductors. The source of decoherence is the fluctuating charge on a capacitively coupled gate, modeled by non-interacting electrons. In this context, we clarify the role of quantum fluctuations and thermal fluctuations and find that quantum fluctuations do not lead to decoherence, while thermal fluctuations do. We explicitly calculate decay times due to thermal noise and give conditions for the gap size in the topological superconductor and the gate temperature. Based on this result, we provide simple rules for gate geometries and materials optimized for reducing the negative effect of thermal charge fluctuations on the gate.

Composite Majorana Fermion Wavefunctions in Nanowires

Jelena Klinovaja and Daniel Loss. Phys. Rev. B 86, 085408 (2012); arXiv:1205.7054. We consider Majorana fermions (MFs) in quasi-one-dimensional nanowire systems containing normal and superconducting sections where the topological phase based on Rashba spin-orbit interaction can be tuned by magnetic fields. We derive explicit analytic solutions of the MF wave function in the weak and strong spin orbit interaction regimes. We find that the wave function for one single MF is a composite object formed by superpositions of different MF wave functions which have nearly disjoint supports in momentum space. These contributions are coming from the extrema of the spectrum, one centered around zero momentum and the other around the two Fermi points. As a result, the various MF wave functions have different localization lengths in real space and interference among them leads to pronounced oscillations of the MF probability density. For a transparent normal-superconducting junction we find that in the topological phase the MF leaks out from the superconducting into the normal section of the wire and is delocalized over the entire normal section, in agreement with numerical results obtained in previous studies.

Hyperfine-induced decoherence in triangular spin-cluster qubits

Filippo Troiani (Modena), Dimitrije Stepanenko, and Daniel Loss. Phys. Rev. B 86, 161409 (2012); arXiv:1205.5629. We investigate hyperfine-induced decoherence in a triangular spin-cluster for different qubit encodings. Electrically controllable eigenstates of spin chirality (C_z) show decoherence times that approach milliseconds, two orders of magnitude longer than those estimated for the eigenstates of the total spin projection (S_z) and of the partial spin sum (S_{12}). The robustness of chirality is due to its decoupling from both the total- and individual-spin components in the cluster. This results in a suppression of the effective interaction between C_z and the nuclear spin bath.

Prospects for Spin-Based Quantum Computing

Christoph Kloeffel and Daniel Loss. Annu. Rev. Condens. Matter Phys. 4, 51 (2013); arXiv:1204.5917. Experimental and theoretical progress toward quantum computation with spins in quantum dots (QDs) is reviewed, with particular focus on QDs formed in GaAs heterostructures, on nanowire-based QDs, and on self-assembled QDs. We report on a remarkable evolution of the field, where decoherence—one of the main challenges for realizing quantum computers—no longer seems to be the stumbling block it had originally been considered. General concepts, relevant quantities, and basic requirements for spin-based quantum computing are explained; opportunities and challenges of spin-orbit interaction and nuclear spins are reviewed. We discuss recent achievements, present current theoretical proposals, and make several suggestions for further experiments.

Cotunneling in the 5/2 fractional quantum Hall regime

Robert Zielke, Bernd Braunecker, and Daniel Loss. Phys. Rev. B 86, 235307 (2012); arXiv:1204.4400. We show that cotunneling in the 5/2 fractional quantum Hall regime allows to test the Moore-Read wave function, proposed for this regime, and to probe the nature of the fractional charge carriers. We calculate the cotunneling current for electrons that tunnel between two quantum Hall edge states via a quantum dot and for quasiparticles with fractional charges e/4 and e/2 that tunnel via an antidot. While electron cotunneling is strongly suppressed, the quasiparticle tunneling shows signatures characteristic for the Moore-Read state. For comparison, we also consider cotunneling between Laughlin states, and find that electron-transport between Moore-Read states and the one between Laughlin states at filling factor 1/3 have identical voltage dependences.

Non-abelian Majoranas and braiding in inhomogeneous spin ladders

Fabio L. Pedrocchi, Stefano Chesi (Montreal), Suhas Gangadharaiah (Bhopal), and Daniel Loss. Phys. Rev. B 86, 205412 (2012); arXiv:1204.3044. We propose an inhomogeneous open spin ladder, related to the Kitaev honeycomb model, which can be tuned between topological and nontopological phases. In extension of Lieb's theorem, we show numerically that the ground state of the spin ladder is either vortex free or vortex full. We study the robustness of Majorana end states (MES) which emerge at the boundary between sections in different topological phases and show that while the MES in the homogeneous ladder are destroyed by single-body perturbations, in the presence of inhomogeneities at least two-body perturbations are required to destabilize MES. Furthermore, we prove that x, y, or z inhomogeneous magnetic fields are not able to destroy the topological degeneracy. Finally, we present a trijunction setup where MES can be braided. A network of such spin ladders provides thus a promising platform for realization and manipulation of MES.

Majorana qubit decoherence by quasiparticle poisoning

Diego Rainis and Daniel Loss. Phys. Rev. B 85, 174533 (2012); arXiv:1204.3326. We consider the problem of quasiparticle poisoning in a nanowire-based realization of a Majorana qubit, where a spin-orbit-coupled semiconducting wire is placed on top of a (bulk) superconductor. By making use of recent experimental data exhibiting evidence of a low-temperature residual nonequilibrium quasiparticle population in superconductors, we show by means of analytical and numerical calculations that the dephasing time due to the tunneling of quasiparticles into the nanowire may be problematically short to allow for qubit manipulation.

Effect of strain on hyperfine-induced hole-spin decoherence in quantum dots

Franziska Maier and Daniel Loss. Phys. Rev. B 85, 195323 (2012); arXiv:1203.3876. We theoretically consider the effect of strain on the spin dynamics of a single heavy hole (HH) confined to a self-assembled quantum dot and interacting with the surrounding nuclei via hyperfine interaction. Confinement and strain hybridize the HH states, which show an exponential decay for a narrowed nuclear spin bath. For different strain configurations within the dot, the dependence of the spin decoherence time T2 on external parameters is shifted and the nonmonotonic dependence of the peak is altered. Application of external strain yields considerable shifts in the dependence of T2 on external parameters. We find that external strain affects mostly the effective hyperfine coupling strength of the conduction band (CB), indicating that the CB admixture of the hybridized HH states plays a crucial role in the sensitivity of T2 on strain.

Frequency dependent transport through a spin chain

Kevin A. van Hoogdalem and Daniel Loss. Phys. Rev. B 85, 054413 (2012); arXiv:1111.4803. Motivated by potential applications in spintronics, we study frequency dependent spin transport in nonitinerant one-dimensional spin chains. We propose a system that behaves as a capacitor for the spin degree of freedom. It consists of a spin chain with two impurities a distance $d$ apart. We find that at low energy (frequency) the impurities flow to strong coupling, thereby effectively cutting the chain into three parts, with the middle island containing a discrete number of spin excitations. At finite frequency spin transport through the system increases. We find a strong dependence of the finite frequency characteristics both on the anisotropy of the spin chain and the applied magnetic field. We propose a method to measure the finite-frequency conductance in this system.

High threshold error correction for the surface code

James R. Wootton and Daniel Loss. Phys. Rev. Lett. 109, 160503 (2012); arXiv:1202.4316. An algorithm is presented for error correction in the surface code quantum memory. This is shown to correct depolarizing noise up to a threshold error rate of 18.5%, exceeding previous results and coming close to the upper bound of 18.9%. The time complexity of the algorithm is found to be sub-exponential, offering a significant speed-up over brute force methods and allowing efficient error correction for codes of realistic sizes.

Electric-Field Induced Majorana Fermions in Armchair Carbon Nanotubes

Jelena Klinovaja, Suhas Gangadharaiah, and Daniel Loss. Phys. Rev. Lett. 108, 196804 (2012); arXiv:1201.0159. We consider theoretically an armchair carbon nanotube (CNT) in the presence of an electric field and in contact with an s-wave superconductor. We show that the proximity effect opens up superconducting gaps in the CNT of different strengths for the exterior and interior branches of the two Dirac points. For strong proximity induced superconductivity the interior gap can be of the p-wave type, while the exterior gap can be tuned by the electric field to be of the s-wave type. Such a setup supports a single Majorana bound state at each end of the CNT. In the case of a weak proximity induced superconductivity, the gaps in both branches are of the p-wave type. However, the temperature can be chosen in such a way that the smallest gap is effectively closed. Using renormalization group techniques we show that the Majorana bound states exist even after taking into account electron-electron interactions.

Thin-Film Magnetization Dynamics on the Surface of a Topological Insulator

Yaroslav Tserkovnyak (UCLA) and Daniel Loss. Phys. Rev. Lett. 108, 187201 (2012); arXiv:1112.5884. We theoretically study the magnetization dynamics of a thin ferromagnetic film exchange-coupled with a surface of a strong three-dimensional topological insulator. We focus on the role of electronic zero modes associated with domain walls (DW's) and other topological textures in the magnetic film. Thermodynamically reciprocal hydrodynamic equations of motion are derived for the DW responding to electronic spin torques, on the one hand, and fictitious electromotive forces in the electronic chiral mode fomented by the DW, on the other. An experimental realization illustrating this physics is proposed based on a ferromagnetic strip, which cuts the topological insulator surface into two gapless regions. In the presence of a ferromagnetic DW, a chiral mode transverse to the magnetic strip acts as a dissipative interconnect, which is itself a dynamic object that controls (and, inversely, responds to) the magnetization dynamics.

Ferromagnetic order of nuclear spins coupled to conduction electrons: a combined effect of the electron-electron and spin-orbit interactions

Robert Andrzej Zak, Dmitrii L. Maslov (Gainesville), and Daniel Loss. Phys. Rev. B 85, 115424 (2012); viewpoint; arXiv:1112.4786. We analyze the ordered state of nuclear spins embedded in an interacting two-dimensional electron gas (2DEG) with Rashba spin-orbit interaction (SOI). Stability of the ferromagnetic nuclear-spin phase is governed by nonanalytic dependences of the electron spin susceptibility $\chi^{ij}$ on the momentum ($\tilde{\mathbf{q}}$) and on the SOI coupling constant ($\alpha$). The uniform ($\tq=0$) spin susceptibility is anisotropic (with the out-of-plane component, $\chi^{zz}$, being larger than the in-plane one, $\chi^{xx}$, by a term proportional to $U^2(2k_F)|\alpha|$, where $U(q)$ is the electron-electron interaction). For $\tq \leq 2m^*|\alpha|$, corrections to the leading, $U^2(2k_F)|\alpha|$, term scale linearly with $\tq$ for $\chi^{xx}$ and are absent for $\chi^{zz}$. This anisotropy has important consequences for the ferromagnetic nuclear-spin phase: $(i)$ the ordered state--if achieved--is of an Ising type and $(ii)$ the spin-wave dispersion is gapped at $\tq=0$. To second order in $U(q)$, the dispersion a decreasing function of $\tq$, and anisotropy is not sufficient to stabilize long-range order. However, renormalization in the Cooper channel for $\tq\ll2m^*|\alpha|$ is capable of reversing the sign of the $\tq$-dependence of $\chi^{xx}$ and thus stabilizing the ordered state. We also show that a combination of the electron-electron and SO interactions leads to a new effect: long-wavelength Friedel oscillations in the spin (but not charge) electron density induced by local magnetic moments. The period of these oscillations is given by the SO length $\pi/m^*|\alpha|$.

Singlet-triplet splitting in double quantum dots due to spin orbit and hyperfine interactions

Dimitrije Stepanenko, Mark Rudner (Harvard), Bertrand I. Halperin (Harvard), and Daniel Loss. Phys. Rev. B 85, 075416 (2012); arXiv:1112.1644. We analyze the low-energy spectrum of a two-electron double quantum dot under a potential bias in the presence of an external magnetic field. We focus on the regime of spin blockade, taking into account the spin-orbit interaction and hyperfine coupling of electron and nuclear spins. Starting from a model for two interacting electrons in a double dot, we derive an effective two-level Hamiltonian in the vicinity of an avoided crossing between singlet and triplet levels, which are coupled by the spin-orbit and hyperfine interactions. We evaluate the level splitting at the anticrossing, and show that it depends on a variety of parameters including the spin-orbit coupling strength, the orientation of the external magnetic field relative to an internal spin-orbit axis, the potential detuning of the dots, and the difference between hyperfine fields in the two dots. We provide a formula for the splitting in terms of the spin-orbit length, the hyperfine fields in the two dots, and the double dot parameters such as tunnel coupling and Coulomb energy. This formula should prove useful for extracting spin-orbit parameters from transport or charge sensing experiments in such systems. We identify a parameter regime where the spin-orbit and hyperfine terms can become of comparable strength, and discuss how this regime might be reached.

Incoherent dynamics in the toric code subject to disorder

Beat Roethlisberger, James R. Wootton, Robert M. Heath (Leeds), Jiannis K. Pachos (Leeds), and Daniel Loss. Phys. Rev. A 85, 022313 (2012); arXiv:1112.1613. We numerically study the effects of two forms of quenched disorder on the anyons of the toric code. Firstly, a new class of codes based on random lattices of stabilizer operators is presented, and shown to be superior to the standard square lattice toric code for certain forms of biased noise. It is further argued that these codes are close to optimal, in that they tightly reach the upper bound of error thresholds beyond which no correctable CSS codes can exist. Additionally, we study the classical motion of anyons in toric codes with randomly distributed onsite potentials. In the presence of repulsive long-range interaction between the anyons, a surprising increase with disorder strength of the lifetime of encoded states is reported and explained by an entirely incoherent mechanism. Finally, the coherent transport of the anyons in the presence of both forms of disorder is investigated, and a significant suppression of the anyon motion is found.

Localized end states in density modulated quantum wires and rings

Suhas Gangadharaiah, Luka Trifunovic, and Daniel Loss. Phys. Rev. Lett. 108, 136803 (2012); arXiv:1111.5273. We study finite quantum wires and rings in the presence of a charge-density wave gap induced by a periodic modulation of the chemical potential. We show that the Tamm-Shockley bound states emerging at the ends of the wire are stable against weak disorder and interactions, for discrete open chains and for continuum systems. The low-energy physics can be mapped onto the Jackiw-Rebbi equations describing massive Dirac fermions and bound end states. We treat interactions via the continuum model and show that they increase the charge gap and further localize the end states. The electrons placed in the two localized states on the opposite ends of the wire can interact via exchange interactions and this setup can be used as a double quantum dot hosting spin qubits. The existence of these states could be experimentally detected through the presence of an unusual 4\pi Aharonov-Bohm periodicity in the spectrum and persistent current as a function of the external flux.

Rashba spin orbit interaction in a quantum wire superlattice

Gunnar Thorgilsson (Reykjavik), J. Carlos Egues (Sao Carlos), Daniel Loss, and Sigurdur I. Erlingsson (Reykjavik). Phys. Rev. B 85, 045306 (2012); Phys. Rev. B 85, 039904(E) (2012); arXiv:1111.1534. In this work we study the effects of a longitudinal periodic potential on a parabolic quantum wire defined in a two-dimensional electron gas with Rashba spin-orbit interaction. For an infinite wire superlattice we find, by direct diagonalization, that the energy gaps are shifted away from the usual Bragg planes due to the Rashba spin-orbit interaction. Interestingly, our results show that the location of the band gaps in energy can be controlled via the strength of the Rashba spin-orbit interaction. We have also calculated the charge conductance through a periodic potential of a finite length via the non-equilibrium Green's function method combined with the Landauer formalism. We find dips in the conductance that correspond well to the energy gaps of the infinite wire superlattice. From the infinite wire energy dispersion, we derive an equation relating the location of the conductance dips as a function of the (gate controllable) Fermi energy to the Rashba spin-orbit coupling strength. We propose that the strength of the Rashba spin-orbit interaction can be extracted via a charge conductance measurement.

Long-distance spin-spin coupling via floating gates

Luka Trifunovic, Oliver Dial (Harvard), Mircea Trif, James R. Wootton, Rediet Abebe (Harvard), Amir Yacoby (Harvard), and Daniel Loss. Synopsis; Phys. Rev. X 2, 011006 (2012); arXiv:1110.1342. The electron spin is a natural two level system that allows a qubit to be encoded. When localized in a gate defined quantum dot, the electron spin provides a promising platform for a future functional quantum computer. The essential ingredient of any quantum computer is entanglement---between electron spin qubits---commonly achieved via the exchange interaction. Nevertheless, there is an immense challenge as to how to scale the system up to include many qubits. Here we propose a novel architecture of a large scale quantum computer based on a realization of long-distance quantum gates between electron spins localized in quantum dots. The crucial ingredients of such a long-distance coupling are floating metallic gates that mediate electrostatic coupling over large distances. We show, both analytically and numerically, that distant electron spins in an array of quantum dots can be coupled sel​ectively, with coupling strengths that are larger than the electron spin decay and with switching times on the order of nanoseconds.

Crossed Andreev Reflection in Quantum Wires with Strong Spin-Orbit Interaction

Koji Sato (UCLA), Daniel Loss, and Yaroslav Tserkovnyak (UCLA). Phys. Rev. B 85, 235433 (2012); arXiv:1109.6357. We theoretically study tunneling of Cooper pairs from an s-wave superconductor into two semiconductor quantum wires with strong spin-orbit interaction under magnetic field, which approximate helical Luttinger liquids. The entanglement of electrons within a Cooper pair can be detected by the electric current cross correlations in the wires. By controlling the relative orientation of the wires, either lithographically or mechanically, on the substrate, the current correlations can be tuned, as dictated by the initial spin entanglement. This proposal of a spin-to-charge readout of quantum correlations is alternative to a recently proposed utilization of the quantum spin Hall insulator.

Strong Spin-Orbit Interaction and Helical Hole States in Ge/Si Nanowires

Christoph Kloeffel, Mircea Trif, and Daniel Loss. Phys. Rev. B 84, 195314 (2011); arXiv:1107.4870. We study theoretically the low-energy hole states of Ge/Si core/shell nanowires. The low-energy valence band is quasi-degenerate, formed by two doublets of different orbital angular momentum, and can be controlled via the relative shell thickness and via external fields. We find that direct (dipolar) coupling to a moderate electric field leads to an unusually large spin-orbit interaction of Rashba-type on the order of meV which gives rise to pronounced helical states enabling electrical spin-control. The system allows for quantum dots and spin-qubits with energy levels that can vary from nearly zero to several meV, depending on the relative shell thickness.

libCreme: An optimization library for evaluating convex-roof entanglement measures

Beat Roethlisberger, Joerg Lehmann (ABB Baden), and Daniel Loss. Comput. Phys. Comm. 183, 155 (2012); arXiv:1107.4497. We present the software library libCreme which we have previously used to successfully calculate convex-roof entanglement measures of mixed quantum states appearing in realistic physical systems. Evaluating the amount of entanglement in such states is in general a non-trivial task requiring to solve a highly non-linear complex optimization problem. The algorithms provided here are able to achieve to do this for a large and important class of entanglement measures. The library is mostly written in the Matlab programming language, but is fully compatible to the free and open-source Octave platform. Some inefficient subroutines are written in C/C++ for better performance. This manuscript discusses the most important theoretical concepts and workings of the algorithms, focussing on the actual implementation and usage within the library. Detailed examples in the end should make it easy for the user to apply libCreme to specific problems.

Absence of spontaneous magnetic order of lattice spins coupled to itinerant interacting electrons in one and two dimensions

Daniel Loss, Fabio L. Pedrocchi, and Anthony J. Leggett (Urbana). Phys. Rev. Lett. 107, 107201 (2011); arXiv:1107.1223; Synopsis. We extend the Mermin-Wagner theorem to a system of lattice spins which are spin coupled to itinerant and interacting charge carriers. We use the Bogoliubov inequality to rigorously prove that neither (anti-) ferromagnetic nor helical long-range order is possible in one and two dimensions at any finite temperature. Our proof applies to a wide class of models including any form of electron-electron and single-electron interactions that are independent of spin. In the presence of Rashba or Dresselhaus spin-orbit interactions (SOI) magnetic order is not excluded and intimately connected to equilibrium spin currents. However, in the special case when Rashba and Dresselhaus SOIs are tuned to be equal, magnetic order is excluded again. This opens up a new possibility to control magnetism electrically.

Carbon nanotubes in electric and magnetic fields

Jelena Klinovaja, Manuel J. Schmidt, Bernd Braunecker, and Daniel Loss. Phys. Rev. B 84, 085452 (2011); arXiv:1106.3332. We derive an effective low-energy theory for metallic (armchair and nonarmchair) single-wall nanotubes in the presence of an electric field perpendicular to the nanotube axis, and in the presence of magnetic fields, taking into account spin-orbit interactions and screening effects on the basis of a microscopic tight-binding model. The interplay between electric field and spin-orbit interaction allows us to tune armchair nanotubes into a helical conductor in both Dirac valleys. Metallic nonarmchair nanotubes are gapped by the surface curvature, yet helical conduction modes can be restored in one of the valleys by a magnetic field along the nanotube axis. Furthermore, we discuss electric dipole spin resonance in carbon nanotubes, and find that the Rabi frequency shows a pronounced dependence on the momentum along the nanotube.

Low Bias Negative Differential Resistance in Graphene Nanoribbon Superlattices

Gerson J. Ferreira (S. Carlos), Michael N. Leuenberger (Orlando), Daniel Loss, and J. Carlos Egues (S. Carlos). Phys. Rev. B 84, 125453 (2011); arXiv:1105.4850. We theoretically investigate negative differential resistance (NDR) for ballistic transport in semiconducting armchair graphene nanoribbon (aGNR) superlattices (5 to 20 barriers) at low bias voltages $V_{SD} < 500$ mV. We combine the graphene Dirac hamiltonian with the Landauer-B\"uttiker formalism to calculate the current $I_{SD}$ through the system. We find three distinct transport regimes in which NDR occurs: (i) a "classical" regime for wide layers, through which the transport across bandgaps is strongly suppressed, leading to alternating regions of nearly unity and zero transmission probabilities as a function of $V_{SD}$ due to crossing of bandgaps from different layers. (ii) a quantum regime dominated by superlattice miniband conduction, with current suppression arising from the misalignment of miniband states with increasing $V_{SD}$; and (iii) a Wannier-Stark ladder regime with current peaks occurring at the crossings of Wannier-Stark rungs from distinct ladders. We observe NDR at voltage biases as low as 10 mV with a high current density, making the aGNR superlattices attractive for device applications.

Physical solutions of the Kitaev honeycomb model

Fabio L. Pedrocchi, Stefano Chesi (McGill Univ.), and Daniel Loss. Phys. Rev. B 84, 165414 (2011); arXiv:1105.4573. We investigate the exact solution of the honeycomb model proposed by Kitaev and derive an explicit formula for the projector onto the physical subspace. The physical states are simply characterized by the parity of the total occupation of the fermionic eigenmodes. We consider a general lattice on a torus and show that the physical fermion parity depends in a nontrivial way on the vortex configuration and the choice of boundary conditions. In the vortex-free case with a constant gauge field we are able to obtain an analytical expression of the parity. For a general configuration of the gauge field the parity can be easily evaluated numerically, which allows the exact diagonalization of large spin models. We consider physically relevant quantities, as in particular the vortex energies, and show that their true value and associated states can be substantially different from the one calculated in the unprojected space, even in the thermodynamic limit.

Schrieffer-Wolff transformation for quantum many-body systems

Sergey Bravyi (IBM Yorktown), David DiVincenzo (Julich), and Daniel Loss. Annals of Physics 326, 2793-2826 (2011); arXiv:1105.0675. The Schrieffer-Wolff (SW) method is a version of degenerate perturbation theory in which the low-energy effective Hamiltonian H_{eff} is obtained from the exact Hamiltonian by a unitary transformation decoupling the low-energy and high-energy subspaces. We give a self-contained summary of the SW method with a focus on rigorous results. We begin with an exact definition of the SW transformation in terms of the so-called direct rotation between linear subspaces. From this we obtain elementary proofs of several important properties of H_{eff} such as the linked cluster theorem. We then study the perturbative version of the SW transformation obtained from a Taylor series representation of the direct rotation. Our perturbative approach provides a systematic diagram technique for computing high-order corrections to H_{eff}. We then specialize the SW method to quantum spin lattices with short-range interactions. We establish unitary equivalence between effective low-energy Hamiltonians obtained using two different versions of the SW method studied in the literature. Finally, we derive an upper bound on the precision up to which the ground state energy of the n-th order effective Hamiltonian approximates the exact ground state energy.

Universal quantum computation with topological spin-chain networks

Yaroslav Tserkovnyak (UCLA) and Daniel Loss. Phys. Rev. A 84, 032333 (2011); arXiv:1104.1210. It is shown that anisotropic spin chains with gapped bulk excitations and magnetically ordered ground states offer a promising platform for quantum computation, which bridges the conventional single-spin-based qubit concept with recently developed topological Majorana-based proposals. We show how to realize the single-qubit Hadamard, phase, and π/8 gates as well as the two-qubit controlled-not (cnot) gate, which together form a fault-tolerant universal set of quantum gates. The gates are implemented by judiciously controlling Ising exchange and magnetic fields along a network of spin chains, with each individual qubit furnished by a spin-chain segment. A subset of single-qubit operations is geometric in nature, relying on control of anisotropy of spin interactions rather than their strength. We contrast topological aspects of the anisotropic spin-chain networks to those of p-wave superconducting wires discussed in the literature.

Rectification of spin currents in spin chains

Kevin A. van Hoogdalem and Daniel Loss. Phys. Rev. B 84, 024402 (2011); arXiv:1102.4801. We study spin transport in non-itinerant one-dimensional quantum spin chains. Motivated by possible applications in spintronics, we consider rectification effects in both ferromagnetic and antiferromagnetic systems. We find that the crucial ingredients in designing a system that displays a non-zero rectification current are an anisotropy in the exchange interaction of the spin chain combined with an offset magnetic field. For both ferromagnetic and antiferromagnetic systems we can exploit the gap in the excitation spectrum that is created by a bulk anisotropy to obtain a measurable rectification effect at realistic magnetic fields. For antiferromagnetic systems we also find that we can achieve a similar effect by introducing a magnetic impurity, obtained by altering two neighboring bonds in the spin Hamiltonian.

Helical modes in carbon nanotubes generated by strong electric fields

Jelena Klinovaja, Manuel J. Schmidt, Bernd Braunecker, and Daniel Loss. Phys. Rev. Lett. 106, 156809 (2011); arXiv:1011.3630. Helical modes, conducting opposite spins in opposite directions, are shown to exist in metallic armchair nanotubes in an all-electric setup. This is a consequence of the interplay between spin-orbit interaction and strong electric fields. The helical regime can also be obtained in chiral metallic nanotubes by applying an additional magnetic field. In particular, it is possible to obtain helical modes at one of the two Dirac points only, while the other one remains gapped. Starting from a tight-binding model we derive the effective low-energy Hamiltonian and the resulting spectrum.

Majorana edge states in interacting one-dimensional systems

Suhas Gangadharaiah, Bernd Braunecker, Pascal Simon (Orsay, Paris), and Daniel Loss. Phys. Rev. Lett. 107, 036801 (2011); arXiv:1101.0094. We show that one-dimensional electron systems in the proximity of a superconductor that support Majorana edge states are extremely susceptible to electron-electron interactions. Strong interactions generically destroy the induced superconducting gap that stabilizes the Majorana edge states. For weak interactions, the renormalization of the gap is nonuniversal and allows for a regime in which the Majorana edge states persist. We present strategies of how this regime can be reached.

Spectrum of an electron spin coupled to an unpolarized bath of nuclear spins

Oleksandr Tsyplyatyev (University of Sheffield) and Daniel Loss. Phys. Rev. Lett. 106, 106803 (2011); arXiv:1102.2426. The main source of decoherence for an electron spin confined to a quantum dot is the hyperfine interaction with nuclear spins. To analyze this process theoretically we diagonalize the central spin Hamiltonian in the high magnetic B-field limit. Then we project the eigenstates onto an unpolarized state of the nuclear bath and find that the resulting density of states has Gaussian tails. The level spacing of the nuclear sublevels is exponentially small in the middle of each of the two electron Zeeman levels but increases super-exponentially away from the center. This suggests to sel​ect states from the wings of the distribution when the system is projected on a single eigenstate by a measurement to reduce the noise of the nuclear spin bath. This theory is valid when the external magnetic field is larger than a typical Overhauser field at high nuclear spin temperature.

Quantum memory coupled to cavity modes

Fabio L. Pedrocchi, Stefano Chesi, and Daniel Loss. Phys. Rev. B.83, 115415 (2011); arXiv:1011.3762. Inspired by spin-electric couplings in molecular magnets, we introduce in the Kitaev honeycomb model a linear modification of the Ising interactions due to the presence of quantized cavity fields. This allows to control the properties of the low-energy toric code Hamiltonian, which can serve as a quantum memory, by tuning the physical parameters of the cavity modes, like frequencies, photon occupations, and coupling strengths. We study the properties of the model perturbatively by making use of the Schrieffer-Wolff transformation and show that, depending on the specific setup, the cavity modes can be useful in several ways. They allow to detect the presence of anyons through frequency shifts and to prolong the lifetime of the memory by enhancing the anyon excitation energy or mediating long-range anyon-anyon interactions with tunable sign. We consider both resonant and largely detuned cavity modes.

Controlling the electron spin-nuclear spin interaction of a quantum dot in the tunneling regime

C. Kloeffel, P. A. Dalgarno (Edinburgh), B. Urbaszek (Toulouse), B. D. Gerardot (Edinburgh), D. Brunner (Edinburgh), P. M. Petroff (UC Santa Barbara), D. Loss, and R. J. Warburton. Phys. Rev. Lett. 106, 046802 (2011); arXiv:1010.3330. We present a technique for manipulating the nuclear spins and the emission polarization from a single optically-active quantum dot. When the quantum dot is tunnel coupled to a Fermi sea, we have discovered a natural cycle in which an electron spin is repeatedly created with resonant optical excitation. The spontaneous emission polarization and the nuclear spin polarization exhibit a bistability. For a sigma(+) pump, the emission switches from sigma(+) to sigma(-) at a particular detuning of the laser. Simultaneously, the nuclear spin polarization switches from positive to negative. Away from the bistability, the nuclear spin polarization can be changed continuously from negative to positive, allowing precise control via the laser wavelength.

Hybridization and spin decoherence in heavy-hole quantum dots

Jan Fischer and Daniel Loss. Phys. Rev. Lett. 105, 266603 (2010); arXiv:1009.5195. We theoretically investigate the spin dynamics of a heavy hole confined to a III-V semiconductor quantum dot interacting with a narrowed nuclear-spin bath. We show that band hybridization leads to an exponential decay of hole-spin superpositions due to hyperfine-mediated nuclear pair flips, and that the accordant single-hole-spin decoherence time T2 can be tuned over many orders of magnitude by changing external parameters. In particular, we show that, under experimentally accessible conditions, it is possible to suppress hyperfine-mediated nuclear-pair-flip processes so strongly that hole-spin quantum dots may be operated beyond the `ultimate limitation' set by the hyperfine interaction which is present in other spin-qubit candidate systems.

Geometric Correlations and Breakdown of Mesoscopic Universality in Spin Transport

I. Adagideli (Istanbul), Ph. Jacquod (Tucson, AZ), M. Scheid (Regensburg), M. Duckheim (Berlin), D. Loss, and K. Richter (Regensburg). Phys. Rev. Lett. 105, 246807 (2010); arXiv:1008.4656. We construct a unified semiclassical theory of charge and spin transport in chaotic ballistic and disordered diffusive mesoscopic systems with spin-orbit interaction. Neglecting dynamic effects of spinorbit interaction, we reproduce the random matrix theory results that the spin conductance fluctuates universally around zero average. Incorporating these effects into the theory, we show that geometric correlations generate finite average spin conductances, but that they do not affect the charge conductance to leading order. The theory, which is confirmed by numerical transport calculations, allows us to investigate the entire range from the weak to the previously unexplored strong spin-orbit regime, where the spin rotation time is shorter than the momentum relaxation time.

Energy spectra for quantum wires and 2DEGs in magnetic fields with Rashba and Dresselhaus spin-orbit interactions

Sigurdur I. Erlingsson (Reykjavik), J. Carlos Egues (Sao Carlos), and Daniel Loss. Phys. Rev. B82, 155456 (2010); arXiv:1008.1317. We introduce an analytical approximation scheme to diagonalize parabolically confined two dimensional electron systems with both the Rashba and Dresselhaus spin-orbit interactions. The starting point of our perturbative expansion is a zeroth-order Hamiltonian for an electron confined in a quantum wire with an effective spin-orbit induced magnetic field along the wire, obtained by properly rotating the usual spin-orbit Hamiltonian. We find that the spin-orbit-related transverse coupling terms can be recast into two parts W and V, which couple crossing and non-crossing adjacent transverse modes, respectively. Interestingly, the zeroth-order Hamiltonian together with W can be solved exactly, as it maps onto the Jaynes-Cummings model of quantum optics. We treat the V coupling by performing a Schrieffer-Wolff transformation. This allows us to obtain an effective Hamiltonian to third order in the coupling strength k_Rl of V, which can be straightforwardly diagonalized via an additional unitary transformation. We also apply our approach to other types of effective parabolic confinement, e.g., 2D electrons in a perpendicular magnetic field. To demonstrate the usefulness of our approximate eigensolutions, we obtain analytical expressions for the n^th Landau-level g_n-factors in the presence of both Rashba and Dresselhaus couplings. For small values of the bulk g-factors, we find that spin-orbit effects cancel out entirely for particular values of the spin-orbit couplings. By solving simple transcendental equations we also obtain the band minima of a Rashba-coupled quantum wire as a function of an external magnetic field. These can be used to describe Shubnikov-de Haas oscillations. This procedure makes it easier to extract the strength of the spin-orbit interaction in these systems via proper fitting of the data.

Poor man's derivation of the Bethe-Ansatz equations for the Dicke model

Oleksandr Tsyplyatyev, Jan von Delft (LMU Munich), and Daniel Loss. Phys. Rev. B 82, 092203 (2010); arXiv:1008.1844. We present an elementary derivation of the exact solution (Bethe-Ansatz equations) of the Dicke model, using only commutation relations and an informed Ansatz for the structure of its eigenstates.

Spin susceptibility of interacting two-dimensional electrons in the presence of spin-orbit coupling

Robert Andrzej Zak, Dmitrii L. Maslov (Gainesville, FL), and Daniel Loss. Phys. Rev. B 82, 115415 (2010); arXiv:1005.1913. A long-range interaction via virtual particle-hole pairs between Fermi-liquid quasiparticles leads to a nonanalytic behavior of the spin susceptibility $\chi$ as a function of the temperature ($T$), magnetic field ($\mathbf{B}$), and wavenumber. In this paper, we study the effect of the Rashba spin-orbit interaction (SOI) on the nonanalytic behavior of $\chi$ for a two-dimensional electron liquid. Although the SOI breaks the $SU(2)$ symmetry, it does not eliminate nonanalyticity but rather makes it anisotropic: while the linear scaling of $\chi_{zz}$ with $T$ and $|\mathbf{B}|$ saturates at the energy scale set by the SOI, that of $\chi_{xx}$ ($=\chi_{yy}$) continues through this energy scale, until renormalization of the electron-electron interaction in the Cooper channel becomes important. We show that the Renormalization Group flow in the Cooper channel has a non-trivial fixed point, and study the consequences of this fixed point for the nonanalytic behavior of $\chi$. An immediate consequence of SOI-induced anisotropy in the nonanalytic behavior of $\chi$ is a possible instability of a second-order ferromagnetic quantum phase transition with respect to a first-order transition to an $XY$ ferromagnetic state.

Tunable edge magnetism at graphene/graphane interfaces

Manuel J. Schmidt and Daniel Loss. Phys. Rev. B 82, 085422 (2010); arXiv:1004.4363. We study the magnetic properties of graphene edges and graphene/graphane interfaces under the influence of electrostatic gates. For this, an effective low-energy theory for the edge states, which is derived from the Hubbard model of the honeycomb lattice, is used. We first study the edge state model in a mean-field approximation for the Hubbard Hamiltonian and show that it reproduces the results of the extended 2D lattice theory. Quantum fluctuations around the mean-field theory of the effective one-dimensional model are treated by means of the bosonization technique in order to check the stability of the mean-field solution. We find that edge magnetism at graphene/graphane interfaces can be switched on and off by means of electrostatic gates. We describe a quantum phase transition between an ordinary and a ferromagnetic Luttinger liquid - a realization of itinerant one-dimensional ferromagnetism. This mechanism may provide means to experimentally discriminate between edge magnetism or disorder as the reason for a transport gap in very clean graphene nanoribbons.

RKKY interaction in a disordered two-dimensional electron gas with Rashba and Dresselhaus spin-orbit couplings

Stefano Chesi and Daniel Loss. Phys. Rev. B.82,165303 (2010); arXiv:1007.3506. We study theoretically the statistical properties of the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between localized magnetic moments in a disordered two-dimensional electron gas with both Rashba and Dresselhaus spin-orbit couplings. Averaging over disorder, the static spin susceptibility tensor is evaluated diagrammatically in the mesoscopic (phase-coherent) regime. The disorder-averaged susceptibility leads to a twisted exchange interaction suppressed exponentially with distance, whereas the second-order correlations, which determine the fluctuations (variance) of the RKKY energy, decay with the same power-law as in the clean case. We obtain analytic expressions in the limits of large/small spin orbit interactions and for equal Rashba and Dresselhaus couplings. Beside these limiting cases, we study numerically the variance of the RKKY interaction in the presence of pure Rashba spin-orbit coupling. Our results are relevant for magnetic or nuclear moments embedded in III-V two-dimensional heterostructures or in contact with surface states of metals and metal alloys, which can display a sizable Rashba spin-orbit coupling.

Spin-selective Peierls transition in interacting one-dimensional conductors with spin-orbit interaction

Bernd Braunecker, George I. Japaridze (Tbilisi, Georgia), Jelena Klinovaja, and Daniel Loss. Phys. Rev. B 82, 045127 (2010); arXiv:1004.0467. Interacting one-dimensional conductors with Rashba spin-orbit coupling are shown to exhibit a spin-sel​ective Peierls-like transition into a mixed spin-charge density wave state. The transition leads to a gap for one-half of the conducting modes, which is strongly enhanced by electron-electron interactions. The other half of the modes remains in a strongly renormalized gapless state and conducts opposite spins in opposite directions, thus providing a perfect spin filter. The transition is driven by magnetic field and by spin-orbit interactions. As an example we show for semiconducting quantum wires and carbon nanotubes that the gap induced by weak magnetic fields or intrinsic spin-orbit interactions can get renormalized by one order of magnitude up to 10-30 Kelvins.

Cooper-Pair Injection into Topological Insulators

Koji Sato (UCLA), Daniel Loss, and Yaroslav Tserkovnyak (UCLA). Phys. Rev. Lett. 105, 226401 (2010); arXiv:1003.4316. We theoretically study tunneling of Cooper pairs (CP's) from a superconductor spanning a two-dimensional topological insulator strip into its helical edge states. The coherent low-energy electron-pair tunneling sets off positive nonlocal current cross-correlations along the edges, which reflect an interplay of two quantum-entanglement mechanisms. First of all, superconducting spin pairing dictates a CP partitioning into the helical edge liquids, which transport electrons in the opposite directions for opposite spin orientations. Luttinger-liquid (LL) correlations for the electron-density fluctuations are, furthermore, forcing paired electrons to enter into opposite insulator-strip edges, revealing CP spin entanglement in the inter-edge current correlations. At the same time, the LL behavior, in the absence of Fermi-liquid leads, fractionalizes electrons injected at a given edge into counter-propagating charge pulses carrying definite fractions of the elementary electron charge. The superconductivity as well as LL correlations thus introduce positive current cross-correlations, which reveal a wealth of information about both subsystems.

The classical and quantum dynamics of the inhomogeneous Dicke model and its Ehrenfest time

Oleksandr Tsyplyatyev and Daniel Loss. Phys. Rev. B 82, 024305 (2010); arXiv:1002.3932. We show that in the few-excitation regime the classical and quantum time-evolution of the inhomogeneous Dicke model for N two-level systems coupled to a single boson mode agree for N>>1. In the presence of a single excitation only, the leading term in an 1/N-expansion of the classical equations of motion reproduces the result of the Schroedinger equation. For a small number of excitations, the numerical solutions of the classical and quantum problems become equal for N sufficiently large. By solving the Schroedinger equation exactly for two excitations and a particular inhomogeneity we obtain 1/N-corrections which lead to a significant difference between the classical and quantum solutions at a new time scale which we identify as an Ehrenferst time, given by tau_E=sqrt{N}, where sqrt{} is an effective coupling strength between the two-level systems and the boson.

Spin electric effects in molecular antiferromagnets

Mircea Trif, Filippo Troiani (Modena), Dimitrije Stepanenko, and Daniel Loss. Phys. Rev. B 82, 045429 (2010); arXiv:1001.3584. Molecular nanomagnets show clear signatures of coherent behavior and have a wide variety of effective low-energy spin Hamiltonians suitable for encoding qubits and implementing spin-based quantum information processing. At the nanoscale, the preferred mechanism for the control of a quantum systems is the application of electric fields, which are strong, can be locally applied, and rapidly switched. In this work, we provide the theoretical tools for identifying molecular nanomagnets suitable for electric control. By group-theoretical symmetry analysis we find that the spin-electric coupling in triangular molecules is governed by the modification of the exchange interaction and is possible even in the absence of spin-orbit coupling. In pentagonal molecules the spin-electric coupling can exist only in the presence of spin-orbit interaction. This kind of coupling is allowed for both s=1/2 and s=3/2 spins at the magnetic centers. Within the Hubbard model, we find a relation between the spin-electric coupling and the properties of the chemical bonds in a molecule, suggesting that the best candidates for strong spin-electric coupling are molecules with nearly degenerate bond orbitals. We also investigate the possible experimental signatures of spin-electric coupling in nuclear magnetic resonance and electron spin resonance spectroscopy, as well as in the thermodynamic measurements of magnetization, electric polarization, and specific heat of the molecules.

Magnetic order in nuclear spin two-dimensional lattices due to electronelectron interactions.

P. Simon, B. Braunecker, and D. Loss, Physica E 42 (2010) 634-638 We focus on nuclear spins embedded in a two-dimensional (2D) electron gas. The nuclear spins interact with each other through the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction, which is carried by the electron gas. We show that a nuclear magnetic order at finite temperature relies on the anomalous behaviour of the 2D static electron spin susceptibility due to electron–electron interactions. This provides a connection between low-dimensional magnetism and non-analyticities in interacting 2D electron systems. We discuss the conditions for nuclear magnetism, and show that the associated Curie temperature increases with the electron–electron interactions and may reach up into the millikelvin regime. We also shortly discussed what happens when the dimensionality is further reduced to one dimension.

Free-induction decay and envelope modulations in a narrowed nuclear spin bath

W. A. Coish (Waterloo), Jan Fischer, and Daniel Loss. Phys. Rev. B 81, 165315 (2010); arXiv:0911.4149. We evaluate free-induction decay for the transverse components of a localized electron spin coupled to a bath of nuclear spins via the Fermi contact hyperfine interaction. Our perturbative treatment is valid for special (narrowed) bath initial conditions and when the Zeeman energy of the electron $b$ exceeds the total hyperfine coupling constant $A$: $b>A$. Using one unified and systematic method, we recover previous results reported at short and long times using different techniques. We find a new and unexpected modulation of the free-induction-decay envelope, which is present even for a purely isotropic hyperfine interaction without spin echoes and for a single nuclear species. We give sub-leading corrections to the decoherence rate, and show that, in general, the decoherence rate has a non-monotonic dependence on electron Zeeman splitting, leading to a pronounced maximum. These results illustrate the limitations of methods that make use of leading-order effective Hamiltonians and re-exponentiation of short-time expansions for a strongly-interacting system with non-Markovian (history-dependent) dynamics.

Edge states and enhanced spin-orbit interaction at graphene/graphane interfaces

Manuel J. Schmidt and Daniel Loss. Phys. Rev. B 81, 165439 (2010); arXiv:0910.5333. We study interfaces between graphene and graphane. If the interface is oriented along a zigzag direction, edge states are found which exhibit a strong amplification of effects related to the spin-orbit interaction. The enhanced spin splitting of the edge states allows a conversion between valley polarization and spin polarization at temperatures near one Kelvin. We show that these edge states give rise to quantum spin and/or valley Hall effects.

One-step multi-qubit GHZ state generation in a circuit QED system

Ying-Dan Wang, Stefano Chesi, Daniel Loss, and Christoph Bruder. Phys. Rev. B 81, 104524 (2010); arXiv:0911.1396. We propose a one-step scheme to generate GHZ states for superconducting flux qubits or charge qubits in a circuit QED setup. The GHZ state can be produced within the coherence time of the multi-qubit system. Our scheme is independent of the initial state of the transmission line resonator and works in the presence of higher harmonic modes. Our analysis also shows that the scheme is robust to various operation errors and environmental noise.

Spin Accumulation in Diffusive Conductors with Rashba and Dresselhaus Spin-Orbit Interaction

Mathias Duckheim, Daniel Loss, Matthias Scheid (Regensburg), Klaus Richter (Regensburg), Inanc Adagideli (Istanbul), and Philippe Jacquod (Tucson). Physical Review B 81 085303 (2010); arXiv:0909.4253. We calculate the electrically induced spin accumulation in diffusive systems due to both Rashba (with strength $\alpha)$ and Dresselhaus (with strength $\beta)$ spin-orbit interaction. Using a diffusion equation approach we find that magnetoelectric effects disappear and that there is thus no spin accumulation when both interactions have the same strength, $\alpha=\pm \beta$. In thermodynamically large systems, the finite spin accumulation predicted by Chaplik, Entin and Magarill, [Physica E 13, 744 (2002)] and by Trushin and Schliemann [Phys. Rev. B 75, 155323 (2007)] is recovered an infinitesimally small distance away from the singular point $\alpha=\pm \beta$. We show however that the singularity is broadened and that the suppression of spin accumulation becomes physically relevant (i) in finite-sized systems of size $L$, (ii) in the presence of a cubic Dresselhaus interaction of strength $\gamma$, or (iii) for finite frequency measurements. We obtain the parametric range over which the magnetoelectric effect is suppressed in these three instances as (i) $|\alpha|-|\beta| \lesssim 1/mL$, (ii)$|\alpha|-|\beta| \lesssim \gamma p\rm F^2$, and (iii) $|\alpha|-|\beta| \lesssiM \sqrt{\omega/m p\rm F\ell}$ with $\ell$ the elastic mean free path and $p\rm F$ the Fermi momentum. We attribute the absence of spin accumulation close to $\alpha=\pm \beta$ to the underlying U (1) symmetry. We illustrate and confirm our predictions numerically.

Dynamic spin-Hall effect and driven spin helix for linear spin-orbit interactions

Mathias Duckheim, Dmitrii L. Maslov (Gainesville, FL), and Daniel Loss. Phys. Rev. B 80, 235327 (2009); arXiv:0909.1892. We derive boundary conditions for the electrically induced spin accumulation in a finite 2D semiconductor channel. While for DC electric fields these boundary conditions sel​ect spatially constant spin profiles equivalent to a vanishing spin-Hall effect, we show that an in-plane ac electric field results in a non-zero ac spin-Hall effect, i.e., it generates a spatially non-uniform out-of-plane polarization even for linear intrinsic spin-orbit interactions. Analyzing different geometries in [001] and [110]-grown quantum wells, we find that although this out-of-plane polarization is typically confined to within a few spin-orbit lengths from the channel edges, it is also possible to generate spatially oscillating spin profiles which extend over the whole channel. The latter is due to the excitation of a driven spin-helix mode in the transverse direction of the channel. We show that while finite frequencies suppress this mode, it can be amplified by a magnetic field tuned to resonance with the frequency of the electric field. In this case, finite size effects at equal strengths of Rashba- and Dresselhaus SOI lead to an enhancement of the magnitude of this helix mode. We comment on the relation between spin currents and boundary conditions. http://prb.aps.org/kaleidoscope

A Self-Correcting Quantum Memory in a Thermal Environment

Stefano Chesi, Beat Röthlisberger, and Daniel Loss. Phys. Rev. A 82, 022305 (2010); arXiv:0908.4264. The ability to store information is of fundamental importance to any computer, be it classical or quantum. To identify systems for quantum memories, which rely, analogously to classical memories, on passive error protection (“self-correction”), is of greatest interest in quantum information science. While systems with topological ground states have been considered to be promising candidates, a large class of them was recently proven unstable against thermal fluctuations. Here, we propose two-dimensional (2D) spin models unaffected by this result. Specifically, we introduce repulsive long-range interactions in the toric code and establish a memory lifetime polynomially increasing with the system size. This remarkable stability is shown to originate directly from the repulsive long-range nature of the interactions. We study the time dynamics of the quantum memory in terms of diffusing anyons and support our analytical results with extensive numerical simulations. Our findings demonstrate that self-correcting quantum memories can exist in 2D at finite temperatures.

Holonomic Quantum Computation with Electron Spins in Quantum Dots

Vitaly N. Golovach (LMU Munich), Massoud Borhani (Buffalo), and Daniel Loss. Phys. Rev. A 81, 022315 (2010); arXiv:0908.2800. With the help of the spin-orbit interaction, we propose a scheme to perform holonomic single qubit gates on the electron spin confined to a quantum dot. The manipulation is done in the absence (or presence) of an applied magnetic field. By adiabatic changing the position of the confinement potential, one can rotate the spin state of the electron around the Bloch sphere in semiconductor heterostructures. The dynamics of the system is equivalent to employing an effective non-Abelian gauge potential whose structure depends on the type of the spin-orbit interaction. As an example, we find an analytic expression for the electron spin dynamics when the dot is moved around a circular path (with radius R) on the two dimensional electron gas (2DEG), and show that all single qubit gates can be realized by tuning the radius and orientation of the circular paths. Moreover, using the Heisenberg exchange interaction, we demonstrate how one can generate two-qubit gates by bringing two quantum dots near each other, yielding a scalable scheme to perform quantum computing on arbitrary N qubits. This proposal shows a way of realizing holonomic quantum computers in solid-state systems.

Nuclear magnetism and electron order in interacting one-dimensional conductors

Bernd Braunecker, Pascal Simon (Orsay, Paris), and Daniel Loss. Phys. Rev. B 80, 165119 (2009); arXiv:0908.0904; News in Physics World, Feb 21 (2014). The interaction between localized magnetic moments and the electrons of a one-dimensional conductor can lead to an ordered phase in which the magnetic moments and the electrons are tightly bound to each other. We show here that this occurs when a lattice of nuclear spins is embedded in a Luttinger liquid. Experimentally available examples of such a system are single wall carbon nanotubes grown entirely from 13C and GaAs-based quantum wires. In these systems the hyperfine interaction between the nuclear spin and the conduction electron spin is very weak, yet it triggers a strong feedback reaction that results in an ordered phase consisting of a nuclear helimagnet that is inseparably bound to an electronic density wave combining charge and spin degrees of freedom. This effect can be interpreted as a strong renormalization of the nuclear Overhauser field and is a unique signature of Luttinger liquid physics. Through the feedback the order persists up into the millikelvin range. A particular signature is the reduction of the electric conductance by the universal factor 2.

Spin Hall effect due to inter-subband-induced spin-orbit interaction in symmetric quantum well

Minchul Lee (Yongin, S. Korea), Marco O. Hachiya (Sao Carlos, Brazil), E. Bernardes (Sao Carlos, Brazil), J. Carlos Egues (Sao Carlos, Brazil), and Daniel Loss. Phys. Rev. B 80, 155314 (2009); arXiv:0907.4078. We investigate the intrinsic spin Hall effect in two-dimensional electron gases in quantum wells with two subbands, where a new intersubband-induced spin-orbit coupling is operative. The bulk spin Hall conductivity $\sigma^zxy$ is calculated in the ballistic limit within the standard Kubo formalism in the presence of a magnetic field $B$ and is found to remain finite in the B=0 limit, as long as only the lowest subband is occupied. Our calculated $\sigma^zxy$ exhibits a non-monotonic behavior and can change its sign as the Fermi energy (the carrier areal density $n2D$) is varied between the subband edges. We determine the magnitude of $\sigma^zxy$ for realistic InSb quantum wells by performing a self-consistent calculation of the intersubband-induced spin-orbit coupling.

Thermodynamic stability criteria for a quantum memory based on stabilizer and subsystem codes

Stefano Chesi, Daniel Loss, Sergey Bravyi (IBM Yorktown), and Barbara M. Terhal (IBM Yorktown). New J. Phys. 12 025013 (2010); arXiv:0907.2807. We discuss and review several thermodynamic criteria that have been introduced to characterize the thermal stability of a self-correcting quantum memory. We first examine the use of symmetry-breaking fields in analyzing the properties of self-correcting quantum memories in the thermodynamic limit: we show that the thermal expectation values of all logical operators vanish for any stabilizer and any subsystem code in any spatial dimension. On the positive side, we generalize the results in [R. Alicki et al., arXiv:0811.0033] to obtain a general upper bound on the relaxation rate of a quantum memory at nonzero temperature, assuming that the quantum memory interacts via a Markovian master equation with a thermal bath. This upper bound is applicable to quantum memories based on either stabilizer or subsystem codes.

Dicke model: entanglement as a finite size effect

Oleksandr Tsyplyatyev and Daniel Loss. J. Phys.: Conf. Ser. 193, 012134(2009); arXiv:0907.2553v1. We analyze Dicke model at zero temperature by matrix diagonalization to determine the entanglement in the ground state. In the infinite system limit the mean field approximation predicts a quantum phase transition from a non-interacting state to a Bose-Einstein condensate at a threshold coupling. We show that in a finite system the spin part of the ground state is a bipartite entangled state, which can be tested by probing two parts of the spin system separately, but only in a narrow regime around the threshold coupling. Around the resonance, the size of this regime is inversely proportional to the number of spins and shrinks down to zero for infinite systems. This spin entanglement is a non-perturbative effect and is also missed by the mean-field approximation.

Quantum Computing with Electron Spins in Quantum Dots

Robert Andrzej Żak, Beat Röhlisberger, Stefano Chesi, and Daniel Loss. Lecture notes for Course CLXXI "Quantum Coherence in Solid State Systems" Int. School of Physics "Enrico Fermi", Varenna, July 2008. Rivista del Nuovo Cimento 033 (Issue 07), 345-399 (2010); arXiv:0906.4045. Several topics on the implementation of spin qubits in quantum dots are reviewed. We first provide an introduction to the standard model of quantum computing and the basic criteria for its realization. Other alternative formulations such as measurement-based and adiabatic quantum computing are briefly discussed. We then focus on spin qubits in single and double GaAs electron quantum dots and review recent experimental achievements with respect to initialization, coherent manipulation and readout of the spin states. We extensively discuss the problem of decoherence in this system, with particular emphasis on its theoretical treatment and possible ways to overcome it.

Hyperfine interaction and electron-spin decoherence in graphene and carbon nanotube quantum dots

Jan Fischer, Bjoern Trauzettel (Wuerzburg), and Daniel Loss. Phys. Rev. B 80, 155401 (2009); arXiv:0906.2800. We analytically calculate the nuclear-spin interactions of a single electron confined to a carbon nanotube or graphene quantum dot. While the conduction-band states in graphene are p-type, the accordant states in a carbon nanotube are sp-hybridized due to curvature. This leads to an interesting interplay between isotropic and anisotropic hyperfine interactions. By using only analytical methods, we are able to show how the interaction strength depends on important physical parameters, such as curvature and isotope abundances. We show that for the investigated carbon structures, the 13C hyperfine coupling strength is less than 1 mu-eV, and that the associated electron-spin decoherence time can be expected to be as long as several tens of microseconds, depending on the abundance of spin-carrying 13C nuclei. Furthermore, we find an unusual hyperfine-induced alignment of the 13C nuclear spins in nanotubes of any chirality.

Dealing with Decoherence

Jan Fischer and Daniel Loss. Perspective article Science 324, 1277 (2009) The dream of building computers that work according to the rules of quantum mechanics has strongly driven research over the past decade in many fields of basic and applied sciences, including physics, chemistry, and computer science. About 10 years ago, it was shown mathematically that the direct use of quantum phenomena such as interference and entanglement could crucially speed up data searching and prime factorization for encryption. To turn quantum computers into reality, however, many issues in engineering and in basic physics need to be addressed.

Numerical evaluation of convex-roof entanglement measures with applications to spin rings

Beat Röthlisberger, Jörg Lehmann (ABB Baden), and Daniel Loss. Phys. Rev. A 80, 042301 (2009); arXiv:0905.3106. We present two ready-to-use numerical algorithms to evaluate convex-roof extensions of arbitrary pure-state entanglement monotones. Their implementation leaves the user merely with the task of calculating derivatives of the respective pure-state measure. We provide numerical tests of the algorithms and demonstrate their good convergence properties. We further employ them in order to investigate the entanglement in particular few-spins systems at finite temperature. Namely, we consider ferromagnetic Heisenberg exchange-coupled spin-1/2 rings subject to an inhomogeneous in-plane field geometry obeying full rotational symmetry around the axis perpendicular to the ring through its center. We demonstrate that highly entangled states can be obtained in these systems at sufficiently low temperatures and by tuning the strength of a magnetic field configuration to an optimal value which is identified numerically.

Nanotubes: Carbon surprises again

Björn Trauzettel (Wuerzburg) and Daniel Loss. News and Views Nature Physics 5, 317 (2009) Experiments in 13C nanotubes reveal surprisingly strong nuclear spin effects that, if properly harnessed, could provide a mechanism for manipulation and storage of quantum information.

Undoing a quantum measurement

Christoph Bruder and Daniel Loss. Physics 1, 34 (2008) Quantum measurements are conventionally thought of as irretrievably "collapsing" a wave function to the observed state. However, experiments with superconducting qubits show that the partial collapse resulting from a weak continuous measurement can be restored.

Spin interactions, relaxation and decoherence in quantum dots

Jan Fischer, Mircea Trif, W. A. Coish (Waterloo), and Daniel Loss. Solid State Communications 149, 1443 (2009); arXiv:0903.0527. We review recent studies on spin decoherence of electrons and holes in quasi-two-dimensional quantum dots, as well as electron-spin relaxation in nanowire quantum dots. The spins of confined electrons and holes are considered major candidates for the realization of quantum information storage and processing devices, provided that sufficently long coherence and relaxation times can be achieved. The results presented here indicate that this prerequisite might be realized in both electron and hole quantum dots, taking one large step towards quantum computation with spin qubits.

Spin orbit-induced anisotropic conductivity of a disordered 2DEG

Oleg Chalaev and Daniel Loss. Phys. Rev. B 80, 035305 (2009); arXiv:0902.3277. We present a semi-automated computer-assisted method to generate and calculate diagrams in the disorder averaging approach to disordered 2D conductors with intrinsic spin-orbit interaction (SOI). As an application, we calculate the effect of the SOI on the charge conductivity for disordered 2D systems and rings in the presence of Rashba and Dresselhaus SOI. In an infinite-size 2D system, anisotropic corrections to the conductivity tensor arise due to phase-coherence and the interplay of Rashba and Dresselhaus SOI. The effect is more pronounced in the quasi-onedimensional case, where the conductivity becomes anisotropic already in the presence of only one type of SOI. The anisotropy further increases if the time-reversal symmetry of the Hamiltonian is broken.

Relaxation of hole spins in quantum dots via two-phonon processes

Mircea Trif, Pascal Simon (Orsay), and Daniel Loss. Phys. Rev. Lett. 103, 106601 (2009); arXiv:0902.2457. We investigate theoretically spin relaxation in heavy hole quantum dots in low external magnetic fields. We demonstrate that two-phonon processes and spin-orbit interaction are experimentally relevant and provide an explanation for the recently observed saturation of the spin relaxation rate in heavy hole quantum dots with vanishing magnetic fields. We propose further experiments to identify the relevant spin relaxation mechanisms in low magnetic fields.

Interference of heavy holes in an Aharonov-Bohm ring

Dimitrije Stepanenko, Minchul Lee (Marseille), Guido Burkard (Konstanz), and Daniel Loss. Phys. Rev. B 79, 235301 (2009); arXiv:0811.4566. We study the coherent transport of heavy holes through a one-dimensional ring in the presence of spin-orbit coupling. Spin-orbit interaction of holes, cubic in the in-plane components of momentum, gives rise to an angular momentum dependent spin texture of the eigenstates and influences transport. We analyze the dependence of the resulting differential conductance of the ring on hole polarization of the leads and the signature of the textures in the Aharonov-Bohm oscillations when the ring is in a perpendicular magnetic field. We find that the polarization-resolved conductance reveals whether the dominant spin-orbit coupling is of Dresselhaus or Rashba type, and that the cubic spin-orbit coupling can be distinguished from the conventional linear coupling by observing the four-peak structure in the Aharonov-Bohm oscillations.

Exact quantum dynamics of the inhomogeneous Dicke model

Oleksandr Tsyplyatyev and Daniel Loss. Phys. Rev. A 80, 023803 (2009); arXiv:0811.2386. We study the time dynamics of a single boson coupled to a bath of two-level systems (spins 1/2) with different excitation energies, described by an inhomogeneous Dicke model. Solving the time-dependent Schroedinger equation exactly we find that at resonance the boson decays in time to an oscillatory state characterized by a single Rabi frequency. In the limit of small inhomogeneity, the decay is suppressed and exhibits a complex (mainly Gaussian-like) behavior, whereas the decay is complete and of exponential form in the opposite limit. For intermediate inhomogeneity, the boson decay is partial and governed by a combination of exponential and power laws.

Momentum dependence of the spin-susceptibility in two dimensions: nonanalytic corrections in the Cooper channel

Stefano Chesi, Robert Andrzej Żak, Pascal Simon (Orsay), and Daniel Loss. Phys. Rev. B 79, 115445 (2009); arXiv:0811.0996. We consider the effect of rescattering of pairs of quasiparticles in the Cooper channel resulting in the strong renormalization of second order corrections to the spin susceptibility in a two-dimensional electron system. We use the Fourier expansion of the scattering potential in the vicinity of the Fermi surface to find that each harmonic becomes renormalized independently. Since some of those harmonics are negative, the slope of the spin susceptibility is bound to be negative at small momenta, in contrast to the lowest order perturbation theory result, which predicts a positive slope. We present in detail an effective method to calculate diagrammatically corrections to the spin susceptibility to infinite order.

Semiconductor spintronics: Snapshots of spins separating

Mathias Duckheim and Daniel Loss. News and Views Nature Physics 4, 836-837 (2008) Theories of the spin Hall effect suggest that spin currents generated by electric fields accumulate spin polarization at the sample edges. Now an experiment has observed this conversion in real time.

Magnetic Order in Kondo-Lattice Systems due to Electron-Electron Interactions

Bernd Braunecker, Pascal Simon (Orsay), and Daniel Loss. AIP Conf. Proc. 1074, 62 (2008); arXiv:0808.4063. The hyperfine interaction between the electron spin and the nuclear spins is one of the main sources of decoherence for spin qubits when the nuclear spins are disordered. An ordering of the latter largely suppresses this source of decoherence. Here we show that such an ordering can occur through a thermodynamic phase transition in two-dimensional (2D) Kondo-lattice type systems. We specifically focus on nuclear spins embedded in a 2D electron gas. The nuclear spins interact with each other through the RKKY interaction, which is carried by the electron gas. We show that a nuclear magnetic order at finite temperature relies on the anomalous behavior of the 2D static electron spin susceptibility due to electron-electron interactions. This provides a connection between low-dimensional magnetism and non-analyticities in interacting 2D electron systems. We discuss the conditions for nuclear magnetism, and show that the associated Curie temperature increases with the electron-electron interactions and may reach up into the millikelvin regime. The further reduction of dimensionality to one dimension is shortly discussed.

Nuclear Magnetism and Electronic Order in 13C Nanotubes

Bernd Braunecker, Pascal Simon, and Daniel Loss. Phys. Rev. Lett. 102, 116403 (2009); arXiv.org:0808.1685. Nuclear spins embedded in correlated metals offer an ideal platform to investigate the effect of electron-electron interactions on magnetism of localized moments. Here we focus on a one-dimensional realization of such a system: Single wall carbon nanotubes grown entirely from the 13C isotope. With this isotope the nanotube forms a nuclear spin lattice which couples through the hyperfine interaction to a Luttinger liquid if the nanotube is in the metallic regime. Even though the hyperfine interaction is very weak, the system is driven into an ordered phase which combines electron and nuclear spin degrees of freedom. This phase persists up into the millikelvin regime and leads to a reduction of the nanotube conductance by a universal factor of 2, allowing for an experimental detection by standard transport measurements.

Intersubband-induced spin-orbit interaction in quantum wells

Rafael S. Calsaverini, Esmerindo Bernardes, J. Carlos Egues, and Daniel Loss. Phys. Rev. B 78, 155313 (2008); arXiv:0807.0771. Recently, we have found an additional spin-orbit (SO) interaction in quantum wells with two subbands [Phys. Rev. Lett. 99, 076603 (2007)]. This new SO term is non-zero even in symmetric geometries, as it arises from the intersubband coupling between confined states of distinct parities, and its strength is comparable to that of the ordinary Rashba. Starting from the $8 \times 8$ Kane model, here we present a detailed derivation of this new SO Hamiltonian and the corresponding SO coupling. In addition, within the self-consistent Hartree approximation, we calculate the strength of this new SO coupling for realistic symmetric modulation-doped wells with two subbands. We consider gated structures with either a constant areal electron density or a constant chemical potential. In the parameter range studied, both models give similar results. By considering the effects of an external applied bias, which breaks the structural inversion symmetry of the wells, we also calculate the strength of the resulting induced Rashba couplings within each subband. Interestingly, we find that for double wells the Rashba couplings for the first and second subbands interchange signs abruptly across the zero bias, while the intersubband SO coupling exhibits a resonant behavior near this symmetric configuration. For completeness we also determine the strength of the Dresselhaus couplings and find them essentially constant as function of the applied bias.

Spin decoherence of a heavy hole coupled to nuclear spins in a quantum dot

Jan Fischer, W. A. Coish (Waterloo), D. V. Bulaev (Chernogolovka), and Daniel Loss. Phys. Rev. B 78, 155329 (2008); arXiv:0807.0368. We theoretically study the interaction of a heavy hole with nuclear spins in a quasi-two-dimensional III-V semiconductor quantum dot and the resulting dephasing of heavy-hole spin states. It has frequently been stated in the literature that heavy holes have a negligible interaction with nuclear spins. We show that this is not the case. In contrast, the interaction can be rather strong and will be the dominant source of decoherence in some cases. We also show that the form of the interaction is Ising-like, resulting in unique and interesting decoherence properties, which might provide a crucial advantage to using dot-confined hole spins for quantum information processing, as compared to electron spins.

Mesoscopic fluctuations in the spin-electric susceptibility due to Rashba spin-orbit interaction

Mathias Duckheim and Daniel Loss. Phys. Rev. Lett. 101, 226602 (2008); arXiv:0805.4143. We investigate mesoscopic fluctuations in the spin polarization generated by a static electric field and by Rashba spin-orbit interaction in a disordered 2D electron gas. In a diagrammatic approach we find that the out-of-plane polarization -- while being zero for self-averaging systems -- exhibits large sample-to-sample fluctuations which are shown to be well within experimental reach. We evaluate the disorder-averaged variance of the susceptibility and find its dependence on magnetic field, spin-orbit interaction, dephasing, and chemical potential difference.

Spin-Electric Coupling in Molecular Magnets

Mircea Trif, Filippo Troiani (Modena), Dimitrije Stepanenko, and Daniel Loss. Phys. Rev. Lett. 101, 217201 (2008); arXiv:0805.1158. We study the triangular antiferromagnet Cu$_3$ in external electric fields, using symmetry group arguments and a Hubbard model approach. We identify a spin-electric coupling caused by an interplay between spin exchange, spin-orbit interaction, and the chirality of the underlying spin texture of the molecular magnet. This coupling allows for the electric control of the spin (qubit) states, e.g. by using an STM tip or a microwave cavity. We propose an experimental test for identifying molecular magnets exhibiting spin-electric effects.

AC magnetization transport and power absorption in non-itinerant spin chains

Bjoern Trauzettel (Wuerzburg), Pascal Simon (Grenoble), and Daniel Loss. Phys. Rev. Lett. 101, 017202 (2008); arXiv:0804.3697. We investigate the ac transport of magnetization in non-itinerant quantum systems such as spin chains described by the XXZ Hamiltonian. Using linear response theory, we calculate the ac magnetization current and the power absorption of such magnetic systems. Remarkably, the difference in the exchange interaction of the spin chain itself and the bulk magnets (i.e. the magnetization reservoirs), to which the spin chain is coupled, strongly influences the absorbed power of the system. This feature can be used in future spintronic devices to control power dissipation. Our analysis allows to make quantitative predictions about the power absorption and we show that magnetic systems are superior to their electronic counter parts.

Quantum Hall ferromagnetic states and spin-orbit interactions in the fractional regime

Stefano Chesi and Daniel Loss. Phys. Rev. Lett. 101, 146803 (2008); arXiv:0804.3332. The competition between the Zeeman energy and the Rashba and Dresselhaus spin-orbit couplings is studied for fractional quantum Hall states by including correlation effects. A transition of the direction of the spin-polarization is predicted at specific values of the Zeeman energy. We show that these values can be expressed in terms of the pair-correlation function, which thus provides a way to obtain experimental access to the corresponding ground state. As specific examples, we consider the Laughlin wavefunctions and the 5/2-Pfaffian state and find indications of non-analytic features around the fractional states. We also include effects of the nuclear bath, becoming relevant in the mK-regime.

Nuclear spin dynamics and Zeno effect in quantum dots and defect centers

D. Klauser, W. A. Coish (Waterloo), and Daniel Loss. Phys. Rev. B 78, 205301 (2008); arXiv:0802.2463. We analyze nuclear spin dynamics in quantum dots and defect centers with a bound electron under electron-mediated coupling between nuclear spins due to the hyperfine interaction ("J-coupling" in NMR). Our analysis shows that the Overhauser field generated by the nuclei at the position of the electron has short-time dynamics quadratic in time for an initial nuclear spin state without transverse coherence. The quadratic short-time behavior allows for an extension of the Overhauser field lifetime through a sequence of projective measurements (quantum Zeno effect). We analyze the requirements on the repetition rate of measurements and the measurement accuracy to achieve such an effect. Further, we calculate the long-time behavior of the Overhauser field for effective electron Zeeman splittings larger than the hyperfine coupling strength and find, both in a Dyson series expansion and a generalized master equation approach, that for a nuclear spin system with a sufficiently smooth polarization the electron-mediated interaction alone leads only to a partial decay of the Overhauser field by an amount on the order of the inverse number of nuclear spins interacting with the electron.

Simulation of Many-Body Hamiltonians using Perturbation Theory with Bounded-Strength Interactions

Sergey Bravyi, David P. DiVincenzo, Daniel Loss, and Barbara M. Terhal. Phys. Rev. Lett. 101, 070503 (2008); arXiv:0803.2686; News & Views, M.M. Wolf, Nature Physics 4, 834 (2008). We show how to map a given n-qubit target Hamiltonian with bounded-strength k-body interactions onto a simulator Hamiltonian with two-body interactions, such that the ground-state energy of the target and the simulator Hamiltonians are the same up to an extensive error O(epsilon n) for arbitrary small epsilon. The strength of interactions in the simulator Hamiltonian depends on epsilon and k but does not depend on n. We accomplish this reduction using a new way of deriving an effective low-energy Hamiltonian which relies on the Schrieffer-Wolff transformation of many-body physics.

Spin-orbit interaction and anomalous spin relaxation in carbon nanotube quantum dots

Denis V. Bulaev, Bjoern Trauzettel (Wuerzburg), and Daniel Loss. Phys. Rev. B 77, 235301 (2008); arXiv:0712.3767 [cond-mat.mes-hall]. We study spin relaxation and decoherence caused by electron-lattice and spin-orbit interaction and predict striking effects induced by magnetic fields B. For particular values of B, destructive interference occurs resulting in ultralong spin relaxation times T1 exceeding tens of seconds. For small phonon frequencies \omega, we find a 1/\sqrt{\omega} spin-phonon noise spectrum --a novel dissipation channel for spins in quantum dots --which can reduce T1 by many orders of magnitude. We show that nanotubes exhibit zero-field level splitting caused by spin-orbit interaction. This enables an all-electrical and phase-coherent control of spin.

Exponential decay in a spin bath

W. A. Coish (Waterloo), Jan Fischer, and Daniel Loss. Phys. Rev. B 77, 125329 (2008); arXiv:0710.3762 [cond-mat.mes-hall]. We show that the coherence of an electron spin interacting with a bath of nuclear spins can exhibit a well-defined purely exponential decay for special (`narrowed') bath initial conditions in the presence of a strong applied magnetic field. This is in contrast to the typical case, where spin-bath dynamics have been investigated in the non-Markovian limit, giving super-exponential or power-law decay of correlation functions. We calculate the relevant decoherence time T_2 explicitly for free-induction decay and find a simple expression with dependence on bath polarization, magnetic field, the shape of the electron wave function, dimensionality, total nuclear spin I, and isotopic concentration for experimentally relevant heteronuclear spin systems.

Anisotropic conductivity of disordered 2DEGs due to spin-orbit interactions

Oleg Chalaev and Daniel Loss. Phys. Rev. B 77, 115352 (2008); arXiv:0708.3504. We show that the conductivity tensor of a disordered two-dimensional electron gas becomes anisotropic in the presence of both Rashba and Dresselhaus spin-orbit interactions (SOI). This anisotropy is a mesoscopic effect and vanishes with vanishing charge dephasing time. Using a diagrammatic approach including zero, one, and two-loop diagrams, we show that a consistent calculation needs to go beyond a Boltzmann equation approach. In the absence of charge dephasing and for zero frequency, a finite anisotropy \sigmaxy~ e^2/lhpf arises even for infinitesimal SOI.

Magnetic Ordering of Nuclear Spins in an Interacting 2D Electron Gas

Pascal Simon, Bernd Braunecker, and Daniel Loss. Phys. Rev. B 77, 045108 (2008); arXiv:0709.0164. We investigate the magnetic behavior of nuclear spins embedded in a 2D interacting electron gas using a Kondo lattice model description. We derive an effective magnetic Hamiltonian for the nuclear spins which is of the RKKY type and where the interactions between the nuclear spins are strongly modified by the electron-electron interactions. We show that the nuclear magnetic ordering at finite temperature relies on the (anomalous) behavior of the 2D static electron spin susceptibility, and thus provides a connection between low-dimensional magnetism and non-analyticities in interacting 2D electron systems. Using various perturbative and non-perturbative approximation schemes in order to establish the general shape of the electron spin susceptibility as function of its wave vector, we show that the nuclear spins locally order ferromagnetically, and that this ordering can become global in certain regimes of interest. We demonstrate that the associated Curie temperature for the nuclear system increases with the electron-electron interactions up to the millikelvin range.

Spin dynamics in InAs-nanowire quantum-dots coupled to a transmission line

Mircea Trif, Vitaly N. Golovach (LMU Munich), and Daniel Loss. Phys. Rev. B 77, 045434 (2008); arXiv:0708.2091v1. We study theoretically electron spins in nanowire quantum dots placed inside a transmission line resonator. Because of the spin-orbit interaction, the spins couple to the electric component of the resonator electromagnetic field and enable coherent manipulation, storage, and read-out of quantum information in an all-electrical fashion. Coupling between distant quantum-dot spins, in one and the same or different nanowires, can be efficiently performed via the resonator mode either in real time or through virtual processes. For the latter case we derive an effective spin-entangling interaction and suggest means to turn it on and off. We consider both transverse and longitudinal types of nanowire quantum-dots and compare their manipulation timescales against the spin relaxation times. For this, we evaluate the rates for spin relaxation induced by the nanowire vibrations (phonons) and show that, as a result of phonon confinement in the nanowire, this rate is a strongly varying function of the spin operation frequency and thus can be drastically reduced compared to lateral quantum dots in GaAs. Our scheme is a step forward to the formation of hybrid structures where qubits of different nature can be integrated in a single device.

CNOT for Multi-Particle Qubits and Topological Quantum Computation based on Parity Measurements

Oded Zilberberg, Bernd Braunecker, and Daniel Loss. Phys. Rev. A 77, 012327 (2008); arXiv:0708.1062v1. We discuss a measurement-based implementation of a Controlled-NOT (CNOT) quantum gate. Such a gate has recently been discussed for free electron qubits. Here we extend this scheme for qubits encoded in product states of two (or more) spins-1/2 or in equivalent systems. The key to such an extension is to find a feasible qubit-parity meter. We present a general scheme for reducing this meter to a local spin-parity measurement performed on two spins, one from each qubit. Two possible realizations of a multi-particle CNOT are further discussed: Electron spins in double quantum dots in the singlet-triplet encoding and nu=5/2 Ising non-Abelian anyons using topological quantum computation braiding operations and nontopological charge measurements.

Theory of spin qubits in nanostructures

B. Trauzettel, M. Borhani, M. Trif, and D. Loss. J. Phys. Soc. Jpn. 77 (2008) 031012; arXiv:0707.4622v1. We review recent advances on the theory of spin qubits in nanostructures. We focus on four sel​ected topics. First, we show how to form spin qubits in the new and promising material graphene. Afterwards, we discuss spin relaxation and decoherence in quantum dots. In particular, we demonstrate how charge fluctations in the surrounding environment cause spin decay via spin--orbit coupling. We then turn to a brief overview of how one can use electron-dipole spin resonance (EDSR) to perform single spin rotations in quantum dots using an oscillating electric field. The final topic we cover is the spin-spin coupling via spin-orbit interaction which is an alternative to the usual spin-spin coupling via the Heisenberg exchange interaction.

Electron and hole spin dynamics and decoherence in quantum dots

D. Klauser, D. V. Bulaev, W. A. Coish, and Daniel Loss. Chapter 10 in Semiconductor Quantum Bits, eds. O. Benson and F. Henneberger, World Scientific, 2008. ISBN 978-981-4241-05-2 arXiv:0706.1514v1 In this article we review our work on the dynamics and decoherence of electron and hole spins in single and double quantum dots. The first part, on electron spins, focuses on decoherence induced via the hyperfine interaction while the second part covers decoherence and relaxation of heavy-hole spins due to spin-orbit interaction as well as the manipulation of heavy-hole spin using electric dipole spin resonance.

Polynomial-time algorithm for simulation of weakly interacting quantum spin systems

Sergey Bravyi (IBM Yorktown), David DiVincenzo (IBM Yorktown), and Daniel Loss. Commun. Math. Phys. 284, 481 (2008); arXiv:0707.1894. We describe an algorithm that computes the ground state energy and correlation functions for 2-local Hamiltonians in which interactions between qubits are weak compared to single-qubit terms. The running time of the algorithm is polynomial in the number of qubits and the required precision. Specifically, we consider Hamiltonians of the form $H=H_0+\epsilon V$, where H_0 describes non-interacting qubits, V is a perturbation that involves arbitrary two-qubit interactions on a graph of bounded degree, and $\epsilon$ is a small parameter. The algorithm works if $|\epsilon|$ is below a certain threshold value that depends only upon the spectral gap of H_0, the maximal degree of the graph, and the maximal norm of the two-qubit interactions. The main technical ingredient of the algorithm is a generalized Kirkwood-Thomas ansatz for the ground state. The parameters of the ansatz are computed using perturbative expansions in powers of $\epsilon$. Our algorithm is closely related to the coupled cluster method used in quantum chemistry.

Highly Entangled Ground States in Tripartite Qubit Systems

Beat Röthlisberger, Jörg Lehmann, D. S. Saraga, Philipp Traber, and Daniel Loss. Phys. Rev. Lett. 100, 100502 (2008); arXiv:0705.1710v1 [quant-ph]. We investigate the creation of highly entangled ground states in a system of three exchange-coupled qubits arranged in a ring geometry. Suitable magnetic field configurations yielding approximate GHZ and exact W ground states are identified. The entanglement in the system is studied at finite temperature in terms of the mixed-state tangle tau. By adapting a steepest-descent optimization algorithm we demonstrate that tau can be evaluated efficiently and with high precision. We identify the parameter regime for which the equilibrium entanglement of the tripartite system reaches its maximum.

Observation of extremely slow hole spin relaxation in self-assembled quantum dots

D. Heiss (1), S. Schaeck (1), H. Huebl (1), M. Bichler (1), G. Abstreiter (1), J. J. Finley (1), D. V. Bulaev, Daniel Loss ((1), and Technische Universität München). Phys. Rev. B 76, 241306 (2007); cond-mat/0705.1466. We report the measurement of extremely slow hole spin relaxation dynamics in self-assembled InGaAs quantum dots. Individual spin orientated holes are optically created in the lowest orbital state of each dot and read out after a defined storage time using spin memory devices. The hole spin relaxation time (T_1h) is measured as a function of the external magnetic field and lattice temperature. As predicted by theory, hole spin relaxation can occur over remarkably long timescales in strongly confined quantum dots (up to T_1h ~270 \mus) comparable to the corresponding time for electrons. Our findings are supported by calculations that reproduce both the observed magnetic field and temperature dependencies. The results show that hole spin relaxation in strongly confined quantum dots is governed by spin-lattice interaction, in marked contrast to higher dimensional nanostructures where it is limited by spin-orbit coupling between valence bands.

Spin qubits with electrically gated polyoxometalate molecules

Jörg Lehmann, Alejandro Gaita-Ariño, Eugenio Coronado (Valencia), and Daniel Loss. Nature Nanotech. 2, 312 (2007); News and Views, Nature Nanotech. 2, 271 (2007); cond-mat/0703501. Spin qubits offer one of the most promising routes to the implementation of quantum computers. Very recent results in semiconductor quantum dots show that electrically-controlled gating schemes are particularly well-suited for the realization of a universal set of quantum logical gates. Scalability to a larger number of qubits, however, remains an issue for such semiconductor quantum dots. In contrast, a chemical bottom-up approach allows one to produce identical units in which localized spins represent the qubits. Molecular magnetism has produced a wide range of systems with tailored properties, but molecules permitting electrical gating have been lacking. Here we propose to use the polyoxometalate [PMo12O40(VO)2]q-, where two localized spins-1/2 can be coupled through the electrons of the central core. Via electrical manipulation of the molecular redox potential, the charge of the core can be changed. With this setup, two-qubit gates and qubit readout can be implemented.

Universal phase shift and non-exponential decay of driven single-spin oscillations

F.H.L. Koppens (TU Delft), D. Klauser, W. A. Coish, K. C. Nowack (TU Delft), L.P. Kouwenhoven (TU Delft), D. Loss, and L.M.K. Vandersypen (TU Delft). Phys. Rev. Lett. 99, 106803 (2007); cond-mat/0703640. We study, both theoretically and experimentally, driven Rabi oscillations of a single electron spin coupled to a nuclear spin bath. Due to the long correlation time of the bath, two unusual features are observed in the oscillations. The decay follows a power law, and the oscillations are shifted in phase by a universal value of ~pi/4. These properties are well understood from a theoretical expression that we derive here in the static limit for the nuclear bath. This improved understanding of the coupled electron-nuclear system is important for future experiments using the electron spin as a qubit.

Spin densities in parabolic quantum wires with Rashba spin-orbit interaction

S. I. Erlingsson (Reykjavik), J. C. Egues (Sao Paulo), and D. Loss. Phys. Stat. Sol. (c) 3, 4317 (2006) (PASPS IV Proceedings); cond-mat/0701564. Using canonical transformations we diagonalize approximately the Hamiltonian of a gaussian wire with Rashba spin-orbit interaction. This proceedure allows us to obtain the energy dispersion relations and the wavefunctions with good accuracy, even in systems with relatively strong Rashba coupling. With these eigenstates one can calculate the non-equilibrium spin densities induced by applying bias voltages across the sample. We focus on the $z$-component of the spin density, which is related to the spin Hall effect.

Spin orbit interaction and zitterbewegung in symmetric wells

E Bernardes (São Paulo), J. Schliemann (Regensburg), J. C. Egues (São Paulo), and D. Loss. Phys. Stat. Sol. (c) 3, No. 12, 4330 - 4333 (2006); (PASPS IV Proceedings). Recently, we have introduced a novel inter-subband-induced spin-orbit (s-o) coupling (cond-mat/0607218) arising in symmetric wells with at least two subbands. This new s-o coupling gives rise to an usual zitterbewegung -- i.e. the semiconductor analog to the relativistic trembling motion of electrons -- with cycloidal motion without magnetic fields. Here we complement these findings by explicitly deriving expressions for the corresponding zitterbewegung in spin space.

Direct Measurement of the Spin-Orbit Interaction in a Two-Electron InAs Nanowire Quantum Dot

C. Fasth, A. Fuhrer, L. Samuelson (Lund University), Vitaly N. Golovach (LMU Munich), and Daniel Loss. Phys. Rev. Lett. 98, 266801 (2007); cond-mat/0701161. We demonstrate control of the electron number down to the last electron in tunable few-electron quantum dots defined in catalytically grown InAs nanowires. Using low temperature transport spectroscopy in the Coulomb blockade regime we propose a simple method to directly determine the magnitude of the spin-orbit interaction in a two-electron artificial atom with strong spin-orbit coupling. Due to a large effective g-factor |g*|=8+/-1 the transition from singlet S to triplet T+ groundstate with increasing magnetic field is dominated by the Zeeman energy rather than by orbital effects. We find that the spin-orbit coupling mixes the T+ and S states and thus induces an avoided crossing with magnitude $\DeltaSO$=0.25+/-0.05 meV. This allows us to calculate the spin-orbit length $\lambdaSO\approx$127 nm in such systems using a simple model.

Resonant spin polarization and spin current in a two-dimensional electron gas

Mathias Duckheim and Daniel Loss. Phys. Rev. B 75, 201305 (2007); cond-mat/0701559. We study the spin polarization and its associated spin-Hall current due to EDSR in disordered two-dimensional electron systems. We show that the disorder induced damping of the resonant spin polarization can be strongly reduced by an optimal field configuration that exploits the interference between Rashba and Dresselhaus spin-orbit interaction. This leads to a striking enhancement of the spin susceptibility while the spin-Hall current vanishes at the same time. We give an interpretation of the spin current in geometrical terms which are associated with the trajectories the polarization describes in spin space.

Spin relaxation at the singlet-triplet transition in a quantum dot

Vitaly N. Golovach (LMU Munich), Alexander Khaetskii (Chernogolovka), and Daniel Loss. Phys. Rev. B 77, 045328 (2008); cond-mat/0703427. We study spin relaxation in a two-electron quantum dot in the vicinity of the singlet-triplet transition. The spin relaxation occurs due to a combined effect of the spin-orbit, Zeeman, and electron-phonon interactions. The singlet-triplet relaxation rates exhibit strong variations as a function of the singlet-triplet splitting. We show that the Coulomb interaction between the electrons has two competing effects on the singlet-triplet spin relaxation. One effect is to enhance the relative strength of spin-orbit coupling in the quantum dot, resulting in larger spin-orbit splittings and thus in a stronger coupling of spin to charge. The other effect is to make the charge density profiles of the singlet and triplet look similar to each other, thus diminishing the ability of charge environments to discriminate between singlet and triplet states. We thus find essentially different channels of singlet-triplet relaxation for the case of strong and weak Coulomb interaction. Finally, for the linear in momentum Dresselhaus and Rashba spin-orbit interactions, we calculate the singlet-triplet relaxation rates to leading order in the spin-orbit interaction, and find that they are proportional to the second power of the Zeeman energy, in agreement with recent experiments on triplet-to-singlet relaxation in quantum dots.

Nuclear spin ferromagnetic phase transition in an interacting 2D electron gas

Pascal Simon and Daniel Loss. Phys. Rev. Lett. 98, 156401 (2007); cond-mat/0611292. Electrons in a two-dimensional semiconducting heterostructure interact with nuclear spins via the hyperfine interaction. Using a a Kondo lattice formulation of the electron-nuclear spin interaction, we show that the nuclear spin system within an interacting two-dimensional electron gas undergoes a ferromagnetic phase transition at finite temperatures. We find that electron-electron interactions and non-Fermi liquid behavior substantially enhance the nuclear spin Curie temperature into the $mK$ range with decreasing electron density.

Spin qubits in graphene quantum dots

B. Trauzettel, Denis V. Bulaev, Daniel Loss, and Guido Burkard. Nature Physics 3, 192 (2007); News and Views; Research Highlights; cond-mat/0611252. We propose how to form spin qubits in graphene. A crucial requirement to achieve this goal is to find quantum dot states where the usual valley degeneracy in bulk graphene is lifted. We show that this problem can be avoided in quantum dots based on ribbons of graphene with semiconducting armchair boundaries. For such a setup, we find the energies and the exact wave functions of bound states, which are required for localized qubits. Additionally, we show that spin qubits in graphene can not only be coupled between nearest neighbor quantum dots via Heisenberg exchange interaction but also over long distances. This remarkable feature is a direct consequence of the quasi-relativistic spectrum of graphene. See also http://www.nature.com/nnano/reshigh/2006/1106/full/nnano.2006.167.html

Quantum vs. classical hyperfine-induced dynamics in a quantum dot

W. A. Coish, E. A. Yuzbashyan (Rutgers University), B. L. Altshuler (Columbia University), and Daniel Loss. J. Appl. Phys. 101, 081715 (2007); cond-mat/0610633. In this article we analyze spin dynamics for electrons confined to semiconductor quantum dots due to the contact hyperfine interaction. We compare mean-field (classical) evolution of an electron spin in the presence of a nuclear field with the exact quantum evolution for the special case of uniform hyperfine coupling constants. We find that (in this special case) the zero-magnetic-field dynamics due to the mean-field approximation and quantum evolution are similar. However, in a finite magnetic field, the quantum and classical solutions agree only up to a certain time scale t<\tau_c, after which they differ markedly.

Exchange-controlled single-electron-spin rotations in quantum dots

W. A. Coish and Daniel Loss. Phys. Rev. B 75, 161302 (2007) (R); cond-mat/0610443. We show theoretically that arbitrary coherent rotations can be performed quickly (with a gating time ~1 ns) and with high fidelity on the spin of a single electron confined to a quantum dot using exchange. These rotations can be performed using experimentally proven techniques for rapid exchange control, without the need for spin-orbit interaction or ac electromagnetic fields. We expect that implementations of this scheme would achieve gate error rates on the order of $\eta\lesssim 10-3$, within reach of several known error-correction schemes.

Transport through a quantum dot with SU(4) Kondo entanglement

Karyn Le Hur (Yale), Pascal Simon, and Daniel Loss. Phys. Rev. B 75, 035332 (2007); cond-mat/0609298. We investigate a mesoscopic setup composed of a small electron droplet (dot) coupled to a larger quantum dot (grain) also subject to Coulomb blockade as well as two macroscopic leads used as source and drain. An exotic Kondo ground state other than the standard SU(2) Fermi liquid unambiguously emerges: an SU(4) Kondo correlated liquid. The transport properties through the small dot are analyzed for this regime, through boundary conformal field theory, and allow a clear distinction with other regimes such as a two-channel spin state or a two-channel orbital state.

Measurement, control, and decay of quantum-dot spins

W. A. Coish, Vitaly N. Golovach, J. Carlos Egues, and Daniel Loss. Physica Status Solidi (b) 243, 3658 (2006); cond-mat/0606782. In this review we discuss a recent proposal to perform partial Bell-state (parity) measurements on two-electron spin states for electrons confined to quantum dots. The realization of this proposal would allow for a physical implementation of measurement-based quantum computing. In addition, we consider the primary sources of energy relaxation and decoherence which provide the ultimate limit to all proposals for quantum information processing using electron spins in quantum dots. We give an account of the Hamiltonians used for the most important interactions (spin-orbit and hyperfine) and survey some of the recent work done to understand dynamics, control, and decoherence under the action of these Hamiltonians. We conclude the review with a table of important decay times found in experiment, and relate these time scales to the potential viability of measurement-based quantum computing.

Sequential Tunneling through Molecular Spin Rings

Jörg Lehmann and Daniel Loss. Phys. Rev. Lett. 98, 117203 (2007); cond-mat/0608642. We consider electrical transport through molecules with Heisenberg-coupled spins arranged in a ring structure in the presence of an easy-axis anisotropy. The molecules are coupled to two metallic leads and a gate. In the charged state of the ring, a Zener double-exchange mechanism links transport properties to the underlying spin structure. This leads to a remarkable contact-site dependence of the current, which for an antiferromagnetic coupling of the spins can lead to a total suppression of the zero-bias conductance when the molecule is contacted at adjacent sites.

Spin-spin coupling in electrostatically coupled quantum dots

Mircea Trif, Vitaly N. Golovach, and Daniel Loss. Phys. Rev. B 75, 085307 (2007); cond-mat/0608512. We study the spin-spin coupling between two single-electron quantum dots due to the Coulomb and spin-orbit interactions, in the absence of tunneling between the dots. We find an anisotropic XY spin-spin interaction that is proportional to the Zeeman splitting produced by the external magnetic field. This interaction is studied both in the limit of weak and strong Coulomb repulsion with respect to the level spacing of the dot. The interaction is found to be a non-monotonic function of inter-dot distance $a_0$ and external magnetic field, and, moreover, vanishes for some special values of $a_0$ and/or magnetic field orientation. This mechanism thus provides a new way to generate and tune spin interaction between quantum dots. We propose a scheme to measure this spin-spin interaction based on the spin-relaxation-measurement technique.

Electric Dipole Spin Resonance for Heavy Holes in Quantum Dots

Denis V. Bulaev and Daniel Loss. Phys. Rev. Lett. 98, 097202 (2007); cond-mat/0608410. We propose and analyze a new method for manipulation of a heavy hole spin in a quantum dot. Due to spin-orbit coupling between states with different orbital momenta and opposite spin orientations, an applied rf electric field induces transitions between spin-up and spin-down states. This scheme can be used for detection of heavy-hole spin resonance signals, for the control of the spin dynamics in two-dimensional systems, and for determining important parameters of heavy-holes such as the effective $g$-factor, mass, spin-orbit coupling constants, spin relaxation and decoherence times.

Spin-orbit interaction in symmetric wells and cycloidal orbits without magnetic fields

Esmerindo S. Bernardes, John Schliemann (Regensburg), J. Carlos Egues, and Daniel Loss. Phys. Rev. Lett. 99, 076603 (2007); cond-mat/0607218. We investigate the spin-orbit (s-o) interaction in two-dimensional electron gases (2DEGs) in quantum wells with two subbands. From the $8\times 8$ Kane model, we derive a new inter-subband-induced s-o coupling which resembles the functional form of the Rashba s-o -- but is non-zero even in \emph{symmetric} structures. This follows from the distinct parity of the confined states (even/odd) which obliterates the need for asymmetric potentials. Interestingly, our s-o interaction gives rise to an unusual \emph{zitterbewegung} of spin-polarized electrons with cycloidal trajectories \textit{without} magnetic fields. We also predict a sizable effective-mass renormalization due to the s-o--induced subband warping.

Quantum computing with spins in solids

W. A. Coish and Daniel Loss. Handbook of Magnetism and Advanced Magnetic Materials. Helmut Kronmüller (editor) and Stuart Parkin (editor). Volume 5: Spintronics and Magnetoelectronics. ; 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-02217-7; cond-mat/0606550. The ability to perform high-precision one- and two-qubit operations is sufficient for universal quantum computation. For the Loss-DiVincenzo proposal to use single electron spins confned to quantum dots as qubits, it is therefore sufficient to analyze only single- and coupled double-dot structures, since the strong Heisenberg exchange coupling between spins in this proposal falls off exponentially with distance and long-ranged dipolar coupling mechanisms can be made significantly weaker. This scalability of the Loss-DiVincenzo design is both a practical necessity for eventual applications of multi-qubit quantum computing and a great conceptual advantage, making analysis of the relevant components relatively transparent and systematic. We review the Loss-DiVincenzo proposal for quantum-dot-confned electron spin qubits, and survey the current state of experiment and theory regarding the relevant single- and double- quantum dots, with a brief look at some related alternative schemes for quantum computing.

Quantum-dot spin qubit and hyperfine interaction

D. Klauser, W. A. Coish, and Daniel Loss. Advances in Solid State Physics 46, p.17-29, (2007); cond-mat/0604252. We review our investigation of the spin dynamics for two electrons confined to a double quantum dot under the influence of the hyperfine interaction between the electron spins and the surrounding nuclei. Further we propose a scheme to narrow the distribution of difference in polarization between the two dots in order to suppress hyperfine induced decoherence.

Electric-dipole-induced spin resonance in disordered semiconductors (Article)

Mathias Duckheim and Daniel Loss. Nature Physics 2, 195-199 (2006); Supplementary Information; cond-mat/0605735. One of the hallmarks of spintronics is the control of magnetic moments by electric fields enabled by strong spin-orbit interaction (SOI) in semiconductors. A powerful way of manipulating spins in such structures is electric-dipole-induced spin resonance (EDSR), where the radio-frequency fields driving the spins are electric, not magnetic as in standard paramagnetic resonance. Here, we present a theoretical study of EDSR for a two-dimensional electron gas in the presence of disorder, where random impurities not only determine the electric resistance but also the spin dynamics through SOI. Considering a specific geometry with the electric and magnetic fields parallel and in-plane, we show that the magnetization develops an out-of-plane component at resonance that survives the presence of disorder. We also discuss the spin Hall current generated by EDSR. These results are derived in a diagrammatic approach, with the dominant effects coming from the spin vertex correction, and the optimal parameter regime for observation is identified. See also, 'Semiconductor physics: Electric fields drive spins', by Emmanuel I. Rashba, Nature Physics 2, 149-150 (01 Mar 2006) News and Views

Molecular states in carbon nanotube double quantum dots

M.R. Graeber, W.A. Coish, C. Hoffmann, M. Weiss, J. Furer, S. Oberholzer, D. Loss, and C. Schoenenberger. Phys. Rev. B 74, 075427 (2006); cond-mat/0603367. We report electrical transport measurements through a semiconducting single-walled carbon nanotube (SWNT) with three additional top-gates. At low temperatures the system acts as a double quantum dot with large inter-dot tunnel coupling allowing for the observation of tunnel-coupled molecular states extending over the whole double-dot system. We precisely extract the tunnel coupling and identify the molecular states by the sequential-tunneling line shape of the resonances in differential conductance.

Electric Dipole Induced Spin Resonance in Quantum Dots

Vitaly N. Golovach, Massoud Borhani, and Daniel Loss. Phys. Rev. B 74, 165319 (2006); cond-mat/0601674. An alternating electric field, applied to a ``spin 1/2'' quantum dot, couples to the electron spin via the spin-orbit interaction. We analyze different types of spin-orbit couplings known in the literature and find that an electric dipole spin resonance (EDSR) scheme for spin manipulation can be realized with the up-to-date experimental setups. In particular, for the Rashba and Dresselhaus spin-orbit couplings, a fully transverse effective magnetic field arises in the presence of a Zeeman splitting in the lowest order of spin-orbit interaction. Spin manipulation and measurement of the spin decoherence time $T_2$ are straightforward in lateral GaAs quantum dots through the use of EDSR.

Dynamics of Coupled Qubits Interacting with an Off-Resonant Cavity

Oliver Gywat (UCSB), Florian Meier (UCSB), Daniel Loss, and D. D. Awschalom (UCSB). Phys. Rev. B 73, 125336 (2006); cond-mat/0511592. We study a model for a pair of qubits which interact with a single off-resonant cavity mode and, in addition, exhibit a direct inter-qubit coupling. Possible realizations for such a system include coupled superconducting qubits in a line resonator as well as exciton states or electron spin states of quantum dots in a cavity. The emergent dynamical phenomena are strongly dependent on the relative energy scales of the inter-qubit coupling strength, the coupling strength between qubits and cavity mode, and the cavity mode detuning. We show that the cavity mode dispersion enables a measurement of the state of the coupled-qubit system in the perturbative regime. We discuss the effect of the direct inter-qubit interaction on a cavity-mediated two-qubit gate. Further, we show that for asymmetric coupling of the two qubits to the cavity, the direct inter-qubit coupling can be controlled optically via the ac Stark effect.

Spin Decay in a Quantum Dot Coupled to a Quantum Point Contact

Massoud Borhani, Vitaly N. Golovach, and Daniel Loss. Phys. Rev. B 73, 155311 (2006); cond-mat/0510758. We consider a mechanism of spin decay for an electron spin in a quantum dot due to coupling to a nearby quantum point contact (QPC) with and without an applied bias voltage. The coupling of spin to charge is induced by the spin-orbit interaction in the presence of a magnetic field. We perform a microscopic calculation of the effective Hamiltonian coupling constants to obtain the QPC-induced spin relaxation and decoherence rates in a realistic system. This rate is shown to be proportional to the shot noise of the QPC in the regime of large bias voltage and scales as $a-6$ where $a$ is the distance between the quantum dot and the QPC. We find that, for some specific orientations of the setup with respect to the crystallographic axes, the QPC-induced spin relaxation and decoherence rates vanish, while the charge sensitivity of the QPC is not changed. This result can be used in experiments to minimize QPC-induced spin decay in read-out schemes.

Zitterbewegung of electrons and holes in III-V semiconductor quantum wells

John Schliemann (Regensburg), Daniel Loss, and R.M. Westervelt (Harvard). Phys. Rev. B 73, 085323 (2006); cond-mat/0512148. The notion of zitterbewegung is a long-standing prediction of relativistic quantum mechanics. Here we extend earlier theoretical studies on this phenomenon for the case of III-V zinc-blende semiconductors which exhibit particularly strong spin-orbit coupling. This property makes nanostructures made of these materials very favorable systems for possible experimental observations of zitterbewegung. Our investigations include electrons in n-doped quantum wells under the influence of both Rashba and Dresselhaus spin-orbit interaction, and also the two-dimensional hole gas. Moreover, we give a detailed anaysis of electron zitterbewegung in quantum wires which appear to be particularly suited for experimentally observing this effect.

A Mesoscopic Resonating Valence Bond system on a triple dot

Karyn Le Hur (Sherbrooke), Patrik Recher (Stanford), Emilie Dupont (Sherbrooke), and Daniel Loss. Phys. Rev. Lett. 96, 106803 (2006); cond-mat/0510450. We theoretically introduce a mesoscopic pendulum from a triple dot. The pendulum is fastened through a singly-occupied dot (spin qubit). Two other strongly capacitively coupled islands form a double-dot charge qubit with one electron in excess oscillating between the two low-energy charge states (1, 0) and (0, 1). The triple dot is placed between two superconducting leads. Under realistic conditions, the main proximity effect stems from the injection of resonating singlet (valence) bonds on the triple dot. This gives rise to a Josephson current that is charge- and spin-dependent and, as a consequence, exhibits a distinct resonance as a function of the superconducting phase difference.

Nuclear spin state narrowing via gate--controlled Rabi oscillations in a double quantum dot

D. Klauser, W.A. Coish, and Daniel Loss. Phys. Rev. B 73, 205302 (2006); cond-mat/0510177. We study spin dynamics for two electrons confined to a double quantum dot under the influence of an oscillating exchange interaction. This leads to driven Rabi oscillations between the $\ket{\uparrow\downarrow}$--state and the $\ket{\downarrow\uparrow}$--state of the two--electron system. The width of the Rabi resonance is proportional to the amplitude of the oscillating exchange. A measurement of the Rabi resonance allows one to narrow the distribution of nuclear spin states and thereby to prolong the spin decoherence time. Further, we study decoherence of the two-electron states due to the hyperfine interaction and give requirements on the parameters of the system in order to initialize in the $\ket{\uparrow\downarrow}$--state and to perform a $\sqrt{\mathrm{SWAP}}$ operation with unit fidelity.

Cotunneling current through quantum dots with phonon-assisted spin-flip processes

Jörg Lehmann and Daniel Loss. Phys. Rev. B 73, 045328 (2006); cond-mat/0509420. We consider cotunneling through a quantum dot in the presence of spin-flip processes induced by the coupling to acoustic phonons of the surrounding. An expression for the phonon-assisted cotunneling current is derived by means of a generalized Schrieffer-Wolff transformation. The influence of the spin-phonon coupling on the heating of the dot is considered. The result is evaluated for the case of a parabolic semiconductor quantum dot with Rashba and Dresselhaus spin-orbit coupling and a novel method for the determination of the spin-phonon relaxation rate is proposed.

Shot noise and spin-orbit coherent control of entangled and spin polarized electrons

J. Carlos Egues (Sao Paulo), Guido Burkard, D. Saraga, John Schliemann, and Daniel Loss. Phys. Rev. B 72, 235326 (2005); cond-mat/0509038. We extend our previous work on shot noise for entangled and spin polarized electrons in a beam-splitter geometry with spin-orbit (\textit{s-o}) interaction in one of the incoming leads (lead 1). Besides accounting for both the Dresselhaus and the Rashba spin-orbit terms, we present general formulas for the shot noise of singlet and triplets states derived within the scattering approach. We determine the full scattering matrix of the system for the case of leads with \textit{two} orbital channels coupled via weak \textit{s-o} interactions inducing channel anticrossings. We show that this interband coupling coherently transfers electrons between the channels and gives rise to an additional modulation angle -- dependent on both the Rashba and Dresselhaus interaction strengths -- which allows for further independent coherent control of the electrons traversing the incoming leads. We derive explicit shot noise formulas for a variety of correlated pairs (e.g., Bell states) and lead spin polarizations. Interestingly, the singlet and \textit{each} of the triplets defined along the quantization axis perpendicular to lead 1 (with the local \textit{s-o} interaction) and in the plane of the beam splitter display distinctive shot noise for injection energies near the channel anticrossings; hence, one can tell apart all the triplets, in addition to the singlet, through noise measurements. We also find that spin-orbit induced backscattering within lead 1 reduces the visibility of the noise oscillations, due to the additional partition noise in this lead. Finally, we consider injection of two-particle wavepackets into leads with multiple discrete states and find that two-particle entanglement can still be observed via noise bunching and antibunching.

Phase coherence in the inelastic cotunneling regime

M. Sigrist, T. Ihn, K. Ensslin (ETH Zurich), D. Loss, M. Reinwald, and W. Wegscheider (Regensburg). Phys. Rev. Lett. 96, 036804 (2006); cond-mat/0508757. Two quantum dots with tunable mutual tunnel coupling have been embedded in a two-terminal Aharonov-Bohm geometry. Aharonov-Bohm oscillations are investigated in the cotunneling regime. Visibilities of more than 0.8 are measured indicating that phase-coherent processes are involved in the elastic and inelastic cotunneling. An oscillation-phase change of pi is detected as a function of bias voltage at the inelastic cotunneling onset.

Fermionic Bell-State Analyzer for Spin Qubits

Hans-Andreas Engel (Harvard Univ.) and Daniel Loss. Science 309, 586 (2005) We propose a protocol and physical implementation for partial Bell-state measurements of Fermionic qubits, allowing for deterministic quantum computing in solid-state systems without the need for two-qubit gates. Our scheme consists of two spin qubits in a double quantum dot where the two dots have different Zeeman splittings and resonant tunneling between the dots is only allowed when the spins are antiparallel. This converts spin parity into charge information by means of a projective measurement and can be implemented with established technologies. This measurement-based qubit scheme greatly simplifies the experimental realization of scalable quantum computers in electronic nanostructures. [See also Science Perspective, Fingerprinting Spin Qubits, J. Carlos Egues, Science , Vol 309, Issue 5734, 565 (2005); Philip Ball, Nature news: Quantum computers go for a spin. ]

Singlet-triplet decoherence due to nuclear spins in a double quantum dot

W. A. Coish and Daniel Loss. Phys. Rev. B 72, 125337 (2005); cond-mat/0506090. We have evaluated hyperfine-induced electron spin dynamics for two electrons confined to a double quantum dot. Our quantum solution accounts for decay of a singlet-triplet correlator even in the presence of a fully static nuclear spin system, with no ensemble averaging over initial conditions. In contrast to an earlier semiclassical calculation, which neglects the exchange interaction, we find that the singlet-triplet correlator shows a long-time saturation value that differs from 1/2, even in the presence of a strong magnetic field. Furthermore, we find that the form of the long-time decay undergoes a transition from a rapid Gaussian to a slow power law ($\sim 1/t3/2$) when the exchange interaction becomes nonzero and the singlet-triplet correlator acquires a phase shift given by a universal (parameter independent) value of $3\pi/4$ at long times. The oscillation frequency and time-dependent phase shift of the singlet-triplet correlator can be used to perform a precision measurement of the exchange interaction and Overhauser field fluctuations in an experimentally accessible system. We also address the effect of orbital dephasing on singlet-triplet decoherence, and find that there is an optimal operating point where orbital dephasing becomes negligible.

Fermi liquid parameters in 2D with spin-orbit interaction

D. S. Saraga and Daniel Loss. Phys. Rev. B 72, 195319 (2005); cond-mat/0504661. We derive analytical expressions for the quasiparticle lifetime \tau, the effective mass m^*, and the Green's function renormalization factor Z for a 2D Fermi liquid with electron-electron interaction in the presence of the Rashba spin-orbit interaction. For the lifetime \tau and the renormalization Z, we find that the modification is independent of the Rashba band index \rho, and occurs in second order of the spin-orbit coupling \alpha . On the contrary, the modification of the effective mass m^* is linear in \alpha and is different for the two Rashba bands, yielding a spin-dependent effective mass. In the derivation of these results, we also discuss the screening of the Coulomb interaction, and the susceptibility.

Determining the spin Hall conductance via charge transport

Sigurdur I. Erlingsson and Daniel Loss. Phys. Rev. B 72, 121310 (2005); cond-mat/0503605. We propose a scheme where transport measurements of charge current and its noise can be used to determine the spin Hall conductance in a four-terminal setup. Starting from the scattering formalism we express the spin current and spin Hall conductance in terms of spin-dependent transmission coefficients. These coefficients are then expressed in terms of charge current and noise. We use the scheme to characterize the spin injection efficiency of a ferromagnetic/semiconductor interface.

Zitterbewegung of electronic wave packets in semiconductor nanostructures

John Schliemann, Daniel Loss, and R.M. Westervelt (Harvard Univ.). Phys. Rev. Lett. 94, 206801 (2005); cond-mat/0410321. We study the zitterbewegung of electronic wave packets in III-V zinc-blende semiconductor quantum wells due to spin-orbit coupling. Our results suggest a direct experimental proof of this fundamental effect, confirming a long-standing theoretical prediction. For electron motion in a harmonic quantum wire, we numerically and analytically find a resonance condition maximizing the zitterbewegung. See also, 'Dirac gets the jitters', M. Buchanan, Nature Physics 1, 5 (2005); http://www.nature.com/nphys/journal/v1/n1/full/nphys132.html

Phonon Bottleneck Effect Leads to Observation of Quantum Tunneling of the Magnetization and Butterfly Hysteresis Loops in (Et4N)3Fe2F9

Ralph Schenker, Michael N. Leuenberger (UC San Diego), Gregory Chaboussant (Uni Bern), Daniel Loss, and Hans U. Gudel (Uni Bern). Phys. Rev. B 72, 184403 (2005); cond-mat/0502548. A detailed investigation of the unusual dynamics of the magnetization of (Et4N)3Fe2F9 (Fe2), containing isolated [Fe2F9]3- dimers, is presented and discussed. Fe2 possesses an S=5 ground state with an energy barrier of 2.40 K due to an axial anisotropy. Poor thermal contact between sample and bath leads to a phonon bottleneck situation, giving rise to butterfly-shaped hysteresis loops below 5 K concomitant with slow decay of the magnetization for magnetic fields Hz applied along the Fe--Fe axis. The butterfly curves are reproduced using a microscopic model based on the interaction of the spins with resonant phonons. The phonon bottleneck allows for the observation of resonant quantum tunneling of the magnetization at 1.8 K, far above the blocking temperature for spin-phonon relaxation. The latter relaxation is probed by AC magnetic susceptibility experiments at various temperatures and bias fields. At H=0, no out-of-phase signal is detected, indicating that at T smaller than 1.8 K Fe2 does not behave as a single-molecule magnet. At 1 kG, relaxation is observed, occurring over the barrier of the thermally accessible S=4 first excited state that forms a combined system with the S=5 state.

Spin relaxation and decoherence of holes in quantum dots

Denis V. Bulaev and Daniel Loss. Phys. Rev. Lett. 95, 076805 (2005); cond-mat/0503181. We investigate heavy-hole spin relaxation and decoherence in quantum dots in perpendicular magnetic fields. We show that at low temperatures the spin decoherence time is two times longer than the spin relaxation time. We find that the spin relaxation time for heavy holes can be comparable to or even longer than that for electrons in strongly two-dimensional quantum dots. We discuss the difference in the magnetic-field dependence of the spin relaxation rate due to Rashba or Dresselhaus spin-orbit coupling for systems with positive (i.e., GaAs quantum dots) or negative (i.e., InAs quantum dots) $g$-factor.

Cluster States From Heisenberg Interaction

Massoud Borhani and Daniel Loss. Phys. Rev. A 71, 034308 (2005); quant-ph/0410145. We show that a special type of entangled states, cluster states, can be created with Heisenberg interactions and local rotations in 2d steps where d is the dimension of the lattice. We find that, by tuning the coupling strengths, anisotropic exchange interactions can also be employed to create cluster states. Finally, we propose electron spins in quantum dots as a possible realization of a one-way quantum computer based on cluster states.

Recipes for spin-based quantum computing

Veronica Cerletti, W. A. Coish, Oliver Gywat, and Daniel Loss. Nanotechnology 16, R27 (2005); cond-mat/0412028. Technological growth in the electronics industry has historically been measured by the number of transistors that can be crammed onto a single microchip. Unfortunately, all good things must come to an end; spectacular growth in the number of transistors on a chip requires spectacular reduction of the transistor size. For electrons in semiconductors, the laws of quantum mechanics take over at the nanometre scale, and the conventional wisdom for progress (transistor cramming) must be abandoned. This realization has stimulated extensive research on ways to exploit the spin (in addition to the orbital) degree of freedom of the electron, giving birth to the field of spintronics. Perhaps the most ambitious goal of spintronics is to realize complete control over the quantum mechanical nature of the relevant spins. This prospect has motivated a race to design and build a spintronic device capable of complete control over its quantum mechanical state, and ultimately, performing computations: a quantum computer. In this tutorial we summarize past and very recent developments which point the way to spin-based quantum computing in the solid-state. After introducing a set of basic requirements for any quantum computer proposal, we offer a brief summary of some of the many theoretical proposals for solid-state quantum computers. We then focus on the Loss-DiVincenzo proposal for quantum computing with the spins of electrons confined to quantum dots. There are many obstacles to building such a quantum device. We address these, and survey recent theoretical, and then experimental progress in the field. To conclude the tutorial, we list some as-yet unrealized experiments, which would be crucial for the development of a quantum-dot quantum computer.

Entanglement transfer from electron spins to photons

Veronica Cerletti, Oliver Gywat, and Daniel Loss. Phys. Rev. B 72, 115316 (2005); cond-mat/0411235. We show that electron recombination in spin light-emitting diodes provides an efficient method to transfer entanglement from electron spins onto pairs of polarization-entangled photons. Because of the interplay of quantum mechanical sel​ection rules and interference, maximally entangled electron pairs are converted into maximally entangled photon pairs with unity fidelity for a continuous set of observation directions. We describe the dynamics of the conversion process using a master-equation approach and show that the implementation of our scheme is feasible with current experimental techniques.

Double Occupancy Errors in Quantum Computing Operations: Corrections to Adiabaticity

Ryan Requist (SUNY), John Schliemann, Alexander G. Abanov (SUNY), and Daniel Loss. Phys. Rev. B 71, 115315 (2005); cond-mat/0409096. We study the corrections to adiabatic dynamics of two coupled quantum dot spin-qubits, each dot singly occupied with an electron, in the context of a quantum computing operation. Tunneling can lead to a double occupancy at the conclusion of an operation and constitutes a processing error. Our model for the dynamics of an effective two-level system is integrable and possesses three independent parameters. We confirm the accuracy of Dykhne's formula, a nonperturbative estimate of transitions, and discuss physically intuitive conditions for its validity. Our semiclassical results are in excellent agreement with numerical simulations of the exact time evolution. A similar approach applies to two-level systems in different contexts.

Spin Relaxation and Anticrossing in Quantum Dots: Rashba versus Dresselhaus Spin-Orbit Coupling

Denis V. Bulaev and Daniel Loss. Phys. Rev. B 71, 205324 (2005); cond-mat/0409614. The spin-orbit splitting of the electron levels in a two-dimensional quantum dot in a perpendicular magnetic field is studied. It is shown that at the point of an accidental degeneracy of the two lowest levels above the ground state the Rashba spin-orbit coupling leads to a level anticrossing and to mixing of spin-up and spin-down states, whereas there is no mixing of these levels due to the Dresselhaus term. We calculate the relaxation and decoherence times of the three lowest levels due to phonons. We find that the spin relaxation rate as a function of a magnetic field exhibits a cusp-like structure for Rashba but not for Dresselhaus spin-orbit interaction.

Spin-Hall conductivity due to Rashba spin-orbit interaction in disordered systems

Oleg Chalaev and Daniel Loss. Phys. Rev. B 71, 245318 (2005); cond-mat/0407342. We consider the spin-Hall current in a disordered two-dimensional electron gas in the presence of Rashba spin-orbit interaction. We derive a generalized Kubo-Greenwood formula for the spin-Hall conductivity $\sigma$ and evaluate it in an systematic way using standard diagrammatic techniques for disordered systems. We find that in the diffusive regime both Boltzmann and the weak localization contributions to the $\sigma$ vanish in the zero frequency limit. We show that the uniform spin current is given by the total time derivative of the magnetization from which we can conclude that the spin current vanishes exactly in the stationary limit. This conclusion is valid for arbitrary spin-independent disorder, external electric field strength, and also for interacting electrons.

Controlling Spin Qubits in Quantum Dots

Hans-Andreas Engel, L.P. Kouwenhoven (Delft), Daniel Loss, and C.M. Marcus (Harvard). Quantum Information Processing, 3, 115-132 (2004); cond-mat/0409294. We review progress on the spintronics proposal for quantum computing where the quantum bits (qubits) are implemented with electron spins. We calculate the exchange interaction of coupled quantum dots and present experiments, where the exchange coupling is measured via transport. Then, experiments on single spins on dots are described, where long spin relaxation times, on the order of a millisecond, are observed. We consider spin-orbit interaction as sources of spin decoherence and find theoretically that also long decoherence times are expected. Further, we describe the concept of spin filtering using quantum dots and show data of successful experiments. We also show an implementation of a read out scheme for spin qubits and define how qubits can be measured with high precision. Then, we propose new experiments, where the spin decoherence time and the Rabi oscillations of single electrons can be measured via charge transport through quantum dots. Finally, all these achievements have promising applications both in conventional and quantum information processing.

Asymmetric Quantum Shot Noise in Quantum Dots

Hans-Andreas Engel and Daniel Loss. Phys. Rev. Lett. 93, 136602 (2004); cond-mat/0312107. We analyze the frequency-dependent noise of a current through a quantum dot which is coupled to Fermi leads and which is in the Coulomb blockade regime. We show that the asymmetric shot noise as function of frequency shows steps and becomes super-Poissonian. This provides experimental access to the quantum fluctuations of the current. We present an exact calculation for a single dot level and a perturbative evaluation of the noise in Born approximation (sequential tunneling regime but without Markov approximation) for the general case of many levels with charging interaction.

Reduced Visibility of Rabi Oscillations in Superconducting Qubits

Florian Meier (UC Santa Barbara) and Daniel Loss. Phys. Rev. B 71, 094519 (2005); cond-mat/0408594. Coherent Rabi oscillations between quantum states of superconducting micro-circuits have been observed in a number of experiments, albeit with a visibility which is typically much smaller than unity. Here, we show that the coherent coupling to background charge fluctuators [R.W. Simmonds et al., Phys. Rev. Lett. 93, 077003 (2004)] leads to a significantly reduced visibility if the Rabi frequency is comparable to the coupling energy of micro-circuit and fluctuator. For larger Rabi frequencies, transitions to the second excited state of the superconducting micro-circuit become dominant in suppressing the Rabi oscillation visibility. We also calculate the probability for Bogoliubov quasi-particle excitations in typical Rabi oscillation experiments.

Probing Single-Electron Spin Decoherence in Quantum Dots using Charged Excitons

Oliver Gywat, Hans-Andreas Engel, and Daniel Loss. Journal of Superconductivity 18 (2), 175 - 183 ( 2005); cond-mat/0408451. We propose to use optical detection of magnetic resonance (ODMR) to measure the decoherence time T2 of a single electron spin in a semiconductor quantum dot. The electron is in one of the spin 1/2 states and a circularly polarized laser can only create an optical excitation for one of the electron spin states due to Pauli blocking. An applied electron spin resonance (ESR) field leads to Rabi spin flips and thus to a modulation of the photoluminescence or, alternatively, of the photocurrent. This allows one to measure the ESR linewidth and the coherent Rabi oscillations, from which the electron spin decoherence can be determined. We study different possible schemes for such an ODMR setup, including cw or pulsed laser excitation.

Coulomb scattering cross-section in a 2D electron gas and production of entangled electrons

D. S. Saraga, B. L. Altshuler (Princeton), Daniel Loss, and R. M. Westervelt (Harvard). Phys. Rev. B 71, 045338 (2005); cond-mat/0408362. We calculate the Coulomb scattering amplitude for two electrons injected with opposite momenta in an interacting 2DEG. We include the effect of the Fermi liquid background by solving the 2D Bethe-Salpeter equation for the two-particle Green function vertex, in the ladder and random phase approximations. This result is used to discuss the feasibility of producing spin EPR pairs in a 2DEG by collecting electrons emerging from collisions at a pi/2 scattering angle, where only the entangled spin-singlets avoid the destructive interference resulting from quantum indistinguishability. Furthermore, we study the effective 2D electron-electron interaction due to the exchange of virtual acoustic and optical phonons, and compare it to the Coulomb interaction. Finally, we show that the 2D Kohn-Luttinger pairing instability for the scattering electrons is negligible in a GaAs 2DEG.

Creation and detection of mobile and non-local spin-entangled electrons

Patrik Recher (Stanford), D.S. Saraga, and Daniel Loss. pp. 179-202, in Fundamental Problems of Mesoscopic Physics, eds. I.V. Lerner et al., NATO Science Ser. II, Vol. 154 (Kluwer, Dordrecht, 2004); cond-mat/0408526. We present electron spin entanglers--devices creating mobile spin-entangled electrons that are spatially separated--where the spin-entanglement in a superconductor present in form of Cooper pairs and in a single quantum dot with a spin singlet groundstate is transported to two spatially separated leads by means of a correlated two-particle tunneling event. The unwanted process of both electrons tunneling into the same lead is suppressed by strong Coulomb blockade effects caused by quantum dots, Luttinger liquid effects or by resistive outgoing leads. In this review we give a transparent description of the different setups, including discussions of the feasibility of the subsequent detection of spin-entanglement via charge noise measurements. Finally, we show that quantum dots in the spin filter regime can be used to perform Bell-type measurements that only require the measurement of zero frequency charge noise correlators.

Spin susceptibilities, spin densities and their connection to spin-currents

Sigurdur I. Erlingsson, John Schliemann, and Daniel Loss. Phys. Rev. B 71, 035319 (2005); cond-mat/0406531. We calculate the frequency dependent spin susceptibilities for a two-dimensional electron gas with both Rashba and Dresselhaus spin-orbit interaction. The resonances of the susceptibilities depends on the relative values of the Rashba and Dresselhaus spin-orbit constants, which could be manipulated by gate voltages. We derive exact continuity equations, with source terms, for the spin density and use those to connect the spin current to the spin density. In the free electron model the susceptibilities play a central role in the spin dynamics since both the spin density and the spin current are proportional to them.

Rigorous Born Approximation and beyond for the Spin-Boson Model

D. P. DiVincenzo (IBM) and D. Loss. The Mathematica file can be obtained directly here. Phys. Rev. B 71, 035318 (2005); cond-mat/0405525. Within the lowest-order Born approximation, we present an exact calculation of the time dynamics of the spin-boson model in the ohmic regime. We observe non-Markovian effects at zero temperature that scale with the system-bath coupling strength and cause qualitative changes in the evolution of coherence at intermediate times of order of the oscillation period. These changes could significantly affect the performance of these systems as qubits. In the biased case, we find a prompt loss of coherence at these intermediate times, whose decay rate is set by $\sqrt{\alpha}$, where $\alpha$ is the coupling strength to the environment. We also explore the calculation of the next order Born approximation: we show that, at the expense of very large computational complexity, interesting physical quantities can be rigorously computed at fourth order using computer algebra, presented completely in an accompanying Mathematica file. We compute the $O(\alpha)$ corrections to the long time behavior of the system density matrix; the result is identical to the reduced density matrix of the equilibrium state to the same order in $\alpha$. All these calculations indicate precision experimental tests that could confirm or refute the validity of the spin-boson model in a variety of systems.

Hyperfine interaction in a quantum dot: Non-Markovian electron spin dynamics

W. A. Coish and Daniel Loss. Phys. Rev. B 70, 195340 (2004); cond-mat/0405676. We have performed a systematic calculation for the non-Markovian dynamics of a localized electron spin interacting with an environment of nuclear spins via the Fermi contact hyperfine interaction. This work applies to an electron in the s -type orbital ground state of a quantum dot or bound to a donor impurity, and is valid for arbitrary polarization p of the nuclear spin system, and arbitrary nuclear spin I in high magnetic fields. In the limit of p=1 and I=1/2, the Born approximation of our perturbative theory recovers the exact electron spin dynamics. We have found the form of the generalized master equation (GME) for the longitudinal and transverse components of the electron spin to all orders in the electron spin--nuclear spin flip-flop terms. Our perturbative expansion is regular, unlike standard time-dependent perturbation theory, and can be carried-out to higher orders. We show this explicitly with a fourth-order calculation of the longitudinal spin dynamics. In zero magnetic field, the fraction of the electron spin that decays is bounded by the smallness parameter \delta=1/p2N, where N is the number of nuclear spins within the extent of the electron wave function. However, the form of the decay can only be determined in a high magnetic field, much larger than the maximum Overhauser field. In general the electron spin shows rich dynamics, described by a sum of contributions with non-exponential decay, exponential decay, and undamped oscillations. There is an abrupt crossover in the electron spin asymptotics at a critical dimensionality and shape of the electron envelope wave function. We propose a scheme that could be used to measure the non-Markovian dynamics using a standard spin-echo technique, even when the fraction that undergoes non-Markovian dynamics is small.

Spin-Hall transport of heavy holes in III-V semiconductor quantum wells

John Schliemann and Daniel Loss. Phys. Rev. B 71, 085308 (2005); cond-mat/0405436. We investigate spin transport of heavy holes in III-V semiconductor quantum wells in the presence of spin-orbit coupling of the Rashba type due to structure-inversion asymmetry. Similarly to the case of electrons, the longitudinal spin conductivity vanishes, whereas the off-diagonal elements of the spin-conductivity tensor are finite giving rise to an intrinsic spin-Hall effect. For a clean system we find a closed expression for the spin-Hall conductivity depending on the length scale of the Rashba coupling and the hole density. In this limit the spin-Hall conductivity is enhanced compared to its value for electron systems, and it vanishes with increasing strength of the impurity scattering. As an aside, we also derive explicit expressions for the Fermi momenta and the densities of holes in the different dispersion branches as a function of the spin-orbit coupling parameter and the total hole density. These results are of relevance for the interpretation of possible Shubnikov-de Haas measurements detecting the Rashba spin splitting.

Grover algorithm with large nuclear spins in semiconductors

Michael N. Leuenberger and Daniel Loss. Phys. Rev. B 68, 165317 (2003); cond-mat/0304674. We show a possible way to implement the Grover algorithm in large nuclear spins 1/2<I<9/2 in semiconductors. The Grover sequence is performed by means of multiphoton transitions that distribute the spin amplitude between the nuclear spin states. They are distinguishable due to the quadrupolar splitting, which makes the nuclear spin levels non-equidistant. We introduce a generalized rotating frame for an effective Hamiltonian that governs the non-perturbative time evolution of the nuclear spin states for arbitrary spin lengths I. The larger the quadrupolar splitting, the better the agreement between our approximative method using the generalized rotating frame and exact numerical calculations.

Spin injection across magnetic/nonmagnetic interfaces with finite magnetic layers

Alexander Khaetskii, J. Carlos Egues, Daniel Loss, Charles Gould, Georg Schmidt, and Laurens W. Molenkamp (Wuerzburg). Phys. Rev. B 71, 235327 (2005); cond-mat/0312705. We have reconsidered the relevant problem of spin injection across ferromagnet/non-magnetic-semiconductor (FM/NMS) and dilute-magnetic-semiconductor/non-magnetic-semiconductor interfaces, for structures with \textit{finite} magnetic layers (FM or DMS). By using appropriate physical boundary conditions, we find new expressions for the resistances of these structures which are in general different from previous results in the literature. The results obtained can be important for the interpretation of the experimental data in the case of DMS/NMS structures.

Molecular spintronics: Coherent spin transfer in coupled quantum dots

Florian Meier (UCSB), Veronica Cerletti, Oliver Gywat, Daniel Loss, and D. D. Awschalom (UCSB). Phys. Rev. B 69, 195315 (2004); cond-mat/0401397. Time-resolved Faraday rotation has recently demonstrated coherent transfer of electron spin between quantum dots coupled by conjugated molecules. Using a transfer Hamiltonian ansatz for the coupled quantum dots, we calculate the Faraday rotation signal as a function of the probe frequency in a pump-probe setup using neutral quantum dots. Additionally, we study the signal of one spin-polarized excess electron in the coupled dots. We show that, in both cases, the Faraday rotation angle is determined by the spin transfer probabilities and the Heisenberg spin exchange energy. By comparison of our results with experimental data, we find that the transfer matrix element for electrons in the conduction band is of order 0.08 eV and the spin transfer probabilities are of order 10%.

Towards Quantum Communication with Electron Spins

D.S. Saraga, G. Burkard, J.C. Egues, H.-A. Engel, P. Recher, and D. Loss. Turk J Phys 27, 427 (2003) Proceedings of the Quantum Computation at the Atomic Scale Conference, (Istanbul, 1-11 June, 2003) We review our recent work towards quantum communication in a solid-state environment with qubits carried by electron spins. We propose three schemes to produce spin-entangled electrons, where the required separation of the partner electrons is achieved via Coulomb interaction. The non-product spin-states originate either from the Cooper pairs found in a superconductor, or in the ground state of a quantum dot with an even number of electrons. In a second stage, we show how spin-entanglement carried by a singlet can be detected in a beam-splitter geometry by an increased (bunching) or decreased (antibunching) noise signal. We also discuss how a local spin-orbit interaction can be used to provide a continuous modulation of the noise as a signature of entanglement. Finally, we review how one can use a quantum dot as a spin- lter, a spin-memory read-out, a probe for single-spin decoherence and ultimately, a single-spin measurement apparatus.

Electron spin dynamics in quantum dots and related nanostructures due to hyperfine interaction with nuclei

John Schliemann, Alexander Khaetskii, and Daniel Loss. J. Phys.: Condens. Matter 15, R1809-R1833 (2003); cond-mat/0311159. We review and summarize recent theoretical and experimental work on electron spin dynamics in quantum dots and related nanostructures due to hyperfine interaction with surrounding nuclear spins. This topic is of particular interest with respect to several proposals for quantum information processing in solid state systems. Specifically, we investigate the hyperfine interaction of an electron spin confined in a quantum dot in an s-type conduction band with the nuclear spins in the dot. This interaction is proportional to the square modulus of the electron wave function at the location of each nucleus leading to an inhomogeneous coupling, i.e. nuclei in different locations are coupled with different strength. In the case of an initially fully polarized nuclear spin system an exact analytical solution for the spin dynamics can be found. For not completely polarized nuclei, approximation-free results can only be obtained numerically in sufficiently small systems. We compare these exact results with findings from several approximation strategies.

Phonon-induced decay of the electron spin in quantum dots

Vitaly N. Golovach, Alexander Khaetskii, and Daniel Loss. Phys. Rev. Lett. 93, 016601 (2004); cond-mat/0310655. We study spin relaxation and decoherence in a GaAs quantum dot due to spin-orbit interaction. We derive an effective Hamiltonian which couples the electron spin to phonons or any other fluctuation of the dot potential. We show that the spin decoherence time $T_2$ is as large as the spin relaxation time $T_1$, under realistic conditions. For the Dresselhaus and Rashba spin-orbit couplings, we find that, in leading order, the effective magnetic field can have only fluctuations transverse to the applied magnetic field. As a result, $T_2=2T_1$ for arbitrarily large Zeeman splittings, in contrast to the naively expected case $T_2\ll T_1$. We show that the spin decay is drastically suppressed for certain magnetic field directions and values of the Rashba coupling constant. Finally, for the spin coupling to acoustic phonons, we show that $T_2=2T_1$ for all spin-orbit mechanisms in leading order in the electron-phonon interaction.

Coulomb scattering in a 2D interacting electron gas and production of EPR pairs

D.S. Saraga, B.L. Altshuler (Princeton), Daniel Loss, and R.M. Westervelt (Harvard). Phys. Rev. Lett. 92, 246803 (2004); cond-mat/0310421. We propose a setup to generate non-local spin-EPR pairs via pair collisions in a 2D interacting electron gas, based on constructive two-particle interference in the spin singlet channel at the pi/2 scattering angle. We calculate the scattering amplitude via the Bethe-Salpeter equation in the ladder approximation and small r_s limit, and find that the Fermi sea leads to a substantial renormalization of the bare scattering process. From the scattering length we estimate the current of spin-entangled electrons and show that it is within experimental reach.

Dissipation effects in spin-Hall transport of electrons and holes

John Schliemann and Daniel Loss. Phys. Rev. B 69, 165315 (2004); cond-mat/0310108. We investigate the spin-Hall effect of both electrons and holes in semiconductors using the Kubo formula in the correct zero-frequency limit taking into account the finite momentum relaxation time of carriers in real semiconductors. This approach allows to analyze the range of validity of recent theoretical findings. In particular, the spin-Hall conductivity vanishes for vanishing spin-orbit coupling if the correct zero-frequency limit is performed.

Measurement efficiency and n-shot read out of spin qubits

Hans-Andreas Engel, Vitaly Golovach, Daniel Loss, L.M.K. Vandersypen (TU Delft), J.M. Elzerman (TU Delft), R. Hanson (TU Delft), and L.P. Kouwenhoven (TU Delft). Phys. Rev. Lett. 93, 106804 (2004); cond-mat/0309023. We consider electron spin qubits in quantum dots and define a measurement efficiency e to characterize reliable measurements via n-shot read outs. We propose various implementations based on a double dot and quantum point contact (QPC) and show that the associated efficiencies e vary between 50% and 100%, allowing single-shot read out in the latter case. We model the read out microscopically and derive its time dynamics in terms of a generalized master equation, calculate the QPC current and show that it allows spin read out under realistic conditions.

Transport through a double quantum dot in the sequential- and co- tunneling regimes

Vitaly N. Golovach and Daniel Loss. Phys. Rev. B 69, 245327 (2004); cond-mat/0308241. We study transport through a double quantum dot, both in the sequential tunneling and cotunneling regimes. Using a master equation approach, we find that, in the sequential tunneling regime, the differential conductance $G$ as a function of the bias voltage $\Delta\mu$ has a number of satellite peaks with respect to the main peak of the Coulomb blockade diamond. The position of these peaks is related to the interdot tunnel splitting and the singlet-triplet splitting. We find satellite peaks with both {\em positive} and {\em negative} values of differential conductance for realistic parameter regimes. Relating our theory to a microscopic (Hund-Mulliken) model for the double dot, we find a temperature regime for which the Hubbard ratio (=tunnel coupling over on-site Coulomb repulsion) can be extracted from $G(\Delta\mu)$ in the cotunneling regime. In addition, we consider a combined effect of cotunneling and sequential tunneling, which leads to new peaks (dips) in $G(\Delta\mu)$ inside the Coulomb blockade diamond below some temperature scales, which we specify.

Optical Detection of Single-Electron Spin Decoherence in a Quantum Dot

Oliver Gywat, Hans-Andreas Engel, Daniel Loss, R. J. Epstein, F. Mendoza, and D. D. Awschalom (UC Santa Barbara). Phys. Rev. B 69, 205303 (2004); cond-mat/0307669. We propose a method based on optically detected magnetic resonance (ODMR) to measure the decoherence time $T2$ of a single electron spin in a semiconductor quantum dot. The electron spin resonance (ESR) of a single excess electron on a quantum dot is probed by circularly polarized laser excitation. The photoluminescence is modulated due to the ESR which enables the measurement of electron spin decoherence. We study different possible schemes for such an ODMR setup.

Dynamical Coulomb blockade and spin-entangled electrons

Patrik Recher and Daniel Loss. Phys. Rev. Lett. 91, 267003 (2003); cond-mat/0307444. We consider the production of mobile and nonlocal pairwise spin-entangled electrons from tunneling of a BCS-superconductor (SC) to two normal Fermi liquid leads. The necessary mechanism to separate the two electrons coming from the same Cooper pair (spin-singlet) is achieved by coupling the SC to leads with a finite resistance. The resulting dynamical Coulomb blockade effect, which we describe phenomenologically in terms of an electromagnetic environment, is shown to be enhanced for tunneling of two spin-entangled electrons into the same lead compared to the process where the pair splits and each electron tunnels into a different lead. On the other hand in the pair-split process, the spatial correlation of a Cooper pair leads to a current suppression as a function of distance between the two tunnel junctions which is weaker for effectively lower dimensional SCs.

Hyperfine interactions and electron spin dynamics in a quantum dot

A. Khaetskii, D. Loss, and L. Glazman (Minnesota). Journal of Superconductivity: Incorporating Novel Magnetism 16, 221 (2003) We show that a wide range of spin clusters with antiferromagnetic intracluster exchange interaction allows one to define a qubit. For these spin cluster qubits, initialization, quantum gate operation, and readout are possible using the same techniques as for single spins. Quantum gate operation for the spin cluster qubit does not require control over the intracluster exchange interaction. Electric and magnetic fields necessary to effect quantum gates need only be controlled on the length scale of the spin cluster rather than the scale for a single spin. Here, we calculate the energy gap separating the logical qubit states from the next excited state and the matrix elements which determine quantum gate operation times. We discuss spin cluster qubits formed by one- and two-dimensional arrays of s=1/2 spins as well as clusters formed by spins s>1/2. We illustrate the advantages of spin cluster qubits for various suggested implementations of spin qubits and analyze the scaling of decoherence time with spin cluster size.

Anisotropic transport in the two-dimensional electron gas in the presence of spin-orbit coupling

John Schliemann and Daniel Loss. Phys. Rev. B 68, 165311 (2003); cond-mat/0306528. In a two-dimensional electron gas as realized by a semiconductor quantum well, the presence of spin-orbit coupling of both the Rashba and Dresselhaus type leads to anisotropic dispersion relations and Fermi contours. We study the effect of this anisotropy on the electrical conductivity in the presence of fixed impurity scatterers. The conductivity also shows in general an anisotropy which can be tuned by varying the Rashba coefficient. This effect provides a method of detecting and investigating spin-orbit coupling by measuring spin-unpolarized electrical currents in the diffusive regime. Our approach is based on an exact solution of the two-dimensional Boltzmann equation and provides also a natural framework for investigating other transport effects including the anomalous Hall effect.

Noise of Spin-Polarized Currents at a Beam Splitter with Local Spin-Orbit Interaction

G. Burkard, J. C. Egues (Sao Paulo), and D. Loss. J. Supercond. 16, 237 (2003) An electronic beam splitter with a local Rashba spin-orbit coupling can serve as a detector for spin-polarized currents. The spin-orbit coupling plays the role of a tunable spin rotator and can be controlled via a gate electrode on top of the conductor. We use spin-resolved scattering theory to calculate the zero-temperature current fluctuations (shot noise) for such a four-terminal device and show that the shot noise is proportional to the spin polarization of the source. Moreover, we analyze the effect of spin-orbit-induced intersubband coupling, leading to an additional spin rotation.

Quantum computing with antiferromagnetic spin clusters

Florian Meier, Jeremy Levy (Pittsburgh), and Daniel Loss. Phys. Rev. B 68, 134417 (2003); cond-mat/0304296. We show that a wide range of spin clusters with antiferromagnetic intracluster exchange interaction allows one to define a qubit. For these spin cluster qubits, initialization, quantum gate operation, and readout are possible using the same techniques as for single spins. Quantum gate operation for the spin cluster qubit does not require control over the intracluster exchange interaction. Electric and magnetic fields necessary to effect quantum gates need only be controlled on the length scale of the spin cluster rather than the scale for a single spin. Here, we calculate the energy gap separating the logical qubit states from the next excited state and the matrix elements which determine quantum gate operation times. We discuss spin cluster qubits formed by one- and two-dimensional arrays of s = 1/2 spins as well as clusters formed by spins s > 1/2. We illustrate the advantages of spin cluster qubits for various suggested implementations of spin qubits and analyze the scaling of decoherence time with spin cluster size.

Exact Born Approximation for the Spin-Boson Model

Daniel Loss and David P. DiVincenzo (IBM Yorktown). cond-mat/0304118 Within the lowest-order Born approximation, we present an exact calculation of the time dynamics of the spin-boson model in the Ohmic regime. We observe non-Markovian effects at zero temperature that scale with the system-bath coupling strength and cause qualitative changes in the evolution of coherence at intermediate times of order of the oscillation period. These changes could significantly affect the performance of these systems as qubits. In the biased case, we find a prompt loss of coherence at these intermediate times, whose decay rate is set by $\sqrt{\alpha}$, where $\alpha$ is the coupling strength to the environment. These calculations indicate precision experimental tests that could confirm or refute the validity of the spin-boson model in a variety of systems.

Shot Noise of Cotunneling Current

Eugene Sukhorukov (Geneva), Guido Burkard (IBM Yorktown), and Daniel Loss. in "Quantum Noise in Mesoscopic Physics", ed. Y.V. Nazarov, pp 149-172, Kluwer, 2003, The Netherlands; cond-mat/0211024. We study the noise of the cotunneling current through one or several tunnel-coupled quantum dots in the Coulomb blockade regime. The various regimes of weak and strong, elastic and inelastic cotunneling are analyzed for quantum-dot systems (QDS) with few-level, nearly-degenerate, and continuous electronic spectra. In the case of weak cotunneling we prove a non-equilibrium fluctuation-dissipation theorem, which leads to a universal expression for the noise-to-current ratio (Fano factor). The noise of strong inelastic cotunneling can be super-Poissonian due to switching between QDS states carrying currents of different strengths. The transport through a double-dot (DD) system shows an Aharonov-Bohm effect both in noise and current. In the case of cotunneling through a QDS with a continuous energy spectrum the Fano factor is very close to one.

Lower bound for electron spin entanglement from beamsplitter current correlations

Guido Burkard (IBM Yorktown Heights) and Daniel Loss. Phys. Rev. Lett. 91, 087903 (2003); cond-mat/0303209. We determine a lower bound for the entanglement of pairs of electron spins injected into a mesoscopic conductor. The bound can be expressed in terms of experimentally accessible quantities, the zero-frequency current correlators (shot noise power or cross-correlators) after transmission through an electronic beam splitter. The effect of spin relaxation (T_1 processes) and decoherence (T_2 processes) during the ballistic coherent transmission of the carriers in the wires is taken into account within Bloch theory. The presence of a variable inhomogeneous magnetic field allows the determination of a useful lower bound for the entanglement of arbitrary entangled states. The decrease in entanglement due to thermally mixed states is studied. Both the entanglement of the output of a source (entangler) and the relaxation (T_1) and decoherence (T_2) times can be determined.

Discrete Fourier Transform in Nanostructures using Scattering

Michael N. Leuenberger (Iowa), Michael E. Flatte (Iowa), Daniel Loss, and D. D. Awschalom (UCSB). J. Appl. Phys. 95, 8167 (2004); cond-mat/0302279. In this paper we show that the discrete Fourier transform can be performed by scattering a coherent particle or laser beam off a two-dimensional potential that has the shape of rings or peaks. After encoding the initial vector into the two-dimensional potential, the Fourier-transformed vector can be read out by detectors surrounding the potential. The wavelength of the laser beam determines the necessary accuracy of the 2D potential, which makes our method very fault-tolerant.

Spin-Orbit Coupling and Time-Reversal Symmetry in Quantum Gates

D. Stepanenko (Florida State), N. E. Bonesteel (Florida State), D.P. DiVincenzo (IBM Yorktown), G. Burkard (IBM Yorktown), and D. Loss. Phys. Rev. B 68, 115306 (2003); cond-mat/0303474. We study the effect of spin-orbit coupling on quantum gates produced by pulsing the exchange interaction between two single electron quantum dots. Spin-orbit coupling enters as a small spin precession when electrons tunnel between dots. For adiabatic pulses the resulting gate is described by a unitary operator acting on the four-dimensional Hilbert space of two qubits. If the precession axis is fixed, time-symmetric pulsing constrains the set of possible gates to those which, when combined with single qubit rotations, can be used in a simple CNOT construction. Deviations from time-symmetric pulsing spoil this construction. The effect of time asymmetry is studied by numerically integrating the Schr\"odinger equation using parameters appropriate for GaAs quantum dots. Deviations of the implemented gate from the desired form are shown to be proportional to dimensionless measures of both spin-orbit coupling and time asymmetry of the pulse.

Spin Stiffness of Mesoscopic Quantum Antiferromagnets

Daniel Loss and Dmitrii L. Maslov. Phys. Rev. Lett. 74, 178 (1995) We study the spin stiffness of a one-dimensional quantum antiferromagnet in the whole range of system sizes L and temperatures T. We show that for integer and half-odd integer spin cases the stiffness differs fundamentally in its L and T dependences, and that in the latter case the stiffness exhibits a striking dependence on the parity of the number of sites. Integer spin chains are treated in terms of the nonlinear sigma model, while half-odd integer spin chains are discussed in a renormalization group approach leading to a Luttinger liquid with Aharonov-Bohm-type boundary conditions.

Quantum Information is Physical

David P. DiVincenzo (IBM Yorktown) and Daniel Loss. Superlattices and Microstructures 23, 419 (1998); cond-mat/9710259. We discuss a few current developments in the use of quantum mechanically coherent systems for information processing. In each of these developments, Rolf Landauer has played a crucial role in nudging us and other workers in the field into asking the right questions, some of which we have been lucky enough to answer. A general overview of the key ideas of quantum error correction is given. We discuss how quantum entanglement is the key to protecting quantum states from decoherence in a manner which, in a theoretical sense, is as effective as the protection of digital data from bit noise. We also discuss five general criteria which must be satisfied to implement a quantum computer in the laboratory, and we illustrate the application of these criteria by discussing our ideas for creating a quantum computer out of the spin states of coupled quantum dots. [Special Issue on the Occasion of Rolf Landauer's 70th Birthday.]

Quantum information processing and communication - Strategic report on current status, visions and goals for research in Europe

Zoller P, Beth T, Binosi D, Blatt R, Briegel H, Bruss D, Calarco T, Cirac JI, Deutsch D, Eisert J, Ekert A, Fabre C, Gisin N, Grangiere P, Grassl M, Haroche S, Imamoglu A, Karlson A, Kempe J, Kouwenhoven L, Kroll S, Leuchs G, Lewenstein M, Loss D, Lutkenhaus N, Massar S, Mooij JE, Plenio MB, Polzik E, Popescu S, Rempe G, Sergienko A, Suter D, Twamley J, Wendin G, Werner R, Winter A, Wrachtrup J, and Zeilinger A. Eur. Phys. J. D 36, 203 (2005) We present an excerpt of the document "Quantum Information Processing and Communication: Strategic report on current status, visions and goals for research in Europe", which has been recently published in electronic form at the website of FET (the Future and Emerging Technologies Unit of the Directorate General Information Society of the European Commission, http://qurope.eu/content/Roadmap). This document has been elaborated, following a former suggestion by FET, by a committee of QIPC scientists to provide input towards the European Commission for the preparation of the Seventh Framework Program. Besides being a document addressed to policy makers and funding agencies (both at the European and national level), the document contains a detailed scientific assessment of the state-of-the-art, main research goals, challenges, strengths, weaknesses, visions and perspectives of all the most relevant QIPC sub-fields, that we report here.

Probing entanglement via Rashba-induced shot noise oscillations

J. Carlos Egues, Guido Burkard, and Daniel Loss. J. Superconductivity, 16, 711 (2003); cond-mat/0207392. We have recently calculated shot noise for entangled and spin-polarized electrons in novel beam-splitter geometries with a local Rashba s-o interaction in the incoming leads. This interaction allows for a gate-controlled rotation of the incoming electron spins. Here we present an alternate simpler route to the shot noise calculation in the above work and focus on only electron pairs. Shot noise for these shows continuous bunching and antibunching behaviors. In addition, entangled and unentangled triplets yield distinctive shot noise oscillations. Besides allowing for a direct way to identify triplet and singlet states, these oscillations can be used to extract s-o coupling constants through noise measurements. Incoming leads with spin-orbit interband mixing give rise an additional modulation of the current noise. This extra rotation allows the design of a spin transistor with enhanced spin control.

Shot noise for entangled and spin-polarized electrons

J. C. Egues, P. Recher, D. S. Saraga, V. N. Golovach, G. Burkard, E. V. Sukhorukov, and D. Loss. Quantum Noise in Mesoscopic Physics, NATO ASI Series II, Vol. 97 (Kluwer, 2003), pp 241-274; Proceedings of the NATO Advanced Research Workshop (Delft, The Netherlands, 2-4 June 2002); cond-mat/0210498. We review our recent contributions on shot noise for entangled electrons and spin-polarized currents in novel mesoscopic geometries. We first discuss some of our recent proposals for electron entanglers involving a superconductor coupled to a double dot in the Coulomb blockade regime, a superconductor tunnel-coupled to Luttinger-liquid leads, and a triple-dot setup coupled to Fermi leads. We calculate current and shot noise for spin-polarized currents and entangled/unentangled electron pairs in a beam-splitter geometry with a \textit{local} Rashba spin-orbit (s-o) interaction in the incoming leads. We find \textit{continuous} bunching and antibunching behaviors for the \textit{entangled} pairs -- triplet and singlet -- as a function of the Rashba rotation angle. In addition, we find that unentangled triplets and the entangled one exhibit distinct shot noise. Shot noise for spin-polarized currents shows sizable oscillations as a function of the Rashba phase. This happens only for electrons injected perpendicular to the Rashba rotation axis; spin-polarized carriers along the Rashba axis are noiseless. We find an additional spin rotation for electrons with energies near the crossing of the bands where s-o induced interband coupling is relevant. This gives rise to an additional modulation of the noise for both electron pairs and spin-polarized currents. Finally, we briefly discuss shot noise for a double dot near the Kondo regime.

Semiconductor Spintronics and Quantum Computation

eds. D.D. Awschalom, D. Loss, and N. Samarth. Springer, Berlin, 2002. The manipulation of electric charge in bulk semiconductors and their heterostructures forms the basis of virtually all contemporary electronic and optoelectronic devices. Recent studies of spin-dependent phenomena in semiconductors have now opened the door to technological possibilities that harness the spin of the electron in semiconductor devices. In addition to providing spin-dependent analogies that extend existing electronic devices into the realm of semiconductor "spintronics" the spin degree of freedom also offers prospects for fundamentally new functionality within the quantum domain, ranging from storage to computation. It is anticipated that the spin degree of freedom in semiconductors will play a crucial role in the development of information technologies in the 21st century. This book brings together a team of experts to provide an overview of emerging concepts in this rapidly developing field. The topics range from spin transport and injection in semiconductors and their heterostructures to coherent processes and quantum computation in semiconductor quantum structures and microcavities.

Dynamics of entanglement between quantum dot spin-qubits

John Schliemann and Daniel Loss. Proceedings of the E. Fermi School, "Quantum Phenomena of Mesoscopic Systems", 9 - 19 July 2002, Varenna, Italy; cond-mat/0212141. We briefly review the physics of gate operations between quantum dot spin-qubits and analyze the dynamics of quantum entanglement in such processes. The indistinguishable character of the electrons whose spins realize the qubits gives rise to further entanglement-like quantum correlations that go beyond simple antisymmetrization effects. We also summarize further recent results concerning this type of quantum correlations of indistinguishable particles. Finally we discuss decoherence properties of spin-qubits when coupled to surrounding nuclear spins in a semiconductor nanostructure

Electron spin evolution induced by interaction with nuclei in a quantum dot

Alexander Khaetskii, Daniel Loss, and Leonid Glazman (Minnesota). Phys. Rev. B 67, 195329 (2003); cond-mat/0211678. We study the decoherence of a single electron spin in an isolated quantum dot induced by hyperfine interaction with nuclei for times smaller than the nuclear spin relaxation time. The decay is caused by the spatial variation of the electron envelope wave function within the dot, leading to a non-uniform hyperfine coupling $A$. We show that the usual treatment of the problem based on the Markovian approximation is impossible because the correlation time for the nuclear magnetic field seen by the electron spin is itself determined by the flip-flop processes. The decay of the electron spin correlation function is not exponential but rather power (inverse logarithm) law-like. For polarized nuclei we find an exact solution and show that the precession amplitude and the decay behavior can be tuned by the magnetic field. The decay time is given by $\hbar N/A$, where $N$ is the number of nuclei inside the dot. The amplitude of precession, reached as a result of the decay, is finite. We show that there is a striking difference between the decoherence time for a single dot and the dephasing time for an ensemble of dots.

Coherent spin quantum dynamics in antiferromagnetic molecular rings

Florian Meier and Daniel Loss. Physica B 329, 1140 (2003) Molecular magnetic clusters with antiferromagnetic exchange interaction and easy axis anisotropy belong to the most promising candidate systems for the observation of coherent spin quantum tunneling on the mesoscopic scale. We point out that both nuclear magnetic resonance and electron spin resonance on doped rings are adequate experimental techniques for the detection of coherent spin quantum tunneling in antiferromagnetic molecular rings. Although challenging, the experiments are feasible with present day techniques.

Non-ballistic spin field-effect transistor

John Schliemann, J. Carlos Egues, and Daniel Loss. Phys. Rev. Lett. 90, 146801 (2003); cond-mat/0211603. We propose a spin field-effect transistor based on spin-orbit (s-o) coupling of both the Rashba and the Dresselhaus types. Differently from earlier proposals, spin transport through our device is tolerant against spin-independent scattering processes. Hence the requirement of strictly ballistic transport can be relaxed. This follows from a unique interplay between the Dresselhaus and the (gate-controlled) Rashba interactions; these can be tuned to have equal strengths thus yielding k-independent eigenspinors even in two dimensions. We discuss implementations with two-dimensional devices and quantum wires. In the latter, our setup presents strictly parabolic dispersions which avoids complications arising from anticrossings of different bands.

Spin qubits in solid-state structures

G. Burkard and D. Loss. Europhysics News 33 5 (2002) 166-170 It is remarkable that today's computers, after the tremendous development during the last 50 years, are still essentially described by the mathematical model formulated by Alan Turing in the 1930's. Turing's model describes computers which operate according to the laws of classical physics. What would happen if a computer was operating according to the quantum laws? Physicists and computer scientists have been interested in this question since the early 1980's, but research in quantum computation really started to flourish after 1994 when Peter Shor discovered a quantum algorithm to find prime factors of large integers efficiently, a problem which is intrinsically hard for any classical computer (see [1] for an introduction into quantum computation). The lack of an algorithm for efficient factoring on a classical machine is actually the basis of the widely used RSA encryption scheme. Phase coherence needs to be maintained for a sufficiently long time in the memory of a quantum computer. This may sound like a harmless requirement, but in fact it is the main reason why the physical implementation of quantum computation is so difficult. Usually, a quantum memory is thought of as a set of two-level systems, named quantum bits, or qubits for short. In analogy to the classical bit, two orthogonal computational basis states |0> and |1> are defined. The textbook example of a quantum two-level system is the spin 1/2 of, say, an electron, where one can identify the "spin up" state with |0> and the "spin down" state with |1>. While several other two-level systems have been proposed for quantum computing, we will devote the majority of our discussion to the potential use of electron spins in nanostructures (such as quantum dots) as qubits.

Magnetization transport and quantized spin conductance

Florian Meier and Daniel Loss. Phys. Rev. Lett. 90, 167204 (2003); cond-mat/0209521. We analyze transport of magnetization in insulating systems described by a spin Hamiltonian. The magnetization current through a quasi one-dimensional magnetic wire of finite length suspended between two bulk magnets is determined by the spin conductance which remains finite in the ballistic limit due to contact resistance. For ferromagnetic systems, magnetization transport can be viewed as transmission of magnons and the spin conductance depends on the temperature T. For antiferromagnetic isotropic spin-1/2 chains, the spin conductance is quantized in units of order $(g \mu_B)^2/h$ at T=0. Magnetization currents produce an electric field and hence can be measured directly. For magnetization transport in electric fields phenomena analogous to the Hall effect emerge.

A Datta-Das transistor with enhanced spin control

J. Carlos Egues, Guido Burkard, and Daniel Loss. Appl. Phys. Lett. 82, 2658 (2003); cond-mat/0209682. We consider a two-channel spin transistor with weak spin-orbit induced interband coupling. We show that the coherent transfer of carriers between the coupled channels gives rise to an additional spin rotation. We calculate the corresponding spin-resolved current in a Datta-Das geometry and show that a weak interband mixing leads to enhanced spin control.

Variational study of the nu=1 quantum Hall ferromagnet in the presence of spin-orbit interaction

John Schliemann, J. Carlos Egues, and Daniel Loss. Phys. Rev. B 67, 085302 (2003); cond-mat/0209185. We investigate the nu=1 quantum Hall ferromagnet in the presence of spin-orbit coupling of the Rashba or Dresselhaus type by means of Hartree-Fock-typed variational states. In the presence of Rashba (Dresselhaus) spin-orbit coupling the fully spin-polarized quantum Hall state is always unstable resulting in a reduction of the spin polarization if the product of the particle charge $q$ and the effective $g$-factor is positive (negative). In all other cases an alternative variational state with O(2) symmetry and finite in-plane spin components is lower in energy than the fully spin-polarized state for large enough spin-orbit interaction. The phase diagram resulting from these considerations differs qualitatively from earlier studies.

Spin decay and quantum parallelism

John Schliemann, Alexander V. Khaetskii, and Daniel Loss. Phys. Rev. B 66, 245303 (2002); cond-mat/0207195. We study the time evolution of a single spin coupled inhomogeneously to a spin environment. Such a system is realized by a single electron spin bound in a semiconductor nanostructure and interacting with surrounding nuclear spins. We find striking dependencies on the type of the initial state of the nuclear spin system. Simple product states show a profoundly different behavior than randomly correlated states whose time evolution provides an illustrative example of quantum parallelism and entanglement in a decoherence phenomenon.

Quantum computing with spin cluster qubits

Florian Meier, Jeremy Levy (Pittsburgh), and Daniel Loss. Phys. Rev. Lett. 90, 047901 (2003); cond-mat/0206310. We study the low energy states of finite spin chains with isotropic (Heisenberg) and anisotropic (XY and Ising-like) exchange interaction with uniform and non-uniform coupling constants. We show that for an odd number of sites a spin cluster qubit can be defined in terms of the ground state doublet. This qubit is remarkably insensitive to the placement and coupling anisotropy of spins within the cluster. One- and two-qubit quantum gates can be generated by magnetic fields and inter-cluster exchange, and leakage during quantum gate operation is small. Spin cluster qubits inherit the long decoherence times and short gate operation times of single spins. Control of single spins is hence not necessary for the realization of universal quantum gates.

Quantum Spin Dynamics in Molecular Magnets

Michael N. Leuenberger, Florian Meier, and Daniel Loss. Monatshefte für Chem. 134, 217(2003); cond-mat/0205457. The detailed theoretical understanding of quantum spin dynamics in various molecular magnets is an important step on the roadway to technological applications of these systems. Quantum effects in both ferromagnetic and antiferromagnetic molecular clusters are, by now, theoretically well understood. Ferromagnetic molecular clusters allow one to study the interplay of incoherent quantum tunneling and thermally activated transitions between states with different spin orientation. The Berry phase oscillations found in Fe_8 are signatures of the quantum mechanical interference of different tunneling paths. Antiferromagnetic molecular clusters are promising candidates for the observation of coherent quantum tunneling on the mesoscopic scale. Although challenging, applications of molecular magnetic clusters for data storage and quantum data processing are within experimental reach already with present day technology.

Spin-entangled currents created by a triple quantum dot

Daniel S. Saraga and Daniel Loss. Phys. Rev. Lett. 90, 166803 (2003); cond-mat/0205553. We propose a simple setup of three coupled quantum dots in the Coulomb blockade regime as a source for spatially separated currents of spin-entangled electrons. The entanglement originates from the singlet ground state of a quantum dot with an even number of electrons. To preserve the entanglement of the electron pair during its extraction to the drain leads, the electrons are transported through secondary dots. This prevents one-electron transport by energy mismatch, while joint transport is resonantly enhanced by conservation of the total two-electron energy.

Rashba spin-orbit interaction and shot noise for spin-polarized and entangled electrons

J. Carlos Egues, Guido Burkard, and Daniel Loss. Phys. Rev. Lett. 89, 176401 (2002); cond-mat/0204639. We study shot noise for spin-polarized currents and entangled electron pairs in a four-probe (beam splitter) geometry with a local Rashba spin-orbit (s-o) interaction in the incoming leads. Within the scattering formalism we find that shot noise exhibits Rashba-induced oscillations with continuous bunching and antibunching. We show that entangled states as well as triplet states can be identified via their Rashba phase in noise measurements. For two-channel leads we find an additional spin rotation due to s-o induced interband coupling which provides additional spin control. We show that the s-o interaction determines the Fano factor which provides a direct way to measure the Rashba coupling constant via noise.

Quantum information processing with large nuclear spins in GaAs semiconductors

Michael N. Leuenberger, Daniel Loss, Martino Poggio (UCSB), and David D. Awschalom (UCSB). Phys. Rev. Lett. 89, 207601 (2002); cond-mat/0204355. We propose an implementation for quantum information processing based on coherent manipulations of nuclear spins I=3/2 in GaAs semiconductors. We describe theoretically an NMR method which involves multiphoton transitions and which exploits the non-equidistance of nuclear spin levels due to quadrupolar splittings. Starting from known spin anisotropies we derive effective Hamiltonians in a generalized rotating frame, valid for arbitrary I, which allow us to describe the non-perturbative time evolution of spin states generated by magnetic rf fields. We identify an experimentally accessible regime where multiphoton Rabi oscillations are observable. In the nonlinear regime, we find Berry phase interference effects.

Superconductor coupled to two Luttinger liquids as an entangler for electron spins

Patrik Recher and Daniel Loss. Phys. Rev. B 65, 165327 (2002); cond-mat/0112298. We consider an s-wave superconductor (SC) which is tunnel-coupled to two spatially separated Luttinger liquid (LL) leads. We demonstrate that such a setup acts as an entangler, i.e. it creates spin-singlets of two electrons which are spatially separated, thereby providing a source of electronic Einstein-Podolsky-Rosen pairs. We show that in the presence of a bias voltage, which is smaller than the energy gap in the SC, a stationary current of spin-entangled electrons can flow from the SC to the LL leads due to Andreev tunneling events. We discuss two competing transport channels for Cooper pairs to tunnel from the SC into the LL leads. On the one hand, the coherent tunneling of two electrons into the same LL lead is shown to be suppressed by strong LL correlations compared to single-electron tunneling into a LL. On the other hand, the tunneling of two spin-entangled electrons into different leads is suppressed by the initial spatial separation of the two electrons coming from the same Cooper pair. We show that the latter suppression depends crucially on the effective dimensionality of the SC. We identify a regime of experimental interest in which the separation of two spin-entangled electrons is favored. We determine the decay of the singlet state of two electrons injected into different leads caused by the LL correlations. Although the electron is not a proper quasiparticle of the LL, the spin information can still be transported via the spin density fluctuations produced by the injected spin-entangled electrons.

Electron spin decoherence in quantum dots due to interaction with nuclei

Alexander Khaetskii, Daniel Loss, and Leonid Glazman (Univ. of Minnesota). Phys. Rev. Lett. 88, 186802 (2002); cond-mat/0201303. We study the decoherence of a single electron spin in an isolated quantum dot induced by hyperfine interaction with nuclei for times smaller than the nuclear spin relaxation time. The decay is caused by the spatial variation of the electron envelope wave function within the dot, leading to a non-uniform hyperfine coupling. We evaluate the spin correlation function with and without magnetic fields and find that the decay of the spin precession amplitude is not exponential but rather power (inverse logarithm) law-like. For fully polarized nuclei we find an exact solution and show that the precession amplitude and the decay behavior can be tuned by the magnetic field. The corresponding decay time is given by $\hbar N/A$, where $A$ is a hyperfine interaction constant and $N$ the number of nuclei inside the dot. The amplitude of precession, reached as a result of the decay, is finite. We show that there is a striking difference between the decoherence time for a single dot and the dephasing time for an ensemble of dots.

Creation of Nonlocal Spin-Entangled Electrons via Andreev Tunneling, Coulomb Blockade, and Resonant Transport

Patrik Recher and Daniel Loss. Journal of Superconductivity: Incorporating Novel Magnetism 15: 49-65 (2002); arXiv:cond-mat/0205484. We discuss several scenarios for the creation of nonlocal spin-entangled electrons which provide a source of electronic Einstein-Podolsky-Rosen (EPR) pairs. Such EPR pairs can be used to test nonlocality of electrons in solid state systems, and they form the basic resources for quantum information processing. The central idea is to exploit the spin correlations naturally present in superconductors in form of Cooper pairs possessing spin-singlet wavefunctions. We show that nonlocal spin-entanglement in form of an effective Heisenberg spin interaction is induced between electron spins residing on two quantum dots with no direct coupling between them, but each of them being tunnel-coupled to the same superconductor. We then discuss a nonequilibrium setup with an applied bias where mobile and nonlocal spin-entanglement can be created by coherent injection of two electrons, in a pair (Andreev) tunneling process, into two spatially separated quantum dots and subsequently into two Fermi liquid leads. The current for injecting two spin-entangled electrons into different leads shows a resonance and allows the injection of electrons at the same orbital energy, which is a crucial requirement for the detection of spin-entanglement via the current noise. On the other hand, tunneling via the same dot into the same lead is suppressed by the Coulomb blockade effect of the quantum dots. We discuss Aharonov-Bohm oscillations in the current and show that they contain h/e and h/2e periods, which provides an experimental means to test the nonlocality of the entangled pair. Finally, we discuss a structure consisting of a superconductor weakly coupled to two separate one-dimensional leads with Luttinger liquid properties. We show that strong correlations again suppress the coherent subsequent tunneling of two electrons into the same lead, thus generating again nonlocal spin-entangled electrons in the Luttinger liquid leads.

Electron Spins in Artificial Atoms and Molecules for Quantum Computing

Vitaly N. Golovach and Daniel Loss. Semicond. Sci. Technol. 17, 355- 366 (2002); cond-mat/0201437. Achieving control over the electron spin in quantum dots (artificial atoms) or real atoms promises access to new technologies in conventional and in quantum information processing. Here we review our proposal for quantum computing with spins of electrons confined to quantum dots. We discuss the basic requirements for implementing spin-qubits, and describe a complete set of quantum gates for single- and two-qubit operations. We show how a quantum dot attached to leads can be used for spin filtering and spin read-out, and as a spin-memory device. Finally, we focus on the experimental characterization of the quantum dot systems, and discuss transport properties of a double-dot and show how Kondo correlations can be used to measure the Heisenberg exchange interaction between the spins of two dots.

Entanglement and Quantum Gate Operations with Spin-Qubits in Quantum Dots

John Schliemann and Daniel Loss. ``Future Trends in Microelectronics: The Nano Millenium", eds. S. Luryi, J. Xu, and A. Zaslavsky, Wiley, 2002, pp. 319-334; cond-mat/0110150. We give an elementary introduction to the notion of quantum entanglement between distinguishable parties and review a recent proposal about solid state quantum computation with spin-qubits in quantum dots. The indistinguishable character of the electrons whose spins realize the qubits gives rise to further entanglement-like quantum correlations. We summarize recent results concerning this type of quantum correlations of indistinguishable particles.

Single Spin Dynamics and Decoherence in a Quantum Dot via Charge Transport

Hans-Andreas Engel and Daniel Loss. Phys. Rev. B 65, 195321-1 (2002); cond-mat/0109470. We investigate the spin dynamics of a quantum dot with a spin-1/2 ground state in the Coulomb blockade regime and in the presence of a magnetic rf field leading to electron spin resonances (ESR). We show that by coupling the dot to leads, spin properties on the dot can be accessed via the charge current in the stationary and non-stationary limit. We present a microscopic derivation of the current and the master equation of the dot using superoperators, including contributions to decoherence and energy shifts due to the tunnel coupling. We give a detailed analysis of sequential and co-tunneling currents, for linearly and circularly oscillating ESR fields, applied in cw and pulsed mode. We show that the sequential tunneling current exhibits a spin satellite peak whose linewidth gives a lower bound on the decoherence time T_2 of the dot-spin. Similarly, the spin decoherence can be accessed also in the cotunneling regime via ESR induced spin flips. We show that the conductance ratio of the spin satellite peak and the conventional peak due to sequential tunneling saturates at the universal conductance ratio of 0.71 for strong ESR fields. We describe a double-dot setup which generates spin dependent tunneling and acts as a current pump (at zero bias), and as a spin inverter which inverts the spin-polarization of the current. We show that Rabi oscillations of the dot-spin induce coherent oscillations in the time-dependent current. These oscillations are observable in the time-averaged current as function of ESR pulse-duration, and they allow one to access the spin coherence directly in the time domain. We analyze the measurement and read-out process of the dot-spin via currents in spin-polarized leads and identify measurement time and efficiency by calculating the counting statistics, noise, and the Fano factor.

Biexcitons in Coupled Quantum Dots as a Source for Entangled Photons

Oliver Gywat, Guido Burkard, and Daniel Loss. Phys. Rev. B 65, 205329 (2002); cond-mat/0109223. We study biexcitonic states in two tunnel-coupled semiconductor quantum dots and show that they provide a source for entangled photons which are spatially separated at production. We distinguish between the various spin configurations and calculate the low-energy biexciton spectrum using the Heitler-London approximation as a function of magnetic and electric fields. We calculate the oscillator strengths for the biexciton recombination involving the sequential emission of two photons with entangled polarizations corresponding to the spin configuration in the biexciton states.

Quantum coherent dynamics in molecular magnetic rings

A. Honecker, F. Meier, Daniel Loss, and B. Normand. Eur. Phys. J. B 27, 487 (2002); cond-mat/0109201. We present detailed calculations of low-energy spin dynamics in the ``ferric wheel'' systems Na:Fe_6 and Cs:Fe_8 in a magnetic field. We compute by exact diagonalization the low-energy spectra and matrix elements for total-spin and N'eel-vector components, and thus the time-dependent correlation functions of these operators. We compare our results with semiclassical tunneling descriptions, and discuss their implications for mesoscopic quantum coherence, as well as for the experimental techniques to observe it, in molecular magnetic rings.

Kondo effect and singlet-triplet splitting in coupled quantum dots in a magnetic field

Vitaly N. Golovach and Daniel Loss. Europhys. Lett. 62, 83 (2003); cond-mat/0109155. We study two tunnel-coupled quantum dots each with a spin 1/2 and attached to leads in the Coulomb blockade regime. We study the interplay between Kondo correlations and the singlet-triplet exchange splitting $K$ between the two spins. We calculate the cotunneling current with elastic and inelastic contributions and its renormalization due to Kondo correlations, away and at the degeneracy point K=0. We show that these Kondo correlations induce pronounced peaks in the conductance as function of magnetic field $B$, inter-dot coupling $t_0$, and temperature. Moreover, the long-range part of the Coulomb interaction becomes visibile due to Kondo correlations resulting in an additional peak in the conductance vs $t_0$ with a strong $B$-field dependence. These conductance peaks thus provide direct experimental access to $K$, and thus to a crucial control parameter for spin-based qubits and entanglement.

Cancellation of spin-orbit effects in quantum gates based on the exchange coupling

Guido Burkard and Daniel Loss. Phys. Rev. Lett. 88, 047903 (2002); cond-mat/0108101. We study the effect of the spin-orbit interaction on quantum gate operations based on the spin exchange coupling where the qubit is represented by the electron spin in a quantum dot or a similar nanostructure. Our main result is the exact cancellation of the spin-orbit effects in the sequence producing the quantum XOR gate for the ideal case where the pulse shapes of the exchange and spin-orbit interactions are identical. For the non-ideal case, we show that the two pulse shapes can be made almost identical and that the gate error is strongly suppressed by two small parameters, the spin-orbit interaction constant and the smallness of the deviation of the two pulse shapes. Similarly, we show that the dipole-dipole interaction leads only to very small errors in the XOR gate.

Electron and Nuclear Spin Dynamics in Ferric Wheels

Florian Meier and Daniel Loss. Phys. Rev. Lett. 86, 5373 (2001); cond-mat/0101073. We study theoretically the spin dynamics of the ferric wheel, an antiferromagnetic molecular ring. For a single nuclear or impurity spin coupled to one of the electron spins of the ring, we calculate nuclear and electronic spin correlation functions and show that nuclear magnetic resonance (NMR) and electron spin resonance (ESR) techniques can be used to detect coherent tunneling of the Neel vector in these rings. The location of the NMR/ESR resonances gives the tunnel splitting and its linewidth an upper bound on the decoherence rate of the electron spin dynamics. We illustrate the experimental feasibility of our proposal with estimates for Fe_10 molecules.

Thermodynamics and Spin Tunneling Dynamics in Ferric Wheels with Excess Spin

Florian Meier and Daniel Loss. Phys. Rev. B 64, 224411 (2001); cond-mat/0107025. We study theoretically the thermodynamic properties and spin dynamics of a class of magnetic rings closely related to ferric wheels, antiferromagnetic ring systems, in which one of the Fe (III) ions has been replaced by a dopant ion to create an excess spin. Using a coherent-state spin path integral formalism, we derive an effective action for the system in the presence of a magnetic field. We calculate the functional dependence of the magnetization and tunnel splitting on the magnetic field and show that the parameters of the spin Hamiltonian can be inferred from the magnetization curve. We study the spin dynamics in these systems and show that quantum tunneling of the Neel vector also results in tunneling of the total magnetization. Hence, the spin correlation function shows a signature of Neel vector tunneling, and electron spin resonance (ESR) techniques or AC susceptibility measurements can be used to measure both the tunneling and the decoherence rate. We compare our results with exact diagonalization studies on small ring systems. Our results can be easily generalized to a wide class of nanomagnets, such as ferritin.

Quantum Correlations in Two-Fermion Systems

John Schliemann (Univ. of Texas), J. I. Cirac (Univ. of Innsbruck), M. Kus (Polish Academy of Science, Warsaw), M. Lewenstein (Univ. of Hannover), and Daniel Loss. Phys. Rev. A 64, 022303-9 (2001); quant-ph/0012094. We characterize and classify quantum correlations in two-fermion systems having 2K single-particle states. For pure states we introduce the Slater decomposition and rank (in analogy to Schmidt decomposition and rank), i.e. we decompose the state into a combination of elementary Slater determinants formed by mutually orthogonal single-particle states. Mixed states can be characterized by their Slater number which is the minimal Slater rank required to generate them. For K=2 we give a necessary and sufficient condition for a state to have a Slater number of 1. We introduce a correlation measure for mixed states which can be evaluated analytically for K=2. For higher K, we provide a method of constructing and optimizing Slater number witnesses, i.e. operators that detect Slater number for some states.

Quantum Computing in Molecular Magnets

Michael N. Leuenberger and Daniel Loss. Nature 410, 789 (2001); cond-mat/0011415. We propose the implementation of Grover's algorithm with molecular magnets with spin s>>1 in a unary representation. Shor and Grover demonstrated that a quantum computer can outperform any classical computer in factoring numbers[1] and in searching a database[2] by exploiting the parallelism of quantum mechanics. Whereas Shor's algorithm requires both superposition and entanglement of a many-particle system[3], the superposition of single-particle quantum states is sufficient for Grover's algorithm[4]. Recently, the latter has been successfully implemented[5] using Rydberg atoms. Here we propose an implementation of Grover's algorithm that uses molecular magnets[6,7,8,9,10], which are solid-state systems with a large spin; their spin eigenstates make them natural candidates for single-particle systems. We show theoretically that molecular magnets can be used to build dense and efficient memory devices based on the Grover algorithm. In particular, one single crystal can serve as a storage unit of a dynamic random access memory device. Fast electron spin resonance pulses can be used to decode and read out stored numbers of up to 105, with access times as short as 10-10 seconds. We show that our proposal should be feasible using the molecular magnets Fe8 and Mn12.

Detection of Single Spin Decoherence in a Quantum Dot via Charge Currents

Hans-Andreas Engel and Daniel Loss. Phys. Rev. Lett. 86, 4648 (2001); cond-mat/0011193. We consider a quantum dot attached to leads in the Coulomb blockade regime which has a spin 1/2 ground state. We show that by applying an ESR field to the dot-spin the stationary current in the sequential tunneling regime exhibits a resonance whose line width is determined by the single-spin decoherence time T_2. The Rabi oscillations of the dot-spin are shown to induce coherent current oscillations from which T_2 can be deduced in the time domain. We describe a spin-inverter which can be used to pump current through a double-dot via spin flips generated by ESR.

Magnetization in Molecular Iron Rings

B. Normand, X. Wang, X. Zotos, and Daniel Loss. Phys. Rev. B 63, 184409 (2001); cond-mat/0011403. The organometallic ring molecules Fe_6 and Fe_10 are leading examples of a class of nanoscopic molecular magnets, which have been of intense recent interest both for their intrinsic magnetic properties and as candidates for the observation of macroscopic quantum coherent phenomena. Torque magnetometry experiments have been performed to measure the magnetization in single crystals of both systems. We provide a detailed interpretation of these results, with a view to full characterization of the material parameters. We present both the most accurate numerical simulations performed to date for ring molecules, using Exact Diagonalization and Density Matrix Renormalization Group techniques, and a semiclassical description for purposes of comparison. The results permit quantitative analysis of the variation of critical fields with angle, of the nature and height of magnetization and torque steps, and of the width and rounding of the plateau regions in both quantities.

Noise of a Quantum-Dot System in the Cotunneling Regime

Eugene V. Sukhorukov, Guido Burkard, and Daniel Loss. Phys. Rev. B 63, 125315 (2001); cond-mat/0010458. We study the noise of the cotunneling current through one or several tunnel-coupled quantum dots in the Coulomb blockade regime. The various regimes of weak and strong, elastic and inelastic cotunneling are analyzed for quantum-dot systems (QDS) with few-level, nearly-degenerate, and continuous electronic spectra. We find that in contrast to sequential tunneling where the noise is either Poissonian (due to uncorrelated tunneling events) or sub-Poissonian (suppressed by charge conservation on the QDS), the noise in inelastic cotunneling can be super-Poissonian due to switching between QDS states carrying currents of different strengths. In the case of weak cotunneling we prove a non-equilibrium fluctuation-dissipation theorem which leads to a universal expression for the noise-to-current ratio (Fano factor). In order to investigate strong cotunneling we develop a microscopic theory of cotunneling based on the density-operator formalism and using the projection operator technique. The master equation for the QDS and the expressions for current and noise in cotunneling in terms of the stationary state of the QDS are derived and applied to QDS with a nearly degenerate and continuous spectrum.

Andreev-Tunneling, Coulomb Blockade, and Resonant Transport of Non-Local Spin-Entangled Electrons

Patrik Recher, Eugene V. Sukhorukov, and Daniel Loss. Phys. Rev. B 63, 165314 (2001); cond-mat/0009452. We propose and analyze a spin-entangler for electrons based on an s-wave superconductor coupled to two quantum dots each of which is tunnel-coupled to normal Fermi leads. We show that in the presence of a voltage bias and in the Coulomb blockade regime two correlated electrons provided by the Andreev process can coherently tunnel from the superconductor via different dots into different leads. The spin-singlet coming from the Cooper pair remains preserved in this process, and the setup provides a source of mobile and nonlocal spin-entangled electrons. The transport current is calculated and shown to be dominated by a two-particle Breit-Wigner resonance which allows the injection of two spin-entangled electrons into different leads at exactly the same orbital energy, which is a crucial requirement for the detection of spin entanglement via noise measurements. The coherent tunneling of both electrons into the same lead is suppressed by the on-site Coulomb repulsion and/or the superconducting gap, while the tunneling into different leads is suppressed through the initial separation of the tunneling electrons. In the regime of interest the particle-hole excitations of the leads are shown to be negligible. The Aharonov-Bohm oscillations in the current are shown to contain single- and two-electron periods with amplitudes that both vanish with increasing Coulomb repulsion albeit differently fast.

Spintronics and Quantum Computing: Switching Mechanisms for Qubits

Michael N. Leuenberger and Daniel Loss. Physica E 10 452-457 (2001); cond-mat/0010434. Quantum computing and quantum communication are remarkable examples of new information processing technologies that arise from the coherent manipulation of spins in nanostructures. We review our theoretical proposal for using electron spins in quantum-confined nanostructures as qubits. We present single- and two-qubit gate mechanisms in laterally as well as vertically coupled quantum dots and discuss the possibility to couple spins in quantum dots via exchange or superexchange. In addition, we propose a new stationary wave switch, which allows to perform quantum operations with quantum dots or spin-1/2 molecules placed on a 1D or 2D lattice.

Double-Occupancy Errors, Adiabaticity, and Entanglement of Spin-Qubits in Quantum Dots

John Schliemann (University of Texas, USA), Daniel Loss, and A.H. MacDonald (University of Texas, USA). Phys. Rev. B 63, 085311 (2001); cond-mat/0009083. Quantum gates that temporarily increase singlet-triplet splitting in order to swap electronic spins in coupled quantum dots, lead inevitably to a finite double-occupancy probability for both dots. By solving the time-dependent Schröodinger equation for a coupled dot model, we demonstrate that this does not necessarily lead to quantum computation errors. Instead, the coupled dot ground state evolves quasi-adiabatically for typical system parameters so that the double-occupancy probability at the completion of swapping is negligibly small. We introduce a measure of entanglement which explicitly takes into account the possibilty of double occupancies and provides a necessary and sufficient criterion for entangled states.

Quantum Dot as Spin Filter and Spin Memory

Patrik Recher, Eugene V. Sukhorukov, and Daniel Loss. Phys. Rev. Lett. 85, 1962 (2000); cond-mat/0003089]. We consider a quantum dot in the Coulomb blockade regime weakly coupled to current leads and show that in the presence of a magnetic field the dot acts as an efficient spin-filter (at the single-spin level) which produces a spin-polarized current. Conversely, if the leads are fully spin-polarized the up or down state of the spin on the dot results in a large sequential or small cotunneling current, and thus, together with ESR techniques, the setup can be operated as a single-spin memory.

Spintronics and Quantum Computing with Quantum Dots

Patrik Recher, Daniel Loss, and Jeremy Levy (University of Pittsburgh, USA). p.293-306, in Macroscopic Quantum Coherence and Quantum Computing", eds. D.V. Averin, B. Ruggiero, and P. Silvestrini, Kluwer Academic/Plenum Publishers, New York, 2001; cond-mat/0009270. The creation, coherent manipulation, and measurement of spins in nanostructures open up completely new possibilities for electronics and information processing, among them quantum computing and quantum communication. We review our theoretical proposal for using electron spins in quantum dots as quantum bits. We present single- and two qubit gate mechanisms in laterally as well as vertically coupled quantum dots and discuss the possibility to couple spins in quantum dots via superexchange. We further present the recently proposed schemes for using a single quantum dot as spin-filter and spin read-out/memory device.

Spin tunneling and topological selection rules for integer spins

Michael N. Leuenberger and Daniel Loss. Phys. Rev. B 63, 054414 (2001); cond-mat/0006075. We present new topological interference effects for the tunneling of a single large spin, which are caused by the symmetry of a general class of magnetic anisotropies. The interference results from spin Berry phases associated with different tunneling paths exposed to the same dynamics. Introducing a generalized path integral for coherent spin states we evaluate transition amplitudes between ground as well as low-lying excited states. We show that these interference effects lead to topological sel​ection rules and spin parity effects for integer spins which agree with quantum sel​ection rules and which thus provide a generalization of the Kramers degeneracy to integer spins.

Coulomb Blockade in the Fractional Quantum Hall Effect Regime

Michael R. Geller (Athens, Georgia, USA) and Daniel Loss. Phys. Rev. B 62, 16298 (2000) (Rapid Communication); cond-mat/0003318. We use chiral Luttinger liquid theory to study transport through a quantum dot in the fractional quantum Hall effect regime and find rich non-Fermi-liquid tunneling characteristics. In particular, we predict a remarkable Coulomb-blockade-type energy gap that is quantized in units of the noninteracting level spacing, new power-law tunneling exponents for voltages beyond threshold, and a line shape as a function of gate voltage that is dramatically different than that for a Fermi liquid. We propose experiments to use these unique spectral properties as a new probe of the fractional quantum Hall effect.

Superconductors, Quantum Dots, and Spin Entanglement

Mahn-Soo Choi, Christoph Bruder, and Daniel Loss Proceedings of 236. WE-Heraeus-Seminar: Interacting Electrons in Nanostructures, Lecture Notes in Physics, Vol. 579, eds. R. Haug and H. Schoeller, Berlin, 2001, pp. 46-66. In this paper, we review a double quantum dot each dot of which is tunnelcoupled to superconducting leads. In the Coulomb blockade regime, a spin-dependent Josephson coupling between two superconductors is induced, as well as an antiferromagnetic Heisenberg exchange coupling between the spins on the double dot which can be tuned by the superconducting phase difference. We show that the correlated spin states—singlet or triplets—on the double dot can be probed via the Josephson current in a dc-SQUID setup. We also briefly review the Andreev entangler, a non-equilibrium setup that provides a source of pairwise entangled electrons.

Spin-Dependent Josephson Current through Double Quantum Dots and Measurement of Entangled Electron States

Mahn-Soo Choi, C. Bruder, and Daniel Loss. Phys. Rev. B 62, 13569 (2000) We study a double quantum dot each dot of which is tunnel-coupled to superconducting leads. In the Coulomb blockade regime, a spin-dependent Josephson coupling between two superconductors is induced, as well as an antiferromagnetic Heisenberg exchange coupling between the spins on the double dot which can be tuned by the superconducting phase difference. We show that the correlated spin states-singlet or triplets-on the double dot can be probed via the Josephson current in a dc-SQUID setup.

Spintronics and Quantum Dots for Quantum Computing and Quantum Communication

Guido Burkard, Hans-Andreas Engel, and Daniel Loss. Fortschr. Phys. 48, 9-11, 965-986 (2000); special issue on Experimental Proposals for Quantum Computation, eds. H.-K. Lo and S. Braunstrein; cond-mat/0004182. Control over electron-spin states, such as coherent manipulation, filtering and measurement promises access to new technologies in conventional as well as in quantum computation and quantum communication. We review our proposal of using electron spins in quantum confined structures as qubits and discuss the requirements for implementing a quantum computer. We describe several realizations of one- and two-qubit gates and of the read-in and read-out tasks. We discuss recently proposed schemes for using a single quantum dot as spin-filter and spin-memory device. Considering electronic EPR pairs needed for quantum communication we show that their spin entanglement can be detected in mesoscopic transport measurements using metallic as well as superconducting leads attached to the dots.

Conductance fluctuations in diffusive rings: Berry phase effects and criteria for adiabaticity

Hans-Andreas Engel and Daniel Loss. Phys. Rev. B 62, 10238-10254 (2000); cond-mat/0002396. We study Berry phase effects on conductance properties of diffusive mesoscopic conductors, which are caused by an electron spin moving through an orientationally inhomogeneous magnetic field. Extending previous work, we start with an exact, i.e. not assuming adiabaticity, calculation of the universal conductance fluctuations in a diffusive ring within the weak localization regime, based on a differential equation which we derive for the diffuson in the presence of Zeeman coupling to a magnetic field texture. We calculate the field strength required for adiabaticity and show that this strength is reduced by the diffusive motion. We demonstrate that not only the phases but also the amplitudes of the h/2e Aharonov-Bohm oscillations are strongly affected by the Berry phase. In particular, we show that these amplitudes are completely suppressed at certain magic tilt angles of the external fields, and thereby provide a useful criterion for experimental searches. We also discuss Berry phase-like effects resulting from spin-orbit interaction in diffusive conductors and derive exact formulas for both magnetoconductance and conductance fluctuations. We discuss the power spectra of the magnetoconductance and the conductance fluctuations for inhomogeneous magnetic fields and for spin-orbit interaction.

Quantum Computers and Quantum Coherence

David P. DiVincenzo (IBM Yorktown Heights, NY, USA) and Daniel Loss. J. of Magnetism and Magnetic Matls. 200, 202 (1999); cond-mat/9901137. If the states of spins in solids can be created, manipulated, and measured at the single-quantum level, an entirely new form of information processing, quantum computing, will be possible. We first give an overview of quantum information processing, showing that the famous Shor speedup of integer factoring is just one of a host of important applications for qubits, including cryptography, counterfeit protection, channel capacity enhancement, distributed computing, and others. We review our proposed spin-quantum dot architecture for a quantum computer, and we indicate a variety of first generation materials, optical, and electrical measurements which should be considered. We analyze the efficiency of a two-dot device as a transmitter of quantum information via the propagation of qubit carriers (i.e. electrons) in a Fermi sea.

Transport and Noise of Entangled Electrons

Eugene V. Sukhorukov, Daniel Loss, and Guido Burkard. arXiv:cond-mat/9909348 We consider a scattering set-up with an entangler and beam splitter where the current noise exhibits bunching behavior for electronic singlet states and antibunching behavior for triplet states. We show that the entanglement of two electrons in the double-dot can be detected in mesoscopic transport measurements. In the cotunneling regime the singlet and triplet states lead to phase-coherent current contributions of opposite signs and to Aharonov-Bohm and Berry phase oscillations in response to magnetic fields. We analyze the Fermi liquid effects in the transport of entangled electrons.

Electron Spins in Quantum Dots as Quantum Bits

Daniel Loss, Guido Burkard, and David P. DiVincenzo. JOURNAL OF NANOPARTICLE RESEARCH 2, 401-411 (2000) The creation, coherent manipulation, and measurement of spins in nanostructures open up completely new possibilities for electronics and information processing, among them quantum computing and quantum communication. We review our theoretical proposal for using electron spins in quantum dots as quantum bits, explaining why this scheme satisfies all the essential requirements for quantum computing. We include a discussion of the recent measurements of surprisingly long spin coherence times in semiconductors. Quantum gate mechanisms in laterally and vertically tunnel-coupled quantum dots and methods for single-spin measurements are introduced. We discuss detection and transport of electronic EPR pairs in normal and superconducting systems.

Spin interactions and switching in vertically tunnel-coupled quantum dots

Guido Burkard, Georg Seelig, and Daniel Loss. Phys. Rev. B 62, 2581 (2000); cond-mat/9910105. We determine the spin exchange coupling J between two electrons located in two vertically tunnel-coupled quantum dots, and its variation when magnetic (B) and electric (E) fields (both in-plane and perpendicular) are applied. We predict a strong decrease of J as the in-plane B field is increased, mainly due to orbital compression. Combined with the Zeeman splitting of the triplet, this leads to a singlet-triplet crossing, which can be observed as a pronounced jump in the magnetization, occurring at in-plane fields of a few Tesla, and perpendicular fields of the order of 10 Tesla for typical self-assembled dots. We use harmonic potentials to model the confining of electrons, and calculate the exchange J using the Heitler-London and Hund-Mulliken technique, including the long-range Coulomb interaction. With our results we provide experimental criteria for the distinction of singlet and triplet states and therefore for a microscopic spin measurement. In the case where quantum dots of different size are coupled, we present a simple method to switch on and off the spin coupling with exponential sensitivity using an in-plane electric field. Switching the spin coupling is essential for quantum computation using electronic spins as qubits.

Probing Entanglement and Non-locality of Electrons in a Double-Dot via Transport and Noise

Daniel Loss and Eugene V. Sukhorukov. Phys. Rev. Lett. 84 1035-1038 (2000); cond-mat/9907129. Addressing the feasibilty of quantum communication with electrons we consider entangled spin states of electrons in a double-dot which is weakly coupled to in--and outgoing leads. We show that the entanglement of two electrons in the double-dot can be detected in mesoscopic transport and noise measurements. In the Coulomb blockade and cotunneling regime the singlet and triplet states lead to phase-coherent current and noise contributions of opposite signs and to Aharonov-Bohm and Berry phase oscillations in response to magnetic fields. These oscillations are a genuine two-particle effect and provide a direct measure of non-locality in entangled states. We show that the ratio of zero-frequency noise to current (Fano factor) is universal and equal to the electron charge.

Quantum Computation and Spin Electronics

D. P. DiVincenzo, G. Burkard, D. Loss, and E. Sukhorukov. cond-mat/9911245. in Quantum Mesoscopic Phenomena and Mesoscopic Devices in Microelectronics, eds. I. O. Kulik and R. Ellialtoglu (NATO ASI), p. 399-428, 2000, Kluwer, Netherlands In this chapter we explore the connection between mesoscopic physics and quantum computing.  After giving a bibliography providing a general introduction to the subject of quantum information processing, we review the various approaches that are being considered for the experimental implementation of quantum computing and quantum communication in atomic physics, quantum optics, nuclear magnetic resonance, superconductivity, and, especially, normal-electron solid state physics.  We discuss five criteria for the realization of a quantum computer and consider the implications that these criteria have for quantum computation using the spin states of single-electron quantum dots.  Finally, we consider the transport of quantum information via the motion of individual electrons in mesoscopic structures; specific transport and noise measurements in coupled quantum dot geometries for detecting and characterizing electron-state entanglement are analyzed.

Spin tunneling and phonon-assisted relaxation in Mn12-acetate

Michael Leuenberger and Daniel Loss. Phys. Rev. B 61, 1286 (2000); cond-mat/9907154. We present a comprehensive theory of the magnetization relaxation in a Mn12-acetate crystal in the high temperature regime (T>1K), which is based on phonon-assisted spin tunneling induced by quartic magnetic anisotropy and weak transverse magnetic fields. The overall relaxation rate as function of the longitudinal magnetic field is calculated and shown to agree well with experimental data including all resonance peaks measured so far. The Lorentzian shape of the resonances, which we obtain via a generalized master equation that includes spin tunneling, is also in good agreement with recent data. We derive a general formula for the tunnel splitting energy of these resonances. We show that fourth-order diagonal terms in the Hamiltonian lead to satellite peaks. A derivation of the effective linewidth of a resonance peak is given and shown to agree well with experimental data. In addition, previously unknown spin-phonon coupling constants are calculated explicitly. The values obtained for these constants and for the sound velocity are also in good agreement with recent data. We show that the spin relaxation in Mn12-acetate takes place via several transition paths of comparable weight. These transition paths are expressed in terms of intermediate relaxation times, which are calculated and which can be tested experimentally.

Noise of Entangled Electrons: Bunching and Antibunching

Guido Burkard, Daniel Loss, and Eugene V. Sukhorukov. Phys. Rev. B 61, R16303 (2000); cond-mat/9906071]. Addressing the feasibility of quantum communication with entangled electrons in an interacting many-body environment, we propose an interference experiment using a scattering set-up with an entangler and a beam splitter. It is shown that, due to electron-electron interaction, the delity of the entangled singlet and triplet states is reduced by z_F^2 in a conductor described by Fermi liquid theory. We calculate the quasiparticle weight factor z_F for a two-dimensional electron system. The current noise for electronic singlet states turns out to be enhanced (bunching behavior), while it is reduced for triplet states (antibunching). Within standard scattering theory, we nd that the Fano factor (noise-to-current ratio) for singlets is twice as large as for independent classical particles and is reduced to zero for triplets.

Quantum Dynamics of Pseudospin Solitons in Double-Layer Quantum Hall, Systems

J. Kyriakidis, D. Loss, and A. MacDonald (Bloomington, IN, USA). Phys. Rev. Lett. 83, 1411-1414 (1999); cond-mat/9904185. Pseudospin solitons in double-layer quantum Hall systems can be introduced by a magnetic field component coplanar with the electrons and can be pinned by applying voltages to external gates. We estimate the temperature below which depinning occurs predominantly via tunneling and calculate low-temperature depinning rates for realistic geometries.  We discuss the local changes in charge and current densities and in spectral functions that can be used to detect solitons and observe their temporal evolution.

Observing the Berry phase in diffusive conductors: Necessary conditions for adiabaticity

D. Loss, H. Schoeller (Karlsruhe, D), and P.M. Goldbart (Urbana, IL, USA). Phys. Rev. B 59 (1999) 13328-13337 We investigate Berry phase effects in the magnetoconductance of diffusive systems and determine the precise criterion for adiabaticity within the weak localization formalism. We show that  the exact solution of the Cooperon propagator for the special case of a cylindrically symmetric texture agrees with the adiabatic approximation in the adiabatic limit characterized by $\tau \gg {1\over d} {l2\over L2}$.  We point out that orientational inhomogeneities in the magnetic field induce dephasing effects that can mask the Berry phase (and any other phase coherent phenomena) for certain parameter values of system and field.

Incoherent Zener tunneling and its application to molecular magnets

Michael Leuenberger and Daniel Loss. Phys. Rev. B 61, 12200 (2000); cond-mat/9911065. We generalize the Landau-Zener theory of coherent tunneling transitions by taking thermal relaxation into account. The evaluation of a new generalized master equation containing a dynamic tunneling rate that includes the interaction between the relevant system and its environment leads to an incoherent Zener transition probability with an exponent that is twice as large as the one of the coherent Zener probability in the limit T->0. We apply our results to molecular clusters, in particular to recent measurements of the tunneling transition of spins in Fe8 crystals performed by Wernsdorfer and Sessoli  [Science 284, 133 (1999)].

Excess Spin and the Dynamics of Antiferromagnetic Ferritin

J.G.E. Harris (UC Santa Barbara, CA, USA), J.E. Grimaldi (UCSB), D.D. Awschalom (UCSB), A. Chiolero, and D. Loss. Phys. Rev. B 60, 3453-3456 (1999); cond-mat/9904051. Temperature-dependent magnetization measurements on a series of synthetic ferritin proteins containing from 100 to 3000 Fe(III) ions are used to determine the uncompensated moment of these antiferromagnetic particles. The results are compared with recent theories of macroscopic quantum coherence which explicitly include the effect of this excess moment. The scaling of the excess moment with protein size is consistent with a simple model of finite size effects and sublattice noncompensation.

Physical Optimization of Quantum Error Correction Circuits

Guido Burkard, Daniel Loss, David P. DiVincenzo (IBM Yorktown Heights, NY, USA), and John A. Smolin (IBM). Phys. Rev. B 60, 11404 (1999); cond-mat/9905230. Quantum error correcting codes have been developed to protect a quantum computer from decoherence due to a noisy environment. In this paper, we present two methods for optimizing the physical implementation of such error correction schemes. First, we discuss an optimal quantum circuit implementation of the smallest error-correcting code (the three bit code). Quantum circuits are physically implemented by serial pulses, i.e. by switching on and off external parameters in the Hamiltonian one after another. In contrast to this, we introduce a new parallel switching method which allows faster gate operation by switching all external parameters simultaneously, and which has potential applications for arbitrary quantum computer architectures. We apply both serial and parallel switching to electron spins in coupled quantum dots subject to a Heisenberg coupling $H=J(t) S1· S2$. We provide a list of steps that can be implemented experimentally and used as a test for the functionality of quantum error correction.

Quantum information processing using electron spins and cavity-QED

A. Imamoglu (UC Santa Barbara, CA, USA), D. D. Awschalom (UCSB), G. Burkard, D. P. DiVincenzo (IBM Yorktown), D. Loss, M. Sherwin (UC Santa Barbara), and A. Small (UC Santa Barbara). Phys. Rev. Lett. 83, 4204 (1999); quant-ph/9904096. The electronic spin degrees of freedom in semiconductors typically have decoherence times that are several orders of magnitude longer than other relevant timescales. A solid-state quantum computer based on localized electron spins as qubits is therefore of potential interest. Here, a scheme that realizes controlled interactions between two distant quantum dot spins is proposed. The effective long-range interaction is mediated by the vacuum field of a high finesse microcavity. By using conduction-band-hole Raman transitions induced by classical laser fields and the cavity-mode, arbitrary single qubit rotations and controlled-not operations can be realized. Optical techniques can also be used to measure the spin-state of each quantum dot.

Coupled quantum dots as quantum gates

G. Burkard, D. Loss, and D.P. DiVincenzo (IBM Yorktown Heights, NY, USA). Phys. Rev. B 59, 2070 (1999); cond-mat/9808026. We consider a new quantum gate mechanism based on electron spins in coupled semiconductor quantum dots.  Such gates provide a general source of spin entanglement and can be used for quantum computers.  We determine the exchange coupling $J$ in the effective Heisenberg model as a function of magnetic ($B$) and electric fields, and of the inter-dot distance $a$ within the Heitler-London approximation of molecular physics. This result is refined by using sp-hybridization, and by the Hund-Mulliken molecular-orbit approach which leads to an extended Hubbard description for the two-dot system that shows a remarkable dependence on $B$ and $a$ due to the long-range Coulomb interaction. We find that the exchange $J$ changes sign at a finite field (leading to a pronounced jump in the magnetization) and then decays exponentially.  The magnetization and the spin susceptibilities of the coupled dots are calculated.  We show that the dephasing due to nuclear spins in GaAs can be strongly suppressed by dynamical nuclear spin polarization and/or by magnetic fields.

Nonlinear sigma Model Treatment of Quantum Antiferromagnets in a Magnetic Field

Bruce Normand, Jordan Kyriakidis, and Daniel Loss Ann. Phys. (Leipzig) 9, 133 (2000); arXiv:cond-mat/9902104; . We present a theoretical analysis of the properties of low-dimensional quantum antiferromagnets in applied magnetic fields. In a nonlinear sigma model description, we use a spin stiffness analysis, a 1/N expansion, and a renormalization group approach to describe the broken-symmetry regimes of finite magnetization, and, in cases of most interest, a low-field regime where symmetry is restored by quantum fluctuations. We compute the magnetization, critical fields, spin correlation functions, and decay exponents accessible by nuclear magnetic resonance experiments. The model is relevant to many systems exhibiting Haldane physics, and provides good agreement with data for the two-chain spin ladder compound CuHpCl.

Noise in Multiterminal Diffusive Conductors: Universality, Nonlocality and Exchange Effects

Eugene V. Sukhorukov and Daniel Loss. Phys. Rev. Lett. 80, 4959 (1998); cond-mat/9802050; Phys. Rev. B 59, 13054-13066 (1999); cond-mat/9809239. We study noise and transport in multiterminal diffusive conductors. Using a Boltzmann-Langevin equation approach we reduce the calculation of shot-noise correlators to the solution of diffusion equations. Within this approach we prove the universality of shot noise in multiterminal diffusive conductors of arbitrary shape and dimension for purely elastic scattering as well as for hot electrons. We show that shot noise in multiterminal conductors is a non-local quantity and that exchange effects can occur in the absence of quantum phase coherence even at zero electron temperature. It is also shown that the exchange effect measured in one contact is always negative -- in agreement with the Pauli principle. We discuss a new phenomenon in which current noise is induced by thermal transport. We propose a possible experiment to measure locally the effective noise temperature. Concrete numbers for shot noise are given that can be tested experimentally.

Quantum computation with quantum dots

D. Loss and D.P. DiVincenzo (IBM Yorktown Heights, NY, USA). Phys. Rev. A 57 120-126 (1998); Milestone paper of Phys. Rev. A (1970-2020); cond-mat/9701055. We propose a new implementation of a universal set of one- and two-qubit gates for quantum computation using the spin states of coupled single-electron quantum dots.  Desired operations are effected by the gating of the tunneling barrier between neighboring dots. Several measures of the gate quality are computed within a newly derived spin master equation incorporating decoherence caused by a prototypical magnetic environment.  Dot-array experiments which would provide an initial demonstration of the desired non-equilibrium spin dynamics are proposed.

Quantum dynamics in mesoscopic magnetism

D. Loss, in Dynamical Properties of, and Unconventional Magnetic Systems. A.T. Skjeltorp and D. Sherrington (eds.), NATO ASI Series E: Applied Sciences, Vol. 349, 1998 Kluwer Academic Publishers, p. 29-75 A review of quantum coherence effects in mesoscopic  spin systems is presented. We  begin with a general introduction to the topic of mesoscopic effects in magnetism and give some specific examples of current interest. We review then theoretical results in single domain magnetism of superparamagnetic type and mention  recent measurements on antiferromagnetic  grains (ferritin) and their interpretation in terms of macroscopic quantum coherence. Introducing the effects originating from  spin parity in the context of ferromagnetic grains, we discuss antiferromagnetic particles with excess spins and  molecular magnets such as the ferric wheel. It is shown that tunneling in such magnets can be tuned by external magnetic fields and is directly observable via the magnetization and the Schottky anomaly in the specific heat. The main part of the review  will be devoted to non-uniform magnets and specifically to the   quantum dynamics of domain walls or magnetic solitons.  In a semiclassical analysis based on coherent spin-state path-integrals an effective model for the  domain wall dynamics is derived which includes the effects of spin-wave dissipation and of quantum spin phases (Berry phases). In the presence of a Peierls potential (e.g. due to the discrete lattice)  the soliton center can tunnel coherently between the lattice sites and form a Bloch band. Integer and half-odd integer spins have different energy dispersions resulting from interference between soliton states of opposite chirality--the internal rotation sense of the soliton. These effects occur in ferro- and antiferromagnets due to the presence of a topological spin phase. For antiferromagnetic chains, this spin phase occurs in addition to the Pontryagin-index phase.We will discuss  experimental consequences of this Bloch band structure and  show that ---in analogy to Bloch oscillations of crystal electrons--- static magnetic fields induce large oscillations in the sample magnetization. We will also discuss the extreme quantum limit of spin-1/2 chains in the Ising regime, and show that, quite remarkably, the semiclassical analysis  is valid even in this regime. In particular, for antiferromagnetic Ising chains the low-energy excitations are solitons (Villain modes) which have  been observed in neutron scattering experiments on CsCoCl3. We show that the prediction of chirality effects could be tested via the measurement of the off-diagonal components of the dynamical structure factor. The concept of chirality is shown to be of universal character in a variety of magnetic systems, a notable example being the motion of a hole in a 2D antiferromagnetic background. A common thread in the discussion of quantum dynamics in magnets is provided by the Berry phases and their associated interference effects which can lead to surprising spin parity effects.

Macroscopic quantum coherence in molecular magnets

A. Chiolero and D. Loss. Phys. Rev. Lett. 80, 169-172 (1998) We study macroscopic quantum coherence in antiferromagnetic molecular magnets in the presence of magnetic fields. Such fields generate artificial tunnel barriers with externally tunable strength.  We give detailed semi-classical predictions for the tunnel splitting in various regimes for low and high magnetic fields. We show that the tunneling dynamics of the N\'eel vector can be directly measured via the static magnetization and the specific heat.  We also report on a new quantum phase arising from fluctuations. The analytic results are complemented by numerical simulations.

Bloch oscillations of magnetic solitons in anisotropic spin-1/2 chains

Jordan Kyriakidis and Daniel Loss. Phys. Rev. B 58 5568-5583 (1998); cond-mat/9803156. We study the quantum dynamics of soliton-like domain walls in anisotropic spin-1/2 chains in the presence of magnetic fields. In the absence of fields, domain walls form a Bloch band of delocalized quantum states while a static field applied along the easy axis localizes them into Wannier wave packets and causes them to execute Bloch oscillations, i.e.\ the domain walls oscillate along the chain with a finite Bloch frequency and amplitude.  In the presence of the field, the Bloch band, with a continuum of extended states, breaks up into the Wannier-Zeeman ladder---a discrete set of equally spaced energy levels.  We calculate the dynamical structure factor $Szz (q,\omega)$ in the one-soliton sector at finite frequency, wave vector, and temperature, and find sharp peaks at frequencies which are integer multiples of the Bloch frequency.  We further calculate the uniform magnetic susceptibility and find that it too exhibits peaks at the Bloch frequency.  We identify several candidate materials where these Bloch oscillations should be observable, for example, via neutron scattering measurements. For the particular compound ${\rm CoCl2 \!  · \!  2H2O}$ we estimate the Bloch amplitude to be on the order of a few lattice constants, and the Bloch frequency on the order of 100\,GHz for magnetic fields in the Tesla range and at temperatures of about 18\,Kelvin.

Macroscopic quantum coherence in ferrimagnets

A. Chiolero and D. Loss. Phys. Rev. B 56, 738 (1997) We study macroscopic quantum coherence (MQC) in small magnetic particles where the magnetization (in ferromagnets) or the N\'eel vector (in antiferromagnets) can tunnel between energy minima.  We consider here the more general case of MQC in ferrimagnets by studying a model for a mesoscopic antiferromagnet with an uncompensated magnetic moment.  Through semi-classical calculations we show that even a small moment has a drastic effect on MQC. In particular, there is a rapid crossover to a regime where the MQC tunnel splitting is equal to that obtained for a ferromagnet, even though the system is still an antiferromagnet for all other aspects.  We calculate this tunnel splitting via instanton methods and compare it with numerical evaluations. As an application we re-examine the experimental evidence for MQC in ferritin and show that even though the  uncompensated  moment of ferritin is small it greatly modifies the MQC behavior. The excess spin allows us to extract values for experimental parameters without making any assumption about the classical attempt frequency, in contrast to previous fits. Finally, we also discuss the implications of our results for MQC in molecular magnets.

Aharonov-Bohm effect in the chiral Luttinger liquid

M. R. Geller (Athens, Georgia, USA) and D. Loss. Phys. Rev. B 56 9692-9706 (1997) Edge states of the quantum Hall fluid provide an almost unparalled opportunity to study mesoscopic effects in a highly correlated electron system. In this paper we develop a bosonization formalism for the finite-size edge state, as described by chiral Luttinger liquid theory, and use it to study the Aharonov-Bohm effect. We study tunneling through an edge state formed around an antidot in the fractional quantum Hall regime using Wen's chiral Luttinger liquid theory extended to include mesoscopic effects. We identify a new regime where the Aharonov-Bohm oscillation amplitude exhibits a distinctive crossover from Luttinger liquid power-law behavior to Fermi-liquid-like behavior as the temperature is increased. Near the crossover temperature the amplitude has a pronounced maximum. This non-monotonic behavior and  novel high-temperature nonlinear phenomena that we also predict provide new ways to distinguish experimentally between Luttinger and Fermi liquids. Finally, we predict new mesoscopic edge-current oscillations, which are similar to the persistent currentoscillations in a mesoscopic ring, except that they are not reduced in amplitude by weak disorder. In the fractional quantum Hall regime, these ``chiral persistent currents'' have a universal non-Fermi-liquid temperature dependence and may be another ideal system to observe a chiral Luttinger liquid.

Chirality correlation of spin solitons: Bloch walls, spin-1/2 solitons and holes in a 2d antiferromagnetic background

Hans-Benjamin Braun and Daniel Loss. Int. J. Mod. Phys. 10, 21 (1996) We consider the quantum dynamics of spin solitons in a variety of low-dimensional magnetic systems in the semiclassical and the extreme quantum limit. Introducing the concept of chirality of the soliton we derive the dispersion of spin solitons moving through a periodic pinning potential and show that for half-odd integer spin the topological part of the Berry phase induces a halving of the Brillouin zone as well as chirality correlations between subsequent band minima. We demonstrate that these chirality and spin parity effects are universal by considering quasi-one-dimensional ferromagnets and antiferromagnets with local anisotropies and large spins, as well as spin-½ ferromagnetic and antiferromagnetic Heisenberg chains in the Ising limit. For large spin systems, the tunneling rate between states of opposite chiralities is derived and shown to provide a novel scenario for macroscopic quantum phenomena. The results are extended to solitons moving as holes in a two-dimensional antiferromagnetic background, leading to a hole spectrum which is in remarkable agreement with recent ARPES measurements on high-Tc compounds.

Macroscopic quantum tunneling of ferromagnetic domain walls

H.-B. Braun (Paul Scherrer Institut, CH), J. Kyriakidis, and D. Loss. Phys. Rev. B 56, 8129-8137 (1997); cond-mat/9710064. Quantum tunneling of domain walls out of an impurity potential in a mesoscopic ferromagnetic sample is investigated.  Using improved expressions for the domain wall mass and for the pinning potential, we find that the cross-over temperature between thermal activation and quantum tunneling is of a different functional form than found previously. In materials like Ni or YIG, the crossover temperatures are around 5 mK. We also find that the WKB exponent is typically two orders of magnitude larger than current estimates. The sources for these discrepancies are discussed, and precise estimates for the transition from three-dimensional to one-dimensional magnetic behavior of a wire are given. The cross-over temperatures from thermal to quantum transitions and tunneling rates are calculated for various materials and sample sizes.

Mesoscopic Effects in the Fractional Quantum Hall Regime: Chiral Luttinger versus Fermi Liquid

Michael R. Geller, Daniel Loss, and George Kirczenow. Phys. Rev. Lett. 77, 5110-5113(1996); cond-mat/9606070. We study tunneling through an edge state formed around an antidot in the fractional quantum Hall regime using Wen's chiral Luttinger liquid theory extended to include mesoscopic effects. We identify a new regime where the Aharonov-Bohm oscillation amplitude exhibits a distinctive crossover from Luttinger liquid power-law behavior to Fermi-liquid-like behavior as the temperature is increased. Near the crossover temperature the amplitude has a pronounced maximum. This non-monotonic behavior and novel high-temperature nonlinear phenomena that we also predict provide new ways to distinguish experimentally between Luttinger and Fermi liquids.

Berry's phase and quantum dynamics of ferromagnetic solitons

Hans-Benjamin Braun and Daniel Loss. Phys. Rev. B 53, 3237 (1996) We study spin parity effects and the quantum propagation of solitons "Bloch walls" in quasi-one-dimensional ferromagnets. Within a coherent state path integral approach we derive a quantum field theory for nonuniform spin configurations. The effective action for the soliton position is shown to contain a gauge potential due to the Berry phase and a damping term caused by the interaction between soliton and spin waves. For tempera- tures below the anisotropy gap this dissipation reduces to a pure soliton mass renormalization. The quantum dynamics of the soliton in a periodic lattice or pinning potential reveals remarkable consequences of the Berry phase. For half-integer spin, destructive interference between opposite chiralities suppresses nearest-neighbor hopping. Thus the Brillouin zone is halved, and for small mixing of the chiralities the dispersion reveals a surprising dynamical correlation: Two subsequent band minima belong to different chirality states of the soliton. For integer spin the Berry phase is inoperative and a simple tight-binding dispersion is obtained. Finally it is shown that external fields can be used to interpolate continuously between the Bloch wall disper- sions for half-integer and integer spin.

Absence of spontaneous persistent current for interacting fermions in a one-dimensional mesoscopic ring

Daniel Loss and Thierry Martin. Phys. Rev. B 47, 4619 (1993) We address the issue of whether a system of interacting electrons confined to a one-dimensional ring can sustain a spontaneous persistent current in the absence of an externally applied flux. The current-current coupling between electrons, describing radiative back-action effects, is exactly treated in the formalism of quantum electrodynamics, where the electrons interact via the exchange of virtual photons. In addition, the instantaneous screened Coulomb potential is taken into account using the Luttinger liquid model including finite-size parity effects. The partition function is calculated exactly, with the result that the system does not possess a spontaneous persistent current. We show that, in the presence of an external flux, the amplitude of the (conventional) persistent current is reduced by quantum fluctuations of the internal transverse electromagnetic field. These corrections can be expressed in terms of the self-induction of the ring and are shown to be of first and higher order in the small dimensionless parameter αvF*/c, where α is the fine-structure constant and vF* the Fermi velocity renormalized through Coulomb interactions.

Suppression of tunneling due to interference in spin sytems

Daniel Loss, David P. DiVincenzo (IBM Watson, NY), and Goeffrey Grinstein (IBM Watson, NY). Phys. Rev. Lett. 69, 3232 (1992) Within a wide class of ferromagnetic and antiferromagnetic systems, quantum tunneling of magnetization direction is spin-parity dependent: it vanishes for magnetic particles with half-integer spin, but is allowed for integer spin. A coherent-state path-integral calculation shows that this topological effect results from interference between tunneling paths.

Macroscopic Quantum Tunneling in Magnetic Proteins

D. D. Awschalom, J. F. Smyth, G. Grinstein, D. P. DiVincenzo, and D. Loss. Phys. Rev. Lett. 68, 3092 (1992); Phys. Rev. Lett. 71, 4279 (1993); Phys. Rev. Lett. 70, 2199 (1993). We report low-temperature measurements of the frequency-dependent magnetic noise and magnetic susceptibility of nanometer-scale antiferromagnetic horse-spleen ferritin particles, using an integrated dc SQUID microsusceptometer. A sharply defined resonance near 1 MHz develops below T∼0.2 K. The behavior of this resonance as a function of temperature, applied magnetic field, and particle concentration indicates that it results from macroscopic quantum tunneling of the Néel vector of the antiferromagnets.

Josephson current and proximity effect in Luttinger liquids

D. L. Maslov, M. Stone, P. M. Goldbart, and D. Loss. Phys. Rev. B53, 1548 (1996) A theory describing a one-dimensional Luttinger liquid in contact with a superconductor is developed. Boundary conditions for the fermion fields describing Andreev reflection at the contacts are derived and used to construct a bosonic representation of the fermions. The Josephson current through a superconductor/Luttinger liquid/superconductor junction is considered for both perfectly and poorly transmitting interfaces. In the former case, the Josephson current at low temperatures is found to be essentially unaffected by electron-electron interactions. In the latter case, significant renormalization of the Josephson current occurs. The profile of the (induced) condensate wave function in a semi-infinite Luttinger liquid in contact with a superconductor is shown to decay as a power law, the exponent depending on the sign and strength of the interactions. In the case of repulsive (attractive) interactions the decay is faster (slower) than in their absence. An equivalent method of calculating the Josephson current through a Luttinger liquid, which employs the bosonization of the system as a whole (i.e., superconductor, as well as Luttinger liquid) is developed and shown to give the results equivalent to those obtained via boundary condition describing Andreev reflection.

Onset of superconducting fluctuations for interacting fermions coupled to acoustic phonons in one dimension

Thierry Martin (Marseille) and Daniel Loss. Int. J. Mod. Phys. B 9, 495 (1995) We consider a one-dimensional system consisting of electrons with short-ranged repulsive interactions and coupled to small-momentum transfer acoustic phonons. The interacting electrons are bosonized and described in terms of a Luttinger liquid which allows us to calculate exactly the one- and two-electron Green function. For non-interacting electrons, the coupling to phonons alone induces a singularity at the Fermi surface which is analogous to that encountered for electrons with an instantaneous attractive interaction. The exponents which determine the presence of singlet/triplet superconducting pairing fluctuations, and spin/charge density wave fluctuations are strongly affected by the presence of the Wentzel-Bardeen singularity, resulting in the favoring of superconducting fluctuations. For the Hubbard model the equivalent of a phase diagram is established, as a function of: the electron-phonon coupling, the electron filling factor, and the on-site repulsion between electrons. The Wentzel-Bardeen singularity can be reached for arbitrary values of the electron-phonon coupling constant by varying the filling factor. This provides an effective mechanism to push the system from the antiferromagnetic into the metallic phase, and finally into the superconducting phase as the electron filling factor is increased towards half-filling.

Wentzel-Bardeen singularity and phase diagram for interacting electrons coupled to acoustic phonons in one dimension

Daniel Loss and Thierry Martin (Marseille). Phys. Rev. B 50, 12160 (1994) We consider strongly correlated electrons coupled to low-energy acoustic phonons in one dimension. Using a Luttinger liquid description we calculate the exponents of various response functions and discuss their remarkable sensitivity to the Wentzel-Bardeen singularity induced by the presence of phonons. For the Hubbard model plus phonons the equivalent of a phase diagram is established. By increasing the filling factor towards half filling the Wentzel-Bardeen singularity is approached. This in turn triggers a simple and efficient mechanism to suppress antiferromagnetic fluctuations and to drive the system via a normal metallic state towards a superconducting phase.

Parity effects in a Luttinger liquid: Diamagnetic and paramagnetic ground states

Daniel Loss Phys. Rev. Lett. 69, 343 (1992) The concept of a Luttinger liquid in 1D is extended to include twisted boundary conditions on a ring and mesoscopic parity effects due to evenness and oddness of the particle number N0. Using Haldane’s notion of topological excitations, a proof of Leggett’s conjecture is presented, stating that the ground state of interacting 1D quantum systems is diamagnetic or paramagnetic depending on the parity of N0. The persistent currents produced by an Aharonov-Bohm flux are calculated and shown to have period and amplitude that are in agreement with recent experiments.

Experimental consequences of persistent currents due to the Berry phase

Daniel Loss and Paul Goldbart. Phys. Lett. A 215, 197 (1996) It has recently been proposed that a mesoscopic ring of normal metal or semiconductor should exhibit equilibrium persistent currents of charge and spin, when embedded in an inhomogeneous magnetic field. The origin of these phenomena lies in the coupling between spin and orbital motion due to the Zeeman interaction, and the resulting geometric phase acquired during orbital motion. We present expressions for the ground state charge and spin currents for systems of many spin-l /2 fermions moving in a one-dimensional ring, and give numerical estimates of the magnitudes of the currents, and also ol‘ the electric field which results from the spin current.

Period and amplitude halving in mesoscopic rings with spin

Daniel Loss and Paul Goldbart. Phys. Rev. B 43, 13762 (1991) We consider the flux dependence of persistent currents in mesoscopic rings threaded by magnetic flux, and extend well-known arguments to include particles of spin 1/2. We find several interesting consequences of spin, such as period and amplitude halving in a single ring, without including electron-electron interactions, transverse channels, or disorder. These consequences depend sensitively on the fixed number (modulo 4) of particles on a given ring, and lead to strong fluctuations between samples containing a small number of rings.

Persistent currents from Berry's phase in mesoscopic systems

Daniel Loss and Paul M. Goldbart (Urbana-Champaign). Phys. Rev. B 45, 13544 (1992) The quantum orbital motion of electrons in mesoscopic normal-metal rings threaded by a magnetic flux produces striking interference phenomena such as persistent currents due to the Aharonov-Bohm effect. Similarly, when a quantum spin adiabatically follows a magnetic field that rotates slowly in time, the phase of its state vector acquires an additional contribution known as the Berry phase. We explore the combination of these two quantum phenomena by examining the interplay between orbital and spin degrees of freedom for a charged spin-1/2 particle moving in a mesoscopic ring embedded in a classical, static inhomogeneous magnetic field, i.e, a texture. As a consequence of its orbital motion through the texture, the spin experiences, via the Zeeman interaction, a varying magnetic field. This results in a Berry—or geometric—phase, leading to persistent (i.e., equilibrium) currents of charge and spin. These mesoscopic phenomena are related to (but should be distinguished from) the conventional persistent currents that result from magnetic flux through a ring. We develop a path-integral approach to decouple the orbital and spin motion and, by using an adiabatic approximation, we compute the equilibrium expectation values of the persistent charge and spin currents and the magnetization. We find that the persistent currents depend on the texture in a striking manner through a geometric phase (related to a surface area characterizing the texture) and a geometric vector (related to the projections of this area). In the special case of a cylindrically symmetric texture we use a spectrum obtained by Kuratsuji and Iida to obtain exact results that confirm, independently, the validity of the path-integral approach in the adiabatic limit. We discuss the connection between the geometric vector and quantum-mechanical correlations, and examine quantum fluctuations and the zero-point energy.

Weak-localization effects and conductance fluctuations: Implications of inhomogeneous magnetic fields

Paul M. Goldbart, Herbert Schoeller, and Daniel Loss. Phys. Rev. B 48, 15218 (1993) Low-temperature transport in disordered conductors exhibits a variety of fascinating quantum-mechanical interference effects associated with the phenomenon of weak localization. Such effects are typically isolated and probed by virtue of their sensitivity to applied homogeneous magnetic fields, which introduce Aharonov-Bohm phase factors into quantum-mechanical amplitudes. Analogous interference effects have been proposed in the context of the quantum transport of (possibly electrically neutral) particles with spin in the presence of inhomogeneous magnetic fields, which have the effect of introducing Berry phases. Thus, the possibility is raised of isolating and probing quantum interference effects through their sensitivity to the inhomogeneity of applied magnetic fields. In this appear we develop an approach to the study of quantum transport in disordered conductors in the presence of almost arbitrarily inhomogeneous magnetic fields, which is based on diagrammatic and semiclassical path-integral techniques and a subsequent adiabatic approximation. We illustrate these ideas with applications to three examples: anomalous weak-field magnetoconductance, conductance oscillations in mesoscopic multiply connected structures, and sample-dependent mesoscopic conductance fluctuations. Among other things, we find that while in the context of the disorder-averaged conductance it is accurate to regard systems as being composed of two independent subsystems (having spins aligned or antialigned with the local external magnetic field) a more interesting and refined structure emerges in the context of conductance fluctuations.

Berry's phase and persistent charge and spin currents in textured mesoscopic rings

Daniel Loss, Paul Goldbart (Urbana-Champaign), and A. V. Balatsky (Los Alamos). Phys. Rev. Lett. 65, 1655 (1990) We consider the motion of electrons through a mesoscopic ring in the presence of a classical, static, inhomogeneous, magnetic field. Zeeman interaction between the electron spin and this texture couples spin and orbital motion, and results in a Berry phase. As a consequence, the system supports persistent equilibrium spin and charge currents, even in the absence of conventional electromagnetic flux through the ring. We mention the possibility of analogous persistent mass and spin currents in normal 3He.

Commutation relations for periodic operators.

D. Loss and K. Mullen J. Phys. A 25 (1992) L235-L239 Although periodic variables are common in quantum systems, there is still some question of their proper commutation relations. The authors show that the standard commutation relations, when applied carefully, do not lead to inconsistencies. They discuss other approaches to the problem in the literature.

The effect of dissipation on phase periodicity and the quantum dynamics of Josephson junctions.

D. Loss and K. Mullen. Phys. Rev. A43 (1991) 2129-2138 We consider systems described by compact, periodic variables, such as Josephson junctions and small normal metal rings. We examine how they can be coupled to a heat bath, via either momentum or position coupling, and how the two descriptions can be related by a unitary transformation. We show how it is critical to transform not only the Hamiltonian, but also the initial conditions. We then demonstrate that for certain types of initial conditions, paths of different winding number can interfere. Still other, ostensibly reasonable initial conditions, have no such interference. We conclude with a discussion of appropriate models to describe periodic systems.

Dephasing by a dynamic environment.

D. Loss and K. Mullen. Phys. Rev. B43 (1991) 13252-13261. We investigate the manner in which quantum interference is suppressed when a particle interacts with a spatially localized, dynamical environment. To do so we examine a model with two classical paths along which an electron can travel and allow it to interact with a bath of harmonic oscillators on one path and travel freely on the other. In particular, we show that the quantum fluctuations of the path of the particle can couple to the environment and thus lead to dephasing and calculate the dephasing time in the high-temperature limit. We compare this result to other views of how propagating electrons lose phase coherence.

Hopping conductivity for localized electronic states-Liouville space formalism.

D. Loss and P. C. W. Holdsworth. Physica B 176 (1992) 319-326. We calculate the conductivity for a system of localized electrons, with conduction via phonon assisted transitions, using a Liouville space formalism. The result is in agreement with lowest order rate equation calculations, leading to an equivalent random resistance network. Our work is in contradiction with a similar work of Capek. We illustrate the differences between the two calculations and show that ours is the correct procedure to follow. We also comment on the criticism that the predictions of the random resistance network model do not agree with experimental data.

Linear quantum Enskog equation. I. Homogeneous quantum uids.

Daniel Loss J. Stat. Phys. 59 (1990) 691-723. The quantum-statistical generalization of the well-known classical, linear revised Enskog equation is derived for spatially uniform systems. This new quantum kinetic equation allows the study of equilibrium time correlation functions and their associated transport coefficients of normal quantum fluids where static correlations and degeneracy effects due to particle statistics (both are treated exactly) are important. Furthermore, we derive the quantum-statistical analog of the classical ring operator. These microscopic and systematic derivations are based on a recently developed superoperator formalism (including cluster expansion techniques) that, as a main feature, allows a clear distinction between static and dynamic correlations, which is crucial in the discussion of the Enskog approximation.

Linear quantum Enskog equation. II. Inhomogeneous quantum fluids.

D. Loss. J. Stat. Phys. 61(1990) 463-493. This is the second part of a work concerned with the quantum-statistical generalization of classical Enskog theory, whereby the first part is extended to spatially inhomogeneous fluids. In particular, working with Liouville operators and using cluster expansions and projection operators, we derive the inhomogeneous linear quantum Enskog equation and express the dynamic structure factor and the nonlocal mobility tensor in terms of the corresponding quantum Enskog collision operator. Thereby static correlations due to excluded volume effects and quantum-statistical correlations due to the fermionic (bosonic) character of the pairwise strongly interacting particles are treated exactly. When static correlations are neglected, this Enskog equation reduces to the inhomogeneous linear quantum Boltzmann equation (containing an exchange-modifiedt-matrix). In the classical limit, the well-known linear revised Enskog theory is recovered for hard spheres.

Quantum Boltzmann-Lorentz model approach to the line shape problem.

D. Loss, A. Thellung and T.A. Turski. Phys. Rev. A41 (1990) 3005-3015. Following our earlier work on quantum kinetic equations and the use of the Boltzmann-Lorentz model to describe the collisional broadening of spectra, we develop a new approach to that problem in which the center-of-mass motion of the emitter as well as the interaction during the collision between the emitter and the buffer gas are treated quantum mechanically. Possible further extensions of our model are discussed.

Comparison between different Markov approximations for open spin systems.

M. Celio and D. Loss. Physica A158 (1989) 769-785. Various Markov approximations obtained from the Zwanzig-Nakajima and the memoryless master equations are investigated. In particular, considering a simple model (spin- in a magnetic field and coupled to a reservoir), we discuss the convergence behaviour and the positivity of these approximations. It is found that a well-known positivity preserving Markov generator, introduced in the rigorous study of the weak coupling limit, does not necessarily represent a good approximation for physically relevant times, especially if the system is nearly degenerate.

Simplified virial expansions in the canonical ensemble.

D. Loss, H. Schoeller and A. Thellung. Physica A155 (1989) 373-384. For a classical monatomic real gas the well-known virial expansion of the equation of state and the reduced distribution function are rederived within the canonical ensemble formalism. In particular, two simplified methods are presented which lead directly to the desired result without making use of fugacity or perturbation expansions. The first method generalizes an argument given by Brout. This generalized argument is also used in the second method, which is based on cluster expansions and which is directly related to van Kampen's approach.

Quantum-statistical kinetic equations.

D. Loss and H. Schoeller. J. Stat. Phys. 56 (1989) 175-201. Considering a homogeneous normal quantum fluid consisting of identical interacting fermions or bosons, we derive an exact quantum-statistical generalized kinetic equation with a collision operator given as explicit cluster series where exchange effects are included through renormalized Liouville operators. This new result is obtained by applying a recently developed superoperator formalism (Liouville operators, cluster expansions, symmetrized projectors,Pq-rule, etc.) to nonequilibrium systems described by a density operator(t) which obeys the von Neumann equation. By means of this formalism a factorization theorem is proven (being essential for obtaining closed equations), and partial resummations (leading to renormalized quantities) are performed. As an illustrative application, the quantum-statistical versions (including exchange effects due to Fermi-Dirac or Bose-Einstein statistics) of the homogeneous Boltzmann (binary collisions) and Choh-Uhlenbeck (triple collisions) equations are derived.

A new quantum-statistical evaluation method for time correlation functions.

D. Loss and H. Schoeller. J. Stat. Phys. 54 (1989) 765-795. Considering a system ofN identical interacting particles, which obey Fermi-Dirac or Bose-Einstein statistics, we derive new formulas for correlation functions of the type C(t)=⟨ΣNi=1Ai(t)ΣNj=1Bj⟩ (whereB j is diagonal in the free-particle states) in the thermodynamic limit. Thereby we apply and extend a superoperator formalism, recently developed for the derivation of long-time tails in semiclassical systems. As an illustrative application, the Boltzmann equation value of the time-integrated correlation functionC(t) is derived in a straightforward manner. Due to exchange effects, the obtained t-matrix and the resulting scattering cross section, which occurs in the Boltzmann collision operator, are now functionals of the Fermi-Dirac or Bose-Einstein distribution.

A new microscopic evaluation method for correlation functions: long time tails.

D. Loss and H. Schoeller. Physica A150 (1988) 199-243. Considering a moderately dense fluid, whose N particles interact with a general short-range repulsive potential, we show that the momentum autocorrelation function (associated with self-diffusion processes) has a long time tail of the form (d = 2, 3: number of dimensions). Thereby new powerful concepts, in particular the Pq-rule and the Pq-singularity (the latter being a generalization of van Hove's diagonal singularity) are introduced in order to select the most divergent terms of the density expansion of the correlation function in the thermodynamic and long time limit. It turns out that this method is more suitable for removing a special class of divergences than the Zwanzig inversion method applied by previous workers in this field.

The effect of dissipation on phase periodicity and the quantum dynamics of Josephson junctions.

D. Loss and K. Mullen. Phys. Rev. A43 (1991) 2129-2138. We consider systems described by compact, periodic variables, such as Josephson junctions and small normal metal rings. We examine how they can be coupled to a heat bath, via either momentum or position coupling, and how the two descriptions can be related by a unitary transformation. We show how it is critical to transform not only the Hamiltonian, but also the initial conditions. We then demonstrate that for certain types of initial conditions, paths of different winding number can interfere. Still other, ostensibly reasonable initial conditions, have no such interference. We conclude with a discussion of appropriate models to describe periodic systems.

Second virial coefficient of an interacting anyon gas.

D. Loss and Y. Fu. Phys. Rev. Lett. 67(1991) 294-297 We consider the statistical mechanics of an interacting anyon gas with a repulsive pair potential of the form g/r2. The second virial coefficient is calculated and shown to be a smooth function of the statistical parameter α for g>0, without the cusps previously found at the Bose point in a noninteracting anyon gas. As g→0 the cusps remain absent but a different type of singularity develops at the Bose point, showing that the order of limits is crucial.

Macroscopic quantum tunneling in antiferromagnetic ferritin particles.

D. Awschalom, J. Smyth, G. Grinstein, D. DiVincenzo, and D. Loss. Phys. Rev. Lett. 68 (1992) 3092-3095. Erratum: 71 (1993) 4279. We report low-temperature measurements of the frequency-dependent magnetic noise and magnetic susceptibility of nanometer-scale antiferromagnetic horse-spleen ferritin particles, using an integrated dc SQUID microsusceptometer. A sharply defined resonance near 1 MHz develops below T∼0.2 K. The behavior of this resonance as a function of temperature, applied magnetic field, and particle concentration indicates that it results from macroscopic quantum tunneling of the Néel vector of the antiferromagnets.

Resonant phenomena in compact and extended systems.

K. Mullen, D. Loss, and H. T. C. Stoof. Phys. Rev. B 47 (1993) 2689-2706. We consider the dynamics of Josephson junctions in two formulations: one where the phase is defined on a compact interval [0,2π], and a second where it is defined on an extended interval (-∞,∞). We find that in general the two approaches are not equivalent: they have different sets of allowable initial conditions, and similar initial conditions can produce different values for observables. However, they do have identical predictions for the appearance of resonances. These resonances can be understood in a framework of ‘‘resonant Zener tunneling,’’ which encompasses resonant tunneling in superlattices, tunneling between Wannier-Stark states in crystals, and Landau-Zener transitions in small metal rings. This framework also reveals a type of resonance in some quantum systems that are strongly driven (i.e., in the sudden limit), wherein the system has a large amplitude to transfer out of the diabatic state.

Quantum tunneling and dissipation in nanometer-scale magnets.

D. Loss, D. DiVincenzo, G. Grinstein, D. Awschalom, and J. Smyth. Physica B 189 (1993) 189-203. We summarize recent low-temperature noise and AC magnetic susceptibility measurements on the nanometer-scale magnetic protein horse-spleen ferritin. The experiments show a narrow resonance peak at about 106 Hz, which is discussed in the framework of recently-developed theories of macroscopic quantum coherence and tunneling in antiferromagnets; theory and experiment are argued to agree qualitatively, though quantitative discrepancies remain. We also review the recent analysis of a rather general spin-parity effect for tunneling in magnetic systems: systems with appropriate symmetry may exhibit quantum tunneling for integer spin, but not for half-odd-integer spin, where destructive interference between different tunneling paths suppresses the tunneling. Finally, we study the effect on this spin-parity phenomenon caused by dissipation, i.e. coupling to an environment consisting of a bath of harmonic oscillators. Using the real-time, Feynman-Vernon path integral formalism, we find models where an arbitrarily small amount of ohmic dissipation completely destroys the spin-parity effect (i.e., produces as such tunneling for half-odd-integer spins as for integer spins), and others where the effect appears to disappear gradually with increasing dissipation. Suprisingly, however, there is a sense in which the spin-parity effect is preserved in both types of models: a Calderia-Leggett type of analysis shows that neither experiences any tunnel splitting of its ground state. We present simple arguments for how this intriguing paradox might be resolved.

Edge-state transport and conductance fluctuations in the metallic phase of the quantum Hall regime.

D. Maslov and D. Loss. Phys.Rev.Lett. 71 (1993) 4222-4225. We study mesoscopic conductance fluctuations in the metallic phase of the integer quantum Hall effect. We derive effective boundary conditions for the diffuson propagator which incorporate the edge-state transport along the sample boundaries. It is shown that the presence of edge states leads to a resonance structure in the conductance fluctuations which occurs as the Fermi level moves across the band of extended states. The result of this analytic treatment is consistent with that of our numerical simulations and can be tested experimentally.

Quantum interference effects in inhomogeneous magnetic fields.

D. Loss, H. Schoeller, and P.M. Goldbart. Physica B 194-196 (1994) 1145-1146. An approach is developed to study quantum transport in disordered conductors in the presence of inhomogeneous magnetic fields, which have the effect of introducing Berry phases. The possibility is raised of isolating and probing quantum interference phenomena through their sensitivity to the inhomogeneity of applied magnetic fields. Using semiclassical and diagrammatic techniques, the ideas are illustrated with applications to three examples: anomalous weak-field magnetoconductance, conductance oscillations in mesoscopic multiply-connected structures, and sample-dependent mesoscopic conductance variations.

Bloch states of a Bloch wall.

H. B. Braun and D. Loss. J. Appl. Phys. 76 (1994) 6177-6179. Bloch walls in mesoscopic ferromagnets can tunnel between periodically arranged pinning sites leading to a Bloch band. The quantum spin phase gives rise to a spin parity dependent shift in the dispersion. Static external magnetic fields induce magnetization oscillations and provide a magnetic analogue of the Josephson effect.

Comment on Have resonance experiments seen macroscopic quantum coherence in magnetic particles? The case from power absorption.

D. Awschalom, J. Smyth, G. Grinstein, D. DiVincenzo, and D. Loss. Phys. Rev. Lett. 71 (1993) 4276. A Comment on the Letter by A. Garg Phys. Rev. Lett. 71, 4442 (1993).

Quantum interference effects in inhomogeneous magnetic fields.

D. Loss, H. Schoeller, and P.M. Goldbart. Physica B 194-196 (1994). An approach is developed to study quantum transport in disordered conductors in the presence of inhomogeneous magnetic fields, which have the effect of introducing Berry phases. The possibility is raised of isolating and probing quantum interference phenomena through their sensitivity to the inhomogeneity of applied magnetic fields. Using semiclassical and diagrammatic techniques, the ideas are illustrated with applications to three examples: anomalous weak-field magnetoconductance, conductance oscillations in mesoscopic multiply-connected structures, and sample-dependent mesoscopic conductance variations.

Dissipation and quantum propagation of Bloch walls.

H. B. Braun and D. Loss. Europhys. Lett. 31 (1995) 555-560. Bloch walls in mesoscopic ferromagnets are shown to tunnel coherently between periodically arranged pinning sites and to form a band similar to a Bloch electron in a crystal lattice. Eliminating the spin wave degrees of freedom we derive an effective action for the Bloch wall position. The dissipative term in this action involves a spectral function with a gap and therefore damping due to spin waves is negligible at low temperatures. It is shown that the Berry phase gives rise to a spin-parity-dependent shift in the dispersion. We show that this band structure can give rise to Bloch oscillations of the magnetization in response to a static external field, which provides a magnetic analog to the Josephson effect.

Quantization of Superflow Circulation and Magnetic Flux with a Tunable Offset.

Y. Lyanda-Geller, P.M. Goldbart, and D. Loss. Phys. Rev. B 53 (1996) 12395-12399. Quantization of superflow circulation and of magnetic flux are considered for systems, such as superfluid A3 and unconventional superconductors, having nonscalar order parameters. The circulation is shown to be the anholonomy in the parallel transport of the order parameter. For multiply-connected samples free of distributed vorticity, circulation and flux are predicted to be quantized, but generically to nonintegral values that are tunably offset from integers. This amounts to a version of Aharonov-Bohm physics. Experimental settings for testing these issues are discussed. © 1996 The American Physical Society.

Andreev reflection and Josephson currents in Luttinger liquids.

D. L. Maslov, M. Stone, P.M. Goldbart, and D. Loss. Phys. Rev. B 53 (1996) 1548-1557. A theory describing a one-dimensional Luttinger liquid in contact with a superconductor is developed. Boundary conditions for the fermion fields describing Andreev reflection at the contacts are derived and used to construct a bosonic representation of the fermions. The Josephson current through a superconductor/Luttinger liquid/superconductor junction is considered for both perfectly and poorly transmitting interfaces. In the former case, the Josephson current at low temperatures is found to be essentially unaffected by electron-electron interactions. In the latter case, significant renormalization of the Josephson current occurs. The profile of the (induced) condensate wave function in a semi-infinite Luttinger liquid in contact with a superconductor is shown to decay as a power law, the exponent depending on the sign and strength of the interactions. In the case of repulsive (attractive) interactions the decay is faster (slower) than in their absence. An equivalent method of calculating the Josephson current through a Luttinger liquid, which employs the bosonization of the system as a whole (i.e., superconductor, as well as Luttinger liquid) is developed and shown to give the results equivalent to those obtained via boundary conditions describing Andreev reflection. © 1996 The American Physical Society.

Luttinger liquids and composite fermions in nanostructures: what is the nature of the edge states in the fractional quantum Hall regime?

M. Geller, D. Loss, and G. Kirzcenow. Superlattices and Microstructures 21, 49 (1997) We study the Aharonov-Bohm conductance oscillations of a constriction with an antidot in the fractional quantum Hall regime using a recently proposed composite-fermion Fermi liquid theory, and also using Wen's chiral Luttinger liquid theory extended to include mesoscopic effects. The predictions of the composite-fermion Fermi liquid theory are very similar to standard Fermi liquid theory and are consistent with recent experiments. In our chiral Luttinger liquid theory, which is valid in an experimentally realizable ‘strong-antidot-coupling’ regime for bulk filling factorsg= 1/q(qodd), the finite size of the antidot introduces a new temperature scaleT0≡hv/ΠkBL, wherevis the Fermi velocity andLis the circumference of the antidot edge state. Chiral Luttinger liquid theory predicts the low-temperature (TT0), however, we predict a new crossover to aT2q–1e–qT/T0temperature dependence, which is qualitatively similar to Fermi liquid behavior. We show how measurements in the strong-antidot-coupling regime, where transmission through the device is weak, should be able to distinguish between Fermi liquid and chiral Luttinger liquid behavior both at low and high temperatures and in the linear and nonlinear response regimes. Finally, we predict new mesoscopic edge-current oscillations, which are similar to persistent current oscillations in a mesoscopic ring, except that they are not reduced in amplitude by disorder. In the fractional regime, these ‘chiral persistent currents’ have a universal non-Fermi-liquid temperature dependence, and may be another ideal system to observe a chiral Luttinger liquid.

Relationship between susceptibility and spin stiffness of mesoscopic quantum antiferromagnets.

A. Scott and D. Loss. Physica A 239, 47 (1997). We investigate the quantum antiferromagnetic Heisenberg spin chain of integer spinS in the semiclassical limit (S≫1) for a non-uniform system of finite-size and temperature. At low temperatures and in the mesoscopic regime with the correlation length being larger than the characteristic system size, boundary conditions significantly affect the scaling behavior of the spin stiffnessϱs and the magnetic response. Using a coherent spin-state path integral formulation and allowing for a non-uniform time varying external magnetic field we determine the dynamic susceptibility tensorχαβ. We find that stiffness and susceptibility satisfy a remarkable relation in the mesoscopic regime, viz. ϱs/c2=χzz. We briefly discuss the experimental consequences of this relationship.

Quantum spin dynamics in mesoscopic magnets.

A. Chiolero and D. Loss. Physica E 1 (1998) 292-296. Molecular magnets present a new class of systems that can exhibit macroscopic quantum coherence. We show that magnetic fields can induce controllable tunnel barriers for the Néel vector in antiferromagnets such as the ferric wheel. The tunneling dynamics manifests itself in e.g. NMR resonances, the static magnetization or the specific heat. Numerical simulations confirm and complement the analytical results based on an extended non-linear sigma model.

Mesoscopic effects in the fractional quantum Hall effect.

M.R. Geller and D. Loss. Physica E 1 (1998) 120-123. Edge states of the quantum Hall fluid provide an opportunity to study mesoscopic effects in a highly correlated electron system that is both experimentally accessible and theoretically tractable. In this paper we review recent work on the persistent current and Aharonov–Bohm magnetoconductance oscillations in the fractional quantum Hall effect regime.

Universality of shot noise in multiterminal diffusive conductors

Eugene Sukhorukov and Daniel Loss Phys. Rev. Lett. 80, 4959 (1998); arXiv:cond-mat/9802050. We prove the universality of shot noise in multiterminal diffusive conductors of arbitrary shape and dimension for purely elastic scattering as well as for hot electrons. Using a Boltzmann-Langevin approach we reduce the calculation of shot-noise correlators to the solution of a diffusion equation. We show that shot noise in multiterminal conductors is a nonlocal quantity and that exchange effects can occur without quantum phase coherence even at zero electron temperature. Concrete numbers for shot noise are given that can be tested experimentally.

Spin relaxation in Mn12-acetate.

M. Leuenberger and D. Loss. Europhys. Lett. 46 (5), 692-698 (1999); cond-mat/9810156. We present a comprehensive theory of the magnetization relaxation in a Mn12-acetate crystal based on thermally assisted spin tunneling induced by quartic anisotropy and weak transverse magnetic fields. The overall relaxation rate as a function of the magnetic field is calculated and shown to agree well with data including all resonance peaks. The Lorentzian shape of the resonances is also in good agreement with recent data. A generalized master equation including resonances is derived and solved exactly. It is shown that many transition paths with comparable weight exist that contribute to the relaxation process. Previously unknown spin-phonon coupling constants are calculated explicitly.

Quantum Computers and Quantum Coherence.

D.P. DiVincenzo and D. Loss. J. Mag. Magn. Matl. 200, 202 (1999). Invited review paper for special issue of J. Mag. Magn. Matl. ("Magnetism beyond 2000"); arXiv:cond-mat/9901137. If the states of spins in solids can be created, manipulated, and measured at the single-quantum level, an entirely new form of information processing, quantum computing, will be possible. We first give an overview of quantum information processing, showing that the famous Shor speedup of integer factoring is just one of a host of important applications for qubits, including cryptography, counterfeit protection, channel capacity enhancement, distributed computing, and others. We review our proposed spin-quantum dot architecture for a quantum computer, and we indicate a variety of first generation materials, optical, and electrical measurements which should be considered. We analyze the efficiency of a two-dot device as a transmitter of quantum information via the propagation of qubit carriers (i.e. electrons) in a Fermi sea.

Electron and Nuclear Spin Dynamics in Antiferromagnetic Molecular Rings

Florian Meier and Daniel Loss. Phys. Rev. Lett. 86, 5373 (2001)

Spin-entangled electrons in mesoscopic systems.

G. Burkard, E.V. Sukhorukov, P. Recher, and D. Loss. Phys.-Usp. 44, 126 (2001) Entanglement acts as a fundamental resource for many applications in quantum communication. We propose and theoretically analyze methods for preparing and detecting entanglement between the spins of electrons in a mesoscopic environment. The entanglement production mechanism which we present is based on two quantum dots coupled to a superconductor from which paired electrons are injected via Andreev tunneling. The spin-correlated electrons can then hop from the quantum dots into normal leads. For detection we propose to measure the shot noise which is produced by the entangled electrons after they have passed a beam splitter. The enhancement of the noise by a factor of two turns out to be a unique signature for the spin singlet, a maximally entangled state. In a different setting, the entangled ground state in two tunnel-coupled quantum dots is detected via the Aharonov–Bohm oscillations in the co-tunneling current.

Quantenphysik hat grossen praktischen Nutzen.

Christoph Bruder and Daniel Loss. Basler Zeitung, Nr. 242, 17. Oktober 1998. Fuer Laien ein Buch mit sieben Siegeln, fuer die Wissenschaft ein weites fruchtbares Feld: Die Quantenphysik hat viele ueberraschende Einsichten in die Struktur der Materie gebracht und ist noch fuer manche Ueberraschung gut. Die beiden Basler Quantenforscher Christoph Bruder und Daniel Loss erklaeren, fuer welche Entdeckungen die diesjaehrigen Nobelpreise fuer Chemie und Physik in Stockholm verliehen wurden. Jeden Oktober schauen Wissenschafter aus aller Welt gespannt nach Stockholm: In dieser Zeit wird die Entscheidung des Nobelkomitees bekanntgegeben. Der von Alfred Nobel gestiftete Preis ist die hoechste Auszeichnung in den Naturwissenschaften ueberhaupt, seine Traeger, die vorher oft nur in Fachkreisen bekannt waren, gelangen zu Weltruhm und koennen sich fuer einige Tage des Ansturms von Journalisten und Fernsehkameras nicht mehr erwehren. Die diesjaehrigen Preise fuer Physik und Chemie wurden beide verliehen fuer Fortschritte in der Quantenphysik, der grundlegendsten und erfolgreichsten aller physikalischen Theorien. Der diesjaehrige Nobelpreis fuer Physik geht an drei amerikanische Wissenschafter: den aus Deutschland stammenden Horst Stoermer, 1949, von der Columbia University in New York, den 1939 in China geborenen Daniel Tsui von der Universitaet Princeton und Bob Laughlin, 1950, von der Universitaet Stanford. Sie wurden ausgezeichnet ''fuer die Entdeckung einer neuen Art von Quantenfluessigkeit mit gebrochenzahlig geladenen Anregungen".

Stability of the conventional fixed point of the non-linear sigma-model in three dimensions.

T. Sun and D. Loss. Europhys. Lett. 34 (1996) 355-359. The stability of the conventional fixed point of the nonlinear sigma-model in (2 + epsilon)-dimensions has been studied by calculating the anomalous dimensions of leading order O(n - 1) symmetric gradient operators. The full dimensions, i.e. the canonical dimensions plus the anomalous dimensions, of these operators at the fixed point are found to be negative and therefore the fixed point is stable against the perturbation of these operators. The results indicate that as far as the O(n) symmetry-breaking regime is concerned, the conventional treatment of this model is adequate.

The electrical conductivity for inhomogeneous electric fields by the Liouville operator method.

D. Loss and A. Thellung. Physica 144A (1987) 17-28. A derivation is given of the electrical conductivity for a metal in a space- and time-dependent electric field. Thereby it is assumed that the electrons are scattered elastically by randomly distributed impurities. The derivation starts from the Kubo-Nakajima formula and is based on a perturbation expansion with Liouville operators, where use is made of van Hove's diagonal singularity property of the scattering potential. The formulae obtained are compact and the method is simpler and more transparent than the perturbation formalism developed by van Hove. It is shown that the lowest order approximation corresponds to the Boltzmann equation for electrons in inhomogeneous electric fields.

Correction terms to the Lamba2t-limit of van Hove by the Liouville operator method. II. Evaluation of the Kubo formula for the electrical conductivity.

D. Loss Physica 139A (1986) 526-542. As an illustrative application of the one-resolvent method with Liouville operators recently developed by the author, the electrical conductivity tensor of metals for spherically symmetric impurity centres is calculated including all terms of order λ−2, λ−1 and λ0 in the interaction strength λ between electron and impurities. For each term the dependence on the impurity concentration is determined. Electron-electron interactions are neglected. It is shown that the very compact and explicit result in terms of Liouville operators can be brought into the form previously obtained for the same case by Janner who used an extension of the double-resolvent formalism of van Hove.

Correction terms to the Lamba2t-limit of van Hove by the Liouville operator method. I. A general perturbation treatment.

Daniel Loss Physica 139A (1986) 505-525 A large isolated quantum many-body system described by a Hamiltonian of the form H = H0 + λV is considered. In view of the calculation of transport coefficients general expressions to arbitrary order in λ are derived for the asymptotic values of the time integrals of matrix elements of Heisenberg operators. Thereby use is made of the superoperator formalism, in particular of the Liouville operator, the resolvent of which is the starting-point for a perturbation treatment to general order in λ. The leading part of order λ−2 in the obtained power series for diagonal operators (i.e. diagonal with respect to the eigenstates of the unperturbed Hamiltonian H0) corresponds to the asymptotic value of a time integral calculated in the λ2t-limit introduced by van Hove. The development is based on a thorough discussion of the considered Hamiltonian, unperturbed part and perturbation. Thereby it turns out that van Hove's diagonal singularity is essential for the formalism to make sense in the thermodynamic limit.