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Motivated by the recent development of the quantum twisting microscope, we formulate a theory of elastic momentum-resolved tunneling across a planar tunnel junction between a monolayer graphene layer situated on a tip and a twisting graphene-based sample. We elucidate features in the dependence of the tunnel current on bias and twist angle, which reflect the sample band structure and allow the tip to probe the momentum-and energy-resolved single-particle excitations of the sample. While the strongest features originate from the Fermi edge of the tip, we argue that features associated with the tip Dirac points provide a more immediate and precise map of the sample band structure. We specifically compute the low-temperature tunneling spectrum of magic angle twisted bilayer graphene (MATBG) rotated relative to the tip by nearly commensurate angles, highlighting the potential of Dirac-point spectroscopy to measure single-particle spectral functions of flat bands along specific lines in reciprocal space. Furthermore, our analysis of tunneling matrix elements suggests a method to extract the ratio of the intra-and inter-sublattice tunneling parameters $w_0/w_1$ of MATBG from the differential tunneling conductance. Finally, we discuss signatures of $C_{3z}$ symmetry breaking in the tunneling spectrum using strained MATBG as an example. Our work establishes a general theoretical framework for Dirac-point spectroscopy of flat-band systems using the quantum twisting microscope., Comment: 26 pages, 12 figures

We show that an s-wave superconductor may become topological in the presence of a magnetic field which leads to formation of an Abrikosov vortex lattice. Below the upper critical field, a new superconducting state with a nontrivial topological number emerges, which we call Abrikosov-Chern superconducting state. Deeper in the superconducting domain, the topological number changes in steps, eventually reaching zero. Our theory uncovers the nature of evolution from an integer quantum Hall state having a cyclotron gap above the upper critical field to the topologically-trivial s-wave superconductor carrying finite-energy Caroli-de Gennes-Matricon levels at low field. Topological transitions manifest as changes in the number of edge modes, detectable through tunneling microscopy and thermal transport measurements., Comment: 5 pages, 3 figures

We develop a theory of the nonlocal conductance $G_{RL}(V)$ for a disordered Majorana wire tuned near the topological transition critical point. We show that the differential conductance is an odd function of bias, $G_{RL}(V) = -G_{RL}(-V)$. We factorize the conductance into terms describing the contacts between the wire and the normal leads, and the term describing quasiparticle propagation along the wire. Topological transition affects only the latter term. At the critical point, the quasiparticle localization length has a logarithmic singularity at the Fermi level, $l(E) \propto \ln(1 / E)$. This singularity directly manifests in the conductance magnitude, as $\ln |G_{RL}(V) / G_Q| \sim L / l(eV)$ for the wire of length $L \gg l(eV)$. Tuning the wire away from the immediate vicinity of the critical point changes the monotonicity of $l(E)$. This change in monotonicty allows us to define the width of the critical region around the transition point., Comment: 26 pages, 9 figues

We develop a theory of probing phonon modes of van-der-Waals materials using the quantum twisting microscope. While elastic tunneling dominates the tunneling current at small twist angles, the momentum mismatch between the K-points of tip and sample at large twist angles can only be bridged by inelastic scattering. This allows for probing phonon dispersions along certain lines in reciprocal space by measuring the tunneling current as a function of twist angle and bias voltage. We illustrate this modality of the quantum twisting microscope by developing a systematic theory for graphene-graphene junctions. We show that beyond phonon dispersions, the tunneling current also encodes the strength of electron-phonon couplings. Extracting the coupling strengths for individual phonon modes requires careful consideration of various inelastic tunneling processes. These processes are associated with the intralayer and interlayer electron-phonon couplings and appear at different orders in a perturbative calculation of the tunneling current. We find that the dominant process depends on the particular phonon mode under consideration. Our results inform the quest to understand the origin of superconductivity in twisted bilayer graphene and provide a case study for quantum-twisting-microscope investigations of collective modes., Comment: 22 pages, 8 figures

We demonstrate transport in 2D arrays of multiterminal $\varphi_0$-junctions. When applying an in-plane magnetic field we observe nonreciprocal vortex depinning currents, induced by a ratchet-like pinning potential. The ratchet effect is explained as a consequence of spontaneous supercurrents that arise in the presence of next-nearest neighbor Josephson couplings. Tuning the density of vortices to commensurate values of the frustration parameter results in an enhancement of the ratchet effect. In addition, we find a surprising sign reversal of the ratchet effect near frustration 1/3., Comment: 15 pages, 11 Figures

We analyze the transition into the most favorable ordered state for a system of 2D fermions with spin and valley degrees of freedom. We show that for a range of rotationally invariant dispersions, the ordering transition is highly unconventional: the associated susceptibility diverges (or almost diverges) at the transition, yet immediately below it the system jumps discontinuously into a fully polarized state. We analyze the dispersion of the longitudinal and transverse collective modes in different parameter regimes above and below the transition. Additionally, we consider ordering in a system with full $\mathrm{SU(4)}$ symmetry and show that there is a cascade of discontinuous transitions into a set of states, which includes a quarter-metal, a half-metal and a three-quarter metal. We compare our results with the data for biased bilayer and tri-layer graphene., Comment: 24 pages, 16 figures

We show that two-dimensional fermions with dispersion $k^2$ or $k^4$ undergo a first-order Stoner transition to a fully spin-polarized state despite that the spin susceptibility diverges at the critical point. We extend our analysis to systems with dispersion $k^{2\alpha}$ and spin and valley isospin and show that there is a cascade of instabilities into fractional-metal states with some electron bands fully depleted; narrow intermediate ranges of partially-depleted bands exist for $\alpha<1$ or $\alpha > 2$. The susceptibility becomes large near each transition. We discuss applications to biased bilayer and tri-layer graphene and moir\'e systems., Comment: 6 pages, 4 figures

At zero temperature, a Josephson junction coupled to an ohmic environment displays a quantum phase transition between superconducting and insulating phases, depending whether the resistance of the environment is below or above the resistance quantum. At finite temperature, this so-called Schmid transition turns into a crossover. We determine the conditions under which the temperature dependence of the thermal conductance, which characterizes heat flow from a hot to cold resistor across the Josephson junction, displays universal scaling characteristic of the Schmid transition. We also discuss conditions for heat rectification to happen in the circuit. Our work can serve as a guide for identifying signatures of the Schmid transition in heat transport experiments., Comment: 5 pages, 2 figures

Recently developed Josephson junction array transmission lines implement strong-coupling circuit electrodynamics compatible with a range of superconducting quantum devices. They provide both the high impedance which allows for strong quantum fluctuations, and photon modes with which to probe a quantum device, such as a small Josephson junction. In this high-impedance environment, current through the junction is accompanied by charge Bloch oscillations analogous to those in crystalline systems. However, the interplay between Bloch oscillations and environmental photon resonances remains largely unexplored. Here we describe the Bloch oscillations in a transmon-type qubit attached to high-impedance transmission lines with discrete photon spectra. Transmons are characterized by well-separated charge bands, favoring Bloch oscillations over Landau-Zener tunneling. We find resonances in the voltage--current relation and the spectrum of photons emitted by the Bloch oscillations. The transmon also scatters photons inelastically; we find the cross-section for a novel anti-Stokes-like process whereby photons gain a Bloch oscillation quantum. Our results outline how Bloch oscillations leave fingerprints for experiments across the DC, MHz, and GHz ranges., Comment: 9 pages, 4 figures

A transmon qubit embedded in a high-impedance environment acts in a way dual to a conventional Josephson junction. In analogy to the AC Josephson effect, biasing of the transmon by a direct current leads to the oscillations of voltage across it. These oscillations are known as the Bloch oscillations. We find the Bloch oscillations spectrum, and show that the zero-point fluctuations of charge make it broad-band. Despite having a broad-band spectrum, Bloch oscillations can be brought in resonance with an external microwave radiation. The resonances lead to steps in the voltage-current relation, which are dual to the conventional Shapiro steps. We find how the shape of the steps depends on the environment impedance $R$, parameters of the transmon, and the microwave amplitude. The Bloch oscillations rely on the insulating state of the transmon which is realized at impedances exceeding the Schmid transition point, $R > R_Q = h / (2e)^2$., Comment: 8 pages, 4 figures

A mechanism to deterministically prepare a nanowire Josephson junction in an odd parity state is proposed. The mechanism involves population of two Andreev levels by a resonant microwave drive breaking a Cooper pair, and a subsequent ionization of one of the levels by the same drive. Robust preparation of the odd state is allowed by a residual Coulomb repulsion in the junction. A similar resonant process can also be used to prepare the junction in the even state. Our theory explains a recent experiment [J. J. Wesdorp, et al., Phys. Rev. Lett. 131, 117001 (2023)].

Schmid transition was introduced first as a superconductor-insulator transition in the zero-frequency response of a shunted Josephson junction in equilibrium at zero temperature. As it is typical for a quantum impurity problem, at finite frequencies the transition is broadened to a crossover. Modern attempts to find Schmid transition rely on finite-frequency measurements of a quantum circuit. We predict the frequency dependence of the admittance and reflection phase shift for a high-impedance transmission line terminated by a Josephson junction for a wide variety of devices, from a charge qubit to a transmon. Our results identify the circuit parameters allowing for the universal scaling of the responses with frequency, thus helping to identify the Schmid transition from the finite-frequency measurements., Comment: 6+5 pages, 2 figures

Two-dimensional Josephson junction arrays frustrated by a perpendicular magnetic field are predicted to form a cascade of distinct vortex lattice states. Here, we show that the resistivity tensor provides both structural and dynamical information on the vortex-lattice states and intervening phase transitions, which allows for experimental identification of these symmetry-breaking ground states. We illustrate our general approach by a microscopic theory of the resistivity tensor for a range of magnetic fields exhibiting a rich set of vortex lattices as well as transitions to liquid-crystalline vortex states., Comment: 5+3 pages, 5 figures

The density of quasiparticles typically observed in superconducting qubits exceeds the value expected in equilibrium by many orders of magnitude. Can this out-of-equilibrium quasiparticle density still possess an energy distribution in equilibrium with the phonon bath? Here, we answer this question affirmatively by measuring the thermal activation of charge-parity switching in a transmon qubit with a difference in superconducting gap on the two sides of the Josephson junction. We then demonstrate how the gap asymmetry of the device can be exploited to manipulate its parity., Comment: Updated acknowledgements, corrected typos

Two-impurity Kondo models are paradigmatic for correlated spin-fermion systems. Working with Mn atoms on Au(111) covered by a monolayer of MoS$_2$, we tune the inter-adatom exchange via the adatom distance and the adatom-substrate exchange via the location relative to a moir\'e structure of the substrate. Differential-conductance measurements on isolated adatoms exhibit Kondo peaks with heights depending on the adatom location relative to the moir\'e structure. Mn dimers spaced by a few atomic lattice sites exhibit split Kondo resonances. In contrast, adatoms in closely spaced dimers couple antiferromagnetically, resulting in a molecular-singlet ground state. Exciting the singlet-triplet transition by tunneling electrons, we find that the singlet-triplet splitting is surprisingly sensitive to the moir\'e structure. We interpret our results theoretically by relating the variations in the singlet-triplet splitting to the heights of the Kondo peaks of single adatoms, finding evidence for coupling of the adatom spin to multiple conduction electron channels.

Acoustic spontaneous emission into bulk dielectrics can be a strong source of decoherence in quantum devices, especially when a qubit is in the presence of piezoelectric materials. We study the dynamics of a qubit coupled to an acoustic resonator by a piezoelectric film. By varying the surface topography of the resonator from rough to polished to shaped, we explore the crossover from fast decay of an excited qubit to quantum-coherent coupling between the qubit and an isolated phonon mode. Our experimental approach may be used for precision measurements of crystalline vibrations, the design of quantum memories, and the study of electro-mechanical contributions to dielectric loss., Comment: 7 pages, 4 figures; 19 pages supplementary, 5 supplementary figures

We develop a theory of the non-local transport of two counter-propagating $\nu = 1$ quantum Hall edges coupled via a narrow disordered superconductor. The system is self-tuned to the critical point between trivial and topological phases by the competition between tunneling processes with or without particle-hole conversion. The critical conductance is a random, sample-specific quantity with a zero average and unusual bias dependence. The negative values of conductance are relatively stable against variations of the carrier density, which may make the critical state to appear as a topological one., Comment: 10 pages, 5 figures; 10 pages supplementary, 1 supplementary figure

Single-charge tunneling is a decoherence mechanism affecting superconducting qubits, yet the origin of excess quasiparticle excitations (QPs) responsible for this tunneling in superconducting devices is not fully understood. We measure the flux dependence of charge-parity (or simply, ``parity'') switching in an offset-charge-sensitive transmon qubit to identify the contributions of photon-assisted parity switching and QP generation to the overall parity-switching rate. The parity-switching rate exhibits a qubit-state-dependent peak in the flux dependence, indicating a cold distribution of excess QPs which are predominantly trapped in the low-gap film of the device. Moreover, we find that the photon-assisted process contributes significantly to both parity switching and the generation of excess QPs by fitting to a model that self-consistently incorporates photon-assisted parity switching as well as inter-film QP dynamics.

We develop a theory of charge transport along the quantum Hall edge proximitized by a "dirty" superconductor. Disorder randomizes the Andreev reflection rendering the conductance of a proximitized segment a stochastic quantity with zero average for a sufficiently long segment. We find the statistical distribution of the conductance and its dependence on electron density, magnetic field, and temperature., Comment: 6 pages, 1 figure; 6 pages supplementary

The energy levels of a quasi-continuous spectrum in mesoscopic systems fluctuate in positions, and the distribution of the fluctuations reveals information about the microscopic nature of the structure under consideration. Here, we investigate mesoscopic fluctuations of the secondary "smile" gap, that appears in the quasiclassical spectrum of a chaotic cavity coupled to one or more superconductors. Utilizing a random matrix model, we compute numerically the energies of Andreev levels and access the distribution of the gap widths. We mostly concentrate on the universal regime $E_{\mathrm{Th}}\gg\Delta$ with $E_{\mathrm{Th}}$ being the Thouless energy of the cavity and $\Delta$ being the superconducting gap. We find that the distribution is determined by an intermediate energy scale $\Delta_g$ with the value between the level spacing in the cavity $\delta_s$ and the quasiclassical value of the gap $E_g$. From our numerics we extrapolate the first two cumulants of the gap distribution in the limit of large level and channel number. We find that the scaled distribution in this regime is the Tracy-Widom distribution: the same as found by Vavilov at al. [Phys. Rev. Lett. \textbf{86}, 874 (2001)] for the distribution of the minigap edge in the opposite limit $E_{\mathrm{Th}}\ll \Delta$. This leads us to the conclusion that the distribution found is a universal property of chaotic proximity systems at the edge of a continuous spectrum., Comment: 9 pages, 5 figures, submitted to Physical Review B

In the presence of external magnetic field the Fermi-liquid state supports oscillatory spin modes known as Silin modes. We predict the existence of the generalized Silin modes in a multivalley system, monolayer graphene. A gauge- and Berry-gauge- invariant kinetic equation for a multivalley Fermi liquid is developed and applied to the case of graphene with extrinsic spin-orbit coupling (SOC). The interplay of SOC and Berry curvature allows for the excitation of generalized Silin modes in the spin and valley-staggered-spin channels via an AC electric field. The resonant contributions from these modes to the optical conductivity are calculated., Comment: 5 pages, 1 figure; 9 pages supplementary

We develop a theory for the dynamics of an Andreev bound state hosted by a weak link of finite length for which charging effects are important. We derive the linear response of both the current through the link and charge accumulated in it with respect to the phase and gate voltage biases. The resulting matrix encapsulates the spectroscopic properties of a weak link embedded in a microwave resonator. In the low-frequency limit, we obtain the response functions analytically using an effective low-energy Hamiltonian, which we derive. This Hamiltonian minimally accounts for Coulomb interaction and is suitable for a phenomenological description of a weak link having a finite length., Comment: 26 pages, 7 figures. Added a footnote

The tuneability and control of quantum nanostructures in two-dimensional materials offer promising perspectives for their use in future electronics. It is hence necessary to analyze quantum transport in such nanostructures. Material properties such as a complex dispersion, topology, and charge carriers with multiple degrees of freedom, are appealing for novel device functionalities but complicate their theoretical description. Here, we study quantum tunnelling transport across a few-electron bilayer graphene quantum dot. We demonstrate how to uniquely identify single- and two-electron dot states' orbital, spin, and valley composition from differential conductance in a finite magnetic field. Furthermore, we show that the transport features manifest splittings in the dot's spin and valley multiplets induced by interactions and magnetic field (the latter splittings being a consequence of bilayer graphene's Berry curvature). Our results elucidate spin- and valley-dependent tunnelling mechanisms and will help to utilize bilayer graphene quantum dots, e.g., as spin and valley qubits., Comment: 18 pages, 12 figures

We find the admittance $Y(\omega)$ of a Josephson junction at or near a topological transition. The dependence of the admittance on frequency and temperature at the critical point is universal and determined by the symmetries of the system. Despite the absence of a spectral gap at the transition, the dissipative response may remain weak at low energies: $\mathrm{Re}\,Y(\omega)\propto \max (\omega, T)^2$. This behavior is strikingly different from the electromagnetic response of a normal metal. Away from the critical point, the scaling functions for the dependence of the admittance on frequency and temperature are controlled by at most two parameters., Comment: 27 pages, 7 figures

We investigate the scattering of elastic waves off a disordered region described by a one-dimensional random-phase sine-Gordon model. The collective pinning results in an effective static disorder potential with universal and non-Gaussian correlations, acting on propagating waves. We find signatures of the correlations in the wave transmission in a wide frequency range, which covers both the weak and strong localization regimes. Our theory elucidates the dynamics of collectively-pinned phases occurring in any natural or synthetic elastic medium. The latter one is exemplified by a one-dimensional array of Josephson junctions, for which we specify our results. The obtained results provide benchmarks for the array-enabled quantum simulations addressing the dynamics in broader and yet-unexplored domains of individual pinning and quantum Bose glass., Comment: 19 pages, 12 figures

We develop the theory of collective modes supported by a Fermi liquid of electrons in pristine graphene. Under reasonable assumptions regarding the electron-electron interaction, all the modes but the plasmon are over-damped. In addition to the $SU(2)$ symmetric spin mode, these include also the valley imbalance modes obeying a $U(1)$ symmetry, and a $U(2)$ symmetric valley spin imbalance mode. We derive the interactions and diffusion constants characterizing the over-damped modes. The corresponding relaxation rates set fundamental constraints on graphene valley- and spintronics applications., Comment: 7 pages + 12 pages 1 figure supplemental material

Spontaneous decay of a single photon is a notoriously inefficient process in nature irrespective of the frequency range. We report that a quantum phase-slip fluctuation in high-impedance superconducting waveguides can split a single incident microwave photon into a large number of lower-energy photons with a near unit probability. The underlying inelastic photon-photon interaction has no analogs in non-linear optics. Instead, the measured decay rates are explained without adjustable parameters in the framework of a new model of a quantum impurity in a Luttinger liquid. Our result connects circuit quantum electrodynamics to critical phenomena in two-dimensional boundary quantum field theories, important in the physics of strongly-correlated systems. The photon lifetime data represents a rare example of verified and useful quantum many-body simulation., Comment: minor revision for clarity, supplementary material is available at www.superconducting-circuits.com

We investigate the joint effect of viscous and Ohmic dissipation on electric current flow through a slit in a barrier dividing a graphene sheet in two. In the case of the no-slip boundary condition, we find that the competition between the viscous and Ohmic types of the charge flow results in the evolution of the current density profile from a concave to convex shape. We provide a detailed analysis of the evolution and identify favorable conditions to observe it in experiment. In contrast, in the case of the no-stress boundary condition, there is no qualitative difference between the current profiles in the Ohmic and viscous limits. The dichotomy between the behavior corresponding to distinct boundary conditions could be tested experimentally., Comment: v1. 11 pages, 3 figures. Corresponds to published version

Extending the qubit coherence times is a crucial task in building quantum information processing devices. In the three-dimensional cavity implementations of circuit QED, the coherence of superconducting qubits was improved dramatically due to cutting the losses associated with the photon emission. Next frontier in improving the coherence includes the mitigation of the adverse effects of superconducting quasiparticles. In these lectures, we review the basics of the quasiparticles dynamics, their interaction with the qubit degree of freedom, their contribution to the qubit relaxation rates, and approaches to control their effect., Comment: 42 pages, 14 figures. Submitted to SciPost Lecture Notes. To appear in 'Quantum Information Machines; Lecture Notes of the Les Houches Summer School 2019', eds. M. Devoret, B. Huard, and I. Pop

Theoretically, a pair of Majorana bound states in a topological superconductor forms a single fermionic level even at large separations, implying that the parity information is stored nonlocally. The nonlocality leads to a long-distance coherence for electrons tunneling through a Coulomb blockaded Majorana wire [Fu, Phys. Rev. Lett. 104, 056402 (2010)], an effect that can be observed, e.g., in an interferometer. Here, we examine theoretically the coherent electron transfer, taking into account that tunneling implies the long-distance transfer of charge, which is carried by one-dimensional plasmons. We show that the charge dynamics does not affect the coherence of the electron tunneling process in a topological superconductor consisting of a semiconductor wire proximitized by a single bulk superconductor. The coherence may be strongly suppressed, however, if the topological superconductivity derives from a semiconductor wire proximitized by a granular superconductor., Comment: 4+ pages, 3 figures

A semiconducting nanowire proximitized by an $s$-wave superconductor can be tuned into a topological state by an applied magnetic field. This quantum phase transition is marked by the emergence of Majorana zero modes at the ends of the wire. The fusion of Majorana modes at a junction between two nanowires results in a $4\pi$-periodic Josephson effect. We elucidate how the $4\pi$-periodicity arises across the topological phase transition in a highly-transparent short nanowire junction. Owing to a high transmission coefficient, Majorana zero modes coming from different wires are strongly coupled, with an energy scale set by the proximity-induced, field-independent pairing potential. At the same time, the topological spectral gap---defined by competition between superconducting correlations and Zeeman splitting---becomes narrow in the vicinity of the transition point. The resulting hybridization of the fused Majorana states with the spectral continuum strongly affects the electron density of states at the junction and its Josephson energy. We study the manifestations of this hybridization in the energy spectrum and phase dependence of the Josephson current. We pinpoint the experimentally observable signatures of the topological phase transition, focusing on junctions with weak backscattering., Comment: 26 pages, 11 figures

Shock waves are an ubiquitous feature of hydrodynamic theories. Given that fermionic quantum many-body systems admit hydrodynamical descriptions on length scales large compared to the Fermi wavelength, it is natural to ask what the status of shock waves is in such systems. Free fermions provide a solvable yet non-trivial example, and here we generalise to include generic (non-integrable) weak interactions to understand how a shock wave decays and changes its shape well after forming., Comment: 10 pages, 4 figures

We generalize the Landauer formula to describe the dissipative electron transport through a superconducting point contact. The finite-temperature, linear-in-bias, dissipative DC conductance is expressed in terms of the phase- and energy-dependent scattering matrix of the Bogoliubov quasiparticles in the quantum point contact. The derived formula is also applicable to hybrid superconducting-normal structures and normal contacts, where it agrees with the known limits of Andreev reflection and normal-state conductance, respectively., Comment: v2: 5+6 pages, 2+1 figures; Corresponds to published version

We propose a method to probe and control the interactions between an ensemble of magnetic impurities in a superconductor via microwave radiation. Our method relies upon the presence of sub-gap Yu-Shiba-Rusinov (YSR) states associated with the impurities. Depending on the sign of the detuning, radiation generates either a ferro- or antiferromagnetic contribution to the exchange interaction. This contribution can bias the statistics of the random exchange constants stemming from the RKKY interaction. Moreover, by measuring the microwave response at the YSR resonance, one gains information about the magnetic order of the impurities. To this end, we estimate the absorption coefficient as well as the achievable strength of the microwave-induced YSR-interactions using off-resonant radiation. The ability to utilize microwave fields to both probe and control impurity spins in a superconducting host may open new paths to studying metallic spin glasses., Comment: 11 pages, 3 figures

We find the charge and heat currents caused by a temperature difference applied to a superconducting point contact or to a quantum point contact between a superconducting and normal conductors. The results are formulated in terms of the properties of the electron scattering matrix of the quantum point contact in its normal state, and are valid at any transmission coefficient. In the low-transmission limit, the new theory provides reliable results, setting the limits for the use of the popular method of tunnel Hamiltonian., Comment: 12+8 pages, 8 figures; v.3; updated text and references; corresponds to the published version

A chain of small Josephson junctions (aka superinductor) emerged recently as a high-inductance, low-loss element of superconducting quantum devices. We notice that the intrinsic parameters of a typical superinductor in fact place it into the Bose glass universality class for which the propagation of waves in a sufficiently long chain is hindered by pinning. Its weakness provides for a broad crossover from the spectrum of well-resolved plasmon standing waves at high frequencies to the low-frequency excitation spectrum of a pinned charge density wave. We relate the scattering amplitude of microwave photons reflected off a superinductor to the dynamics of a Bose glass. The dynamics at long and short scales compared to the Larkin pinning length determines the low- and high-frequency asymptotes of the reflection amplitude., Comment: 6+8 pages, 2+3 figures

We develop a unified numerical approach for modeling semiconductor-superconductor heterostructures. Our approach takes into account on equal footing important key ingredients: proximity-induced superconductivity, orbital and Zeeman effect of an applied magnetic field, spin-orbit coupling as well as the electrostatic environment. As a model system, we consider indium arsenide (InAs) nanowires with epitaxial aluminum (Al) shell and demonstrate qualitative agreement of the obtained results with the existing experimental data. Finally, we characterize the topological superconducting phase emerging in a finite magnetic field and calculate the corresponding topological phase diagram., Comment: 16 pages, 18 figures

We consider a new model system supporting Majorana zero modes based on semiconductor nanowires with a full superconducting shell. We demonstrate that, in the presence of spin-orbit coupling in the semiconductor induced by a radial electric field, the winding of the superconducting order parameter leads to a topological phase supporting Majorana zero modes. The topological phase persists over a large range of chemical potentials and can be induced by a predictable and weak magnetic field piercing the cylinder. The system can be readily realized in semiconductor nanowires covered by a full superconducting shell, opening a pathway for realizing topological quantum computing proposals., Comment: 12 pages, 9 figures

We consider the ground-state energy and the spectrum of the low-energy excitations of a Majorana island formed of topological superconductors connected by a single-mode junction of arbitrary transmission. Coulomb blockade results in $e$-periodic modulation of the energies with the gate-induced charge. We find the amplitude of modulation as a function of reflection coefficient ${\cal R}$. The amplitude scales as $\sqrt{\cal R}$ in the limit ${\cal R}\to 0$. At larger ${\cal R}$, the dependence of the amplitude on the Josephson and charging energies is similar to that of a conventional-superconductor Cooper-pair box. The crossover value of ${\cal R}$ is small and depends on the ratio of the charging energy to superconducting gap., Comment: 5+11 pages, 3+1 figures

Quasiparticles are an intrinsic source of relaxation and decoherence for superconducting qubits. Recent works have shown that normal-metal traps may be used to evacuate quasiparticles, and potentially improve the qubit life time. Here, we investigate how far the normal metals themselves may introduce qubit relaxation. We identify the ohmic losses inside the normal metal and the tunnelling current through the normal metal-superconductor interface as the relevant relaxation mechanisms. We show that the ohmic loss contribution depends strongly on the device and trap geometry, as a result of the inhomogeneous electric fields in the qubit. The correction of the quality factor due to the tunnelling current on the other hand is highly sensitive to the nonequilibrium distribution function of the quasiparticles. Overall, we show that even when choosing less than optimal parameters, the presence of normal-metal traps does not affect the quality factor of state-of-the-art qubits., Comment: 13 pages, 4 figures

In the presence of Rashba spin-orbit coupling, magnetic field can drive a proximitized nanowire into a topological superconducting phase. We study transport properties of such nanowires in the Coulomb blockade regime. The associated with the topological superconductivity Majorana modes significantly modify transport and lead to single-electron coherent transmission through the nanowire - a non-local signature of topological superconductivity. In this work, we focus on the case of strong hybridization of the Majorana modes with the normal leads. The induced by hybridization broadening of the Majorana zero-energy states competes with the charging energy, leading to a considerable modification of the Coulomb blockade in a nanowire contacted by two normal leads. We evaluate the two-terminal conductance as a function of the gate voltage, junctions transmission coefficients, the geometric capacitance of and the induced superconducting gap in the nanowire., Comment: 6 pages, 1 figure

We calculate the two-terminal current noise generated by a magnetic moment coupled to a helical edge of a two-dimensional topological insulator. When the system is symmetric with respect to in-plane spin rotation, the noise is dominated by the Nyquist component even in the presence of a voltage bias $V$. The corresponding noise spectrum $S(V,\omega)$ is determined by a modified fluctuation-dissipation theorem with the differential conductance $G(V,\omega)$ in place of the linear one. The differential noise $\partial S/ \partial V$, commonly measured in experiments, is strongly dependent on frequency on a small scale $\tau_{K}^{-1}\ll T$ set by the Korringa relaxation rate of the local moment. This is in stark contrast with the case of conventional mesoscopic conductors where $\partial S/ \partial V$ is frequency-independent and defined by the shot noise. In a helical edge, a violation of the spin-rotation symmetry leads to the shot noise, which becomes important only at a high bias. Uncharacteristically for a fermion system, this noise in the backscattered current is super-Poissonian., Comment: Improved presentation and added references, now 6+3 pages, 2 figures

We study the Josephson effect in a quantum spin Hall system coupled to a localized magnetic impurity. As a consequence of the fermion parity anomaly, the spin of the combined system of impurity and spin-Hall edge alternates between half-integer and integer values when the superconducting phase difference across the junction advances by $2\pi$. This leads to characteristic differences in the splittings of the spin multiplets by exchange coupling and single-ion anisotropy at phase differences, for which time-reserval symmetry is preserved. We discuss the resulting $8\pi$-periodic (or $\mathbb{Z}_4$) fractional Josephson effect in the context of recent experiments., Comment: 5 pages + supplemental materials, 4 figures

The superconducting proximity effect in semiconductor nanowires has recently enabled the study of new superconducting architectures, such as gate-tunable superconducting qubits and multiterminal Josephson junctions. As opposed to their metallic counterparts, the electron density in semiconductor nanosystems is tunable by external electrostatic gates providing a highly scalable and in-situ variation of the device properties. In addition, semiconductors with large $g$-factor and spin-orbit coupling have been shown to give rise to exotic phenomena in superconductivity, such as $\varphi_0$ Josephson junctions and the emergence of Majorana bound states. Here, we report microwave spectroscopy measurements that directly reveal the presence of Andreev bound states (ABS) in ballistic semiconductor channels. We show that the measured ABS spectra are the result of transport channels with gate-tunable, high transmission probabilities up to $0.9$, which is required for gate-tunable Andreev qubits and beneficial for braiding schemes of Majorana states. For the first time, we detect excitations of a spin-split pair of ABS and observe symmetry-broken ABS, a direct consequence of the spin-orbit coupling in the semiconductor., Comment: Raw data and calculations are available at http://dx.doi.org/10.4121/uuid:8c4a0604-ac00-4164-a37a-dad8b9d2f580

Motivated by recent experiment, we consider charging of a nanowire which is proximitized by a superconductor and connected to a normal-state lead by a single-channel junction. The charge $Q$ of the nanowire is controlled by gate voltage $e{\cal N}_g/C$. A finite conductance of the contact allows for quantum charge fluctuations, making the function $Q(\mathcal{N}_g)$ continuous. It depends on the relation between the superconducting gap $\Delta$ and the effective charging energy $E^*_C$. The latter is determined by the junction conductance, in addition to the geometrical capacitance of the proximitized nanowire. We investigate $Q(\mathcal{N}_g)$ at zero magnetic field $B$, and at fields exceeding the critical value $B_c$ corresponding to the topological phase transition. Unlike the case of $\Delta = 0$, the function $Q(\mathcal{N}_g)$ is analytic even in the limit of negligible level spacing in the nanowire. At $B=0$ and $\Delta>E^*_C$, the maxima of $dQ/d\mathcal{N}_g$ are smeared by $2e$-fluctuations described by a single-channel "charge Kondo" physics, while the $B=0$, $\Delta

We investigate the Josephson critical current $I_c(\Phi)$ of a wide superconductor-normal metal-superconductor (SNS) junction as a function of the magnetic flux $\Phi$ threading it. Electronic trajectories reflected from the side edges alter the function $I_c(\Phi)$ as compared to the conventional Fraunhofer-type dependence. At weak magnetic fields, $B\lesssim \Phi_0/d^2$, the edge effect lifts zeros in $I_c(\Phi)$ and gradually shifts the minima of that function toward half-integer multiples of the flux quantum. At $B>\Phi_0/d^2$, the edge effect leads to an accelerated decay of the critical current $I_c(\Phi)$ with increasing $\Phi$. At larger fields, eventually, the system is expected to cross into a regime of "classical" mesoscopic fluctuations that is specific for wide ballistic SNS junctions with rough edges., Comment: 14 pages, 8 figures

We study the effect of localized magnetic moments on the conductance of a helical edge. Interaction with a local moment is an effective backscattering mechanism for the edge electrons. We evaluate the resulting differential conductance as a function of temperature $T$ and applied bias $V$ for any value of $V/T$. Backscattering off magnetic moments, combined with the weak repulsion between the edge electrons results in a power-law temperature and voltage dependence of the conductance; the corresponding small positive exponent is indicative of insulating behavior. Local moments may naturally appear due to charge disorder in a narrow-gap semiconductor. Our results provide an alternative interpretation of the recent experiment by Li et al. \cite{Li15} where a power-law suppression of the conductance was attributed to strong electron repulsion within the edge, with the value of Luttinger liquid parameter $K$ fine-tuned close to $1/4$., Comment: Updated to match published version, now 6+8 pages

We study the effects of a phase difference on Yu-Shiba-Rusinov (YSR) states in a spinful Coulomb-blockaded quantum dot contacted by a superconducting loop. In the limit where charging energy is larger than the superconducting gap, we determine the subgap excitation spectrum, the corresponding supercurrent, and the differential conductance as measured by a normal-metal tunnel probe. In absence of a phase difference only one linear combination of the superconductor lead electrons couples to the spin, which gives a single YSR state. With finite phase difference, however, it is effectively a two-channel scattering problem and therefore an additional state emerges from the gap edge. The energy of the phase-dependent YSR states depend on the gate voltage and one state can cross zero energy twice inside the valley with odd occupancy. These crossings are shifted by the phase difference towards the charge degeneracy points, corresponding to larger exchange couplings. Moreover, the zero-energy crossings give rise to resonant peaks in the differential conductance with magnitude $4e^2/h$. Finally, we demonstrate that the quantum fluctuations of the dot spin do not alter qualitatively any of the results., Comment: 13 pages, 7 figures