Search

We show that the quasicontinuous gapless spectrum of Andreev bound states in multiterminal semi-classical superconducting nanostructures exhibits a large number of topological singularities. We concentrate on Weyl points in a four-terminal nanostructure and compute their density and correlations in three-dimensional parameter space for a universal random matrix theory model as well as for the concrete nanostructures described by the quantum circuit theory. We mention the opportunities for experimental observation of the effect in a quasicontinuous spectrum., QN/Nazarov Group

The technology of superconductor-semiconductor nanowire devices has matured in recent years. This makes it feasible to make more complex and sophisticated devices. We investigate multiterminal superconductor-semiconductor wires to access the feasibility of another topological phenomenon: Weyl singularities in their spectrum. We have found an abundance of Weyl singularities for devices with an intermediate size of the electrodes. We describe their properties and the ways the singularities emerge and disappear upon variation of the setup parameters., QN/Nazarov Group

It has been shown that a Weyl point in a superconducting nanostructure may give rise to a Weyl disk where two quantum states are almost degenerate in a two-dimensional manifold in the parametric space. This opens up the possibility of a holonomic quantum manipulation: A transformation of the wave function upon an adiabatic change of the parameters within the degenerate manifold. In this paper, we investigate in detail the opportunities for holonomic manipulation in Weyl disks. We compute the connection at the manifold in quasiclassical approximation to show it is Abelian and can be used for a phase gate. To provide a closed example of quantum manipulation that includes a state preparation and readout, we augment the holonomic gate with a change of parameters that brings the system out of the degenerate subspace. For numerical illustrations, we use a finite value of quasiclassical parameter and exact quantum dynamics. We investigate the fidelity of an example gate for different execution times. We evaluate the decoherence rate and show it can be made small to ensure a wide frequency range where an adiabatic manipulation remains coherent., QN/Nazarov Group

We propose to synchronize Bloch oscillations in a double phase-slip junction by modulating the gate voltage rather than the bias voltage. We show this is advantageous, and the relatively small ac modulation of the gate voltage gives rise to the pronounced plateaus of quantized current of the width of the order of Coulomb blockade threshold. We theoretically investigate the setup distinguishing three regimes of strong, weak, and intermediate coupling, defined by the ratio of the gate capacitance C and the effective capacitance of the phase-slip junctions. An important feature of the intermediate-coupling regime is the occurrence of the fractional plateaus of the quantized current. We investigate the finite temperature effects, finding an empirical scaling for the smoothing of integer plateaus., QN/Nazarov Group

It has been shown that a Weyl point in a superconducting nanostructure may give rise to a Weyl disk where two quantum states are almost degenerate in a two-dimensional manifold in the parametric space. This opens up the possibility of a holonomic quantum manipulation: A transformation of the wave function upon an adiabatic change of the parameters within the degenerate manifold. In this paper, we investigate in detail the opportunities for holonomic manipulation in Weyl disks. We compute the connection at the manifold in quasiclassical approximation to show it is Abelian and can be used for a phase gate. To provide a closed example of quantum manipulation that includes a state preparation and readout, we augment the holonomic gate with a change of parameters that brings the system out of the degenerate subspace. For numerical illustrations, we use a finite value of quasiclassical parameter and exact quantum dynamics. We investigate the fidelity of an example gate for different execution times. We evaluate the decoherence rate and show it can be made small to ensure a wide frequency range where an adiabatic manipulation remains coherent., QN/Nazarov Group

The technology of superconductor-semiconductor nanowire devices has matured in recent years. This makes it feasible to make more complex and sophisticated devices. We investigate multiterminal superconductor-semiconductor wires to access the feasibility of another topological phenomenon: Weyl singularities in their spectrum. We have found an abundance of Weyl singularities for devices with an intermediate size of the electrodes. We describe their properties and the ways the singularities emerge and disappear upon variation of the setup parameters., QN/Nazarov Group

A Weyl point in a superconducting nanostructure is a generic minimum model of a topological singularity at low energies. We connect the nanostructure to normal leads thereby immersing the topological singularity in the continuous spectrum of the electron states in the leads. This sets another simple and generic model useful to comprehend the modification of low-energy singularity in the presence of a continuous spectrum. The tunnel coupling to the leads gives rise to new low-energy scale Γ at which all topological features are smoothed. We investigate superconducting and normal currents in the nanostructure at this scale. We show how the tunnel currents can be used for detection of the Weyl point. Importantly, we find that the topological charge is not concentrated in a point but rather is spread over the parameter space in the vicinity of the point. We introduce and compute the resulting topological charge density. We also reveal that the pumping to the normal leads helps to detect and investigate the topological effects in the vicinity of the point., QN/Nazarov Group

We investigate transport in a superconducting nanostructure housing a Weyl point in the spectrum of Andreev bound states. A minimum magnet state is realized in the vicinity of the point. One or more normal-metal leads are tunnel-coupled to the nanostructure. We have shown that this minimum magnetic setup is suitable for realization of all common goals of spintronics: detection of a magnetic state, conversion of electric currents into spin currents, potentially reaching the absolute limit of one spin per charge transferred, and detection of spin accumulation in the leads. The peculiarity and possible advantage of the setup is the ability to switch between magnetic and nonmagnetic states by tiny changes of the control parameters: superconducting phase differences. We employ this property to demonstrate the feasibility of less common spintronic effects: spin on demand and alternative spin current., QN/Nazarov Group

We propose a scheme to perform braiding and all other unitary operations with Majorana modes in one dimension that, in contrast to previous proposals, is solely based on resonant manipulation involving the first excited state extended over the modes. The detection of the population of the excited state also enables initialization and read-out. We provide an elaborated illustration of the scheme with a concrete device., QN/Nazarov Group

We investigate transport in a superconducting nanostructure housing a Weyl point in the spectrum of Andreev bound states. A minimum magnet state is realized in the vicinity of the point. One or more normal-metal leads are tunnel-coupled to the nanostructure. We have shown that this minimum magnetic setup is suitable for realization of all common goals of spintronics: detection of a magnetic state, conversion of electric currents into spin currents, potentially reaching the absolute limit of one spin per charge transferred, and detection of spin accumulation in the leads. The peculiarity and possible advantage of the setup is the ability to switch between magnetic and nonmagnetic states by tiny changes of the control parameters: superconducting phase differences. We employ this property to demonstrate the feasibility of less common spintronic effects: spin on demand and alternative spin current., QN/Nazarov Group

We propose a scheme to perform braiding and all other unitary operations with Majorana modes in one dimension that, in contrast to previous proposals, is solely based on resonant manipulation involving the first excited state extended over the modes. The detection of the population of the excited state also enables initialization and read-out. We provide an elaborated illustration of the scheme with a concrete device., QN/Nazarov Group

The recent proposals of devices with overlapping Andreev bound states (ABS) open up opportunities to control and fine tune their spectrum that can be used in various applications in quantum sensing and manipulation. In this paper, we study the ABS in a device consisting of a semiconducting nanowire covered with three superconducting leads. The ABS are formed at two junctions where the wire is not covered. They overlap in the wire where the electron propagation is 1D and in one of the leads where the propagation is 3D. We identify a number of regimes where these two overlaps either dominate or compete, depending on the junction separation L as compared to the correlation lengths ζw,ζs in the wire and in the lead, respectively. We utilize a simple model of 1D electron spectrum in the nanowire and take into account the quality of the contact between the nanowire and the superconducting lead. We present the spectra for different L, detailing the transition from a single ABS in the regime of strong 1D hybridization to two almost independent ABS hybridized at the degeneracy points, in the regime of weak 1D hybridization. We present the details of merging the upper ABS with the continuous spectrum upon decreasing L. We study in detail the effect of quantum interference due to the phase accumulated during the electron passage between the junctions. We develop a perturbation theory for analytical treatment of hybridization. We address an interesting separate case of fully transparent junctions. We derive and exemplify a perturbation theory suitable for the competition regime demonstrating the interference of 1D and two 3D transmission amplitudes., QN/Nazarov Group

We show that the annihilation dynamics of excess quasiparticles in superconductors may result in the spontaneous formation of large spin-polarized clusters. This presents a novel scenario for spontaneous spin polarization. We estimate the relevant scales for aluminum, finding the feasibility of clusters with total spin S≃10^{4}ℏ that could be spread over microns. The fluctuation dynamics of such large spins may be detected by measuring the flux noise in a loop hosting a cluster., QN/Nazarov Group

We show that the annihilation dynamics of excess quasiparticles in superconductors may result in the spontaneous formation of large spin-polarized clusters. This presents a novel scenario for spontaneous spin polarization. We estimate the relevant scales for aluminum, finding the feasibility of clusters with total spin S≃10^{4}ℏ that could be spread over microns. The fluctuation dynamics of such large spins may be detected by measuring the flux noise in a loop hosting a cluster., QN/Nazarov Group

We address the interaction between two quantum systems (A and B) that is mediated by their common linear environment. If the environment is out of equilibrium, the resulting interaction violates Onsager relations and cannot be described by a Hamiltonian. In simple terms, the action of system A on system B does not necessarily produce a back action. We derive general quantum equations describing the situation, and we analyze in detail their classical correspondence. Changing the properties of the environment, one can easily change and engineer the resulting interaction. It is tempting to use this for quantum manipulation of the systems. However, the resulting quantum gate is not always unitary and may induce a loss of quantum coherence. For a relevant example, we consider systems A and B to be spins of arbitrary values and arrange the interaction to realize an analog of the two-qubit cnot gate. The direction of spin A controls the rotation of spin B while spin A is not rotated experiencing no back action from spin B. We solve the quantum dynamics equations and analyze the purity of the resulting density matrix. The resulting purity essentially depends on the initial states of the systems. We attempt to find a universal characteristic of the purity, optimizing it for the worst choice of initial states. For both spins sA=sB=1/2, the optimized purity is bounded by 1/2 irrespective of the details of the gate. We also study in detail the semiclassical limit of large spins. In this case, the optimized purity is bounded by (1+π/2)-1≈0.39. This is much better than the typical purity of a large spin state ∼s-1. We conclude that although the quantum manipulation without back action inevitably causes decoherence of the quantum states, the actual purity of the resulting state can be optimized and made relatively high., QuTech, QCD/Vandersypen Lab, QN/Nazarov Group

We address the statistics of a simultaneous continuous weak linear measurement of two noncommuting variables on a few-state quantum system subject to a postselected evolution. The results of both postselected quantum measurement and simultaneous monitoring of two noncommuting variables differ drastically from the results of either classical or quantum projective measurement. We explore the peculiarities arising from the combination of the two. We concentrate on the distribution function of two measurement outcomes integrated over a time interval. We formulate a proper formalism for the evaluation of such distribution, and further compute and discuss the resulting statistics for idealized and experimentally relevant setups. We demonstrate the visibility and manifestations of the interference between initial and final states in the statistics of measurement outcomes for both variables in various regimes. We analytically predict the peculiarities at the circle O12+O22=1 in the distribution of measurement outcomes O1,2 in the limit of short measurement times and confirm this by numerical calculation at longer measurement times. We demonstrate analytically the anomalously large values of the time-integrated output cumulants in the limit of short measurement times and zero overlap between initial and final states, and give the detailed distributions for this case. We term this situation sudden jump. We present the numerical evaluation of the probability distributions for experimentally relevant parameters in several regimes and demonstrate that interference effects in the postselected measurement can be accurately predicted even if they are small., QN/Nazarov Group

Recently, we have proposed an unusual mechanism of superconducting current that is specific for quantum Hall edge channels connected to superconducting electrodes. We have shown that the supercurrent can be mediated by a nonlocal electron-electron interaction that provides an opportunity for a long-distance information transfer in the direction opposite to the electron flow. A convenient model for such interaction is that of an external circuit. The consideration has been performed for the case of a single channel. In order to facilitate the experimental verification and the observation of peculiar features of the effect, in this paper, we provide a more detailed description of the phenomenon and extend the results to more sophisticated setups. We establish that the dynamical phase contributes to superconducting interference; this being the manifestation of the channel chirality. We consider setups that include the scattering between quantum Hall channels of opposite direction and multiple superconducting contacts. For a single quantum Hall constriction, we derive a general and comprehensive relation for the interaction-induced supercurrent in terms of scattering amplitudes and demonstrate the nonlocal nature of the current by considering its sensitivity to scattering. In multiterminal setups, we reveal the characteristic phase dependences of the supercurrents explaining those in terms of interference of Andreev reflection processes. For more complex setups encompassing, at least, two constrictions, we find an interplay between noninteracting and interaction-induced currents and contributions of more complex interference processes., QN/Nazarov Group

Currently, 'time' does not play any essential role in quantum information theory. In this sense, quantum information theory is underdeveloped similarly to how quantum physics was underdeveloped before Erwin Schrödinger introduced his famous equation for the evolution of a quantum wave function. In this review article, we cope with the problem of time for one of the central quantities in quantum information theory: entropy. Recently, a replica trick formalism, the so-called 'multiple parallel world' formalism, has been proposed that revolutionizes entropy evaluation for quantum systems. This formalism is one of the first attempts to introduce 'time' in quantum information theory. With the total entropy being conserved in a closed system, entropy can flow internally between subsystems; however, we show that this flow is not limited only to physical correlations as the literature suggest. The nonlinear dependence of entropy on the density matrix introduces new types of correlations with no analogue in physical quantities. Evolving a number of replicas simultaneously makes it possible for them to exchange particles between different replicas. We will summarize some of the recent news about entropy in some example quantum devices. Moreover, we take a quick look at a new correspondence that was recently proposed that provides an interesting link between quantum information theory and quantum physics. The mere existence of such a correspondence allows for exploring new physical phenomena as the result of controlling entanglement in a quantum device., QN/Nazarov Group

We show theoretically that in the superconducting nanostructures the gapped states of different topology are not always protected by separating gapless states. Depending on the structure design parameters, they can be either protected or not, with a protection-unprotection transition separating these two distinct situations. We build up a general theoretical description of the transition vicinity in the spirit of Landau theory. We speculate that similar protection-unprotection transitions may also occur for other realizations of topological protection in condensed matter systems., QN/Nazarov Group

Recently, it has been shown that multiterminal superconducting nanostructures may possess topological properties that involve Berry curvatures in the parametric space of the superconducting phases of the terminals, and associated Chern numbers that are manifested in quantized transconductances of the nanostructure. In this paper, we investigate how the continuous spectrum that is intrinsically present in superconductors, affects these properties. We model the nanostructure within scattering formalism deriving the action and the response function that permits a redefinition of Berry curvature for continuous spectrum. We have found that the redefined Berry curvature may have a nontopological phase-independent contribution that adds a nonquantized part to the transconductances. This contribution vanishes for a time-reversible scattering matrix. We have found compact expressions for the redefined Berry curvature for the cases of weak energy dependence of the scattering matrix and investigated the vicinity of Weyl singularities in the spectrum., QN/Nazarov Group

The recent proposals of experiments with single Andreev bound states make relevant a detailed analysis of these states in multiterminal superconducting nanostructures. We evaluate the energy splitting of degenerate Andreev bound states that overlap in a superconducting lead, and find that the splitting is reduced in comparison with their energy by a small factor RGQ, with RGQ being the dimensionless resistance of the overlap region in the normal state. This permits quantum manipulation of the quasiparticles in these states. We provide a simple scheme of such manipulation., QN/Nazarov Group

We address the statistics of a simultaneous continuous weak linear measurement of two noncommuting variables on a few-state quantum system subject to a postselected evolution. The results of both postselected quantum measurement and simultaneous monitoring of two noncommuting variables differ drastically from the results of either classical or quantum projective measurement. We explore the peculiarities arising from the combination of the two. We concentrate on the distribution function of two measurement outcomes integrated over a time interval. We formulate a proper formalism for the evaluation of such distribution, and further compute and discuss the resulting statistics for idealized and experimentally relevant setups. We demonstrate the visibility and manifestations of the interference between initial and final states in the statistics of measurement outcomes for both variables in various regimes. We analytically predict the peculiarities at the circle O12+O22=1 in the distribution of measurement outcomes O1,2 in the limit of short measurement times and confirm this by numerical calculation at longer measurement times. We demonstrate analytically the anomalously large values of the time-integrated output cumulants in the limit of short measurement times and zero overlap between initial and final states, and give the detailed distributions for this case. We term this situation sudden jump. We present the numerical evaluation of the probability distributions for experimentally relevant parameters in several regimes and demonstrate that interference effects in the postselected measurement can be accurately predicted even if they are small., QN/Nazarov Group

Recently, we have proposed an unusual mechanism of superconducting current that is specific for quantum Hall edge channels connected to superconducting electrodes. We have shown that the supercurrent can be mediated by a nonlocal electron-electron interaction that provides an opportunity for a long-distance information transfer in the direction opposite to the electron flow. A convenient model for such interaction is that of an external circuit. The consideration has been performed for the case of a single channel. In order to facilitate the experimental verification and the observation of peculiar features of the effect, in this paper, we provide a more detailed description of the phenomenon and extend the results to more sophisticated setups. We establish that the dynamical phase contributes to superconducting interference; this being the manifestation of the channel chirality. We consider setups that include the scattering between quantum Hall channels of opposite direction and multiple superconducting contacts. For a single quantum Hall constriction, we derive a general and comprehensive relation for the interaction-induced supercurrent in terms of scattering amplitudes and demonstrate the nonlocal nature of the current by considering its sensitivity to scattering. In multiterminal setups, we reveal the characteristic phase dependences of the supercurrents explaining those in terms of interference of Andreev reflection processes. For more complex setups encompassing, at least, two constrictions, we find an interplay between noninteracting and interaction-induced currents and contributions of more complex interference processes., QN/Nazarov Group

Currently, 'time' does not play any essential role in quantum information theory. In this sense, quantum information theory is underdeveloped similarly to how quantum physics was underdeveloped before Erwin Schrödinger introduced his famous equation for the evolution of a quantum wave function. In this review article, we cope with the problem of time for one of the central quantities in quantum information theory: entropy. Recently, a replica trick formalism, the so-called 'multiple parallel world' formalism, has been proposed that revolutionizes entropy evaluation for quantum systems. This formalism is one of the first attempts to introduce 'time' in quantum information theory. With the total entropy being conserved in a closed system, entropy can flow internally between subsystems; however, we show that this flow is not limited only to physical correlations as the literature suggest. The nonlinear dependence of entropy on the density matrix introduces new types of correlations with no analogue in physical quantities. Evolving a number of replicas simultaneously makes it possible for them to exchange particles between different replicas. We will summarize some of the recent news about entropy in some example quantum devices. Moreover, we take a quick look at a new correspondence that was recently proposed that provides an interesting link between quantum information theory and quantum physics. The mere existence of such a correspondence allows for exploring new physical phenomena as the result of controlling entanglement in a quantum device., QN/Nazarov Group

We show theoretically that in the superconducting nanostructures the gapped states of different topology are not always protected by separating gapless states. Depending on the structure design parameters, they can be either protected or not, with a protection-unprotection transition separating these two distinct situations. We build up a general theoretical description of the transition vicinity in the spirit of Landau theory. We speculate that similar protection-unprotection transitions may also occur for other realizations of topological protection in condensed matter systems., QN/Nazarov Group

Recently, it has been shown that multiterminal superconducting nanostructures may possess topological properties that involve Berry curvatures in the parametric space of the superconducting phases of the terminals, and associated Chern numbers that are manifested in quantized transconductances of the nanostructure. In this paper, we investigate how the continuous spectrum that is intrinsically present in superconductors, affects these properties. We model the nanostructure within scattering formalism deriving the action and the response function that permits a redefinition of Berry curvature for continuous spectrum. We have found that the redefined Berry curvature may have a nontopological phase-independent contribution that adds a nonquantized part to the transconductances. This contribution vanishes for a time-reversible scattering matrix. We have found compact expressions for the redefined Berry curvature for the cases of weak energy dependence of the scattering matrix and investigated the vicinity of Weyl singularities in the spectrum., QN/Nazarov Group

A variety of quantum systems exhibit Weyl points in their spectra where two bands cross in a point of three-dimensional parameter space with conical dispersion in the vicinity of the point. We consider theoretically the soft constraint regime where the parameters are dynamical quantum variables. We have shown that in general the soft constraints, in the quasiclassical limit, result in Weyl disks where two states are (almost) degenerate in a finite two-dimensional region of the three-dimensional parameter space. We provide concrete calculations for two setups: Weyl point in a four-terminal superconducting structure and a Weyl exciton, i.e., a bound state of Weyl electron and a massive hole., QN/Nazarov Group

A variety of quantum systems exhibit Weyl points in their spectra where two bands cross in a point of three-dimensional parameter space with conical dispersion in the vicinity of the point. We consider theoretically the soft constraint regime where the parameters are dynamical quantum variables. We have shown that in general the soft constraints, in the quasiclassical limit, result in Weyl disks where two states are (almost) degenerate in a finite two-dimensional region of the three-dimensional parameter space. We provide concrete calculations for two setups: Weyl point in a four-terminal superconducting structure and a Weyl exciton, i.e., a bound state of Weyl electron and a massive hole., QN/Nazarov Group

We study the Zeeman splitting in lateral quantum dots that are defined in GaAs-AlGaAs heterostructures by means of split gates. We demonstrate a nonlinear dependence of the splitting on magnetic field and its substantial variations from dot to dot and from heterostructure to heterostructure. These phenomena are important in the context of information processing since the tunability and dot-dependence of the Zeeman splitting allow for a selective manipulation of spins. We show that spin-orbit effects related to the GaAs band structure quantitatively explain the observed magnitude of the nonlinear dependence of the Zeeman splitting. Furthermore, spin-orbit effects result in a dependence of the Zeeman splitting on predominantly the out-of-plane quantum dot confinement energy. We also show that the variations of the confinement energy due to charge disorder in the heterostructure may explain the dependence of Zeeman splitting on the dot position. This position may be varied by changing the gate voltages, which leads to an electrically tunable Zeeman splitting., QCD/Vandersypen Lab, QN/Nazarov Group

We present an exhaustive theoretical analysis of a double-loop Josephson proximity interferometer, such as the one recently realized by Strambini et al. for control of the Andreev spectrum via an external magnetic field. This system, called ω-SQUIPT, consists of a T-shaped diffusive normal metal (N) attached to three superconductors (S) forming a double-loop configuration. By using the quasiclassical Green-function formalism, we calculate the local normalized density of states, the Josephson currents through the device, and the dependence of the former on the length of the junction arms, the applied magnetic field, and the S/N interface transparencies. We show that by tuning the fluxes through the double loop, the system undergoes transitions from a gapped to a gapless state. We also evaluate the Josephson currents flowing in the different arms as a function of magnetic fluxes, and we explore the quasiparticle transport by considering a metallic probe tunnel-coupled to the Josephson junction and calculating its I-V characteristics. Finally, we study the performances of the ω-SQUIPT and its potential applications by investigating its electrical and magnetometric properties., QN/Nazarov Group

Recently we predicted that the Andreev bound-state spectrum of four-terminal Josephson junctions may possess topologically protected zero-energy Weyl singularities, which manifest themselves in a quantized transconductance in units of 4e2/h when two of the terminals are voltage biased [R.-P. Riwar, M. Houzet, J. S. Meyer, and Y. V. Nazarov, Nature Commun. 7, 11167 (2016)2041-172310.1038/ncomms11167]. Here, using the Landauer-Büttiker scattering theory, we compute numerically the currents flowing through such a structure in order to assess the conditions for observing this effect. We show that the voltage below which the transconductance becomes quantized is determined by the interplay of nonadiabatic transitions between Andreev bound states and inelastic relaxation processes. We demonstrate that the topological quantization of the transconductance can be observed at voltages of the order of 10-2Δ/e,Δ being the the superconducting gap in the leads., QN/Nazarov Group

It has been thought that the long chiral edge channels cannot support any supercurrent between the superconducting electrodes. We show theoretically that the supercurrent can be mediated by a nonlocal interaction that facilitates a long-distance information transfer in the direction opposite of electron flow. We compute the supercurrent for several interaction models, including that of an external circuit., QN/Nazarov Group

We address the statistics of continuous weak linear measurement on a few-state quantum system that is subject to a conditioned quantum evolution. For a conditioned evolution, both the initial and final states of the system are fixed: the latter is achieved by the postselection in the end of the evolution. The statistics may drastically differ from the nonconditioned case, and the interference between initial and final states can be observed in the probability distributions of measurement outcomes as well as in the average values exceeding the conventional range of nonconditioned averages. We develop a proper formalism to compute the distributions of measurement outcomes, and evaluate and discuss the distributions in experimentally relevant setups. We demonstrate the manifestations of the interference between initial and final states in various regimes. We consider analytically simple examples of nontrivial probability distributions. We reveal peaks (or dips) at half-quantized values of the measurement outputs. We discuss in detail the case of zero overlap between initial and final states demonstrating anomalously big average outputs and sudden jump in time-integrated output. We present and discuss the numerical evaluation of the probability distribution aiming at extending the analytical results and describing a realistic experimental situation of a qubit in the regime of resonant fluorescence., QN/Nazarov Group

We consider the spectrum of Andreev bound states (ABSs) in an exemplary four-terminal superconducting structure where four chaotic cavities are connected by quantum point contacts to the terminals and to each other forming a ring. We nickname the resulting device 4T-ring. Such a tunable device can be realized in a 2D electron gas-superconductor or a graphene-based hybrid structure. We concentrate on the limit of a short structure and large conductance of the point contacts where there are many ABS in the device forming a quasicontinuous spectrum. The energies of the ABS can be tuned by changing the superconducting phases of the terminals. We observe the opening and closing of gaps in the spectrum upon changing the phases. This concerns the usual proximity gap that separates the levels from zero energy as well as less usual "smile" gaps that split the levels of the quasicontinuous spectrum. We demonstrate a remarkable crossover in the overall spectrum that occurs upon changing the ratio of conductances of the inner and outer point contacts. At big values of the ratio (closed limit), the levels exhibit a generic behavior expected for the spectrum of a disordered system manifesting level repulsion and Brownian "motion" upon changing the phases. At small values of the ratio (open limit), the levels are squeezed into narrow bunches separated by wide smile gaps. Each bunch consists of almost degenerate ABS formed by Andreev reflection between two adjacent terminals. We study in detail the properties of the spectrum in the limit of a small ratio, paying special attention to the crossings of bunches. We distinguish two types of crossings: (i) with a regular phase dependence of the levels and (ii) crossings where the Brownian motion of the levels leads to an apparently irregular phase dependence. We work out a perturbation theory that explains the observations both at a detailed level of random scattering in the device and at a phenomenological level of positively defined, QN/Theoretical Physics, QN/Nazarov Group

We consider the spectrum of Andreev bound states (ABSs) in an exemplary four-terminal superconducting structure where four chaotic cavities are connected by quantum point contacts to the terminals and to each other forming a ring. We nickname the resulting device 4T-ring. Such a tunable device can be realized in a 2D electron gas-superconductor or a graphene-based hybrid structure. We concentrate on the limit of a short structure and large conductance of the point contacts where there are many ABS in the device forming a quasicontinuous spectrum. The energies of the ABS can be tuned by changing the superconducting phases of the terminals. We observe the opening and closing of gaps in the spectrum upon changing the phases. This concerns the usual proximity gap that separates the levels from zero energy as well as less usual "smile" gaps that split the levels of the quasicontinuous spectrum. We demonstrate a remarkable crossover in the overall spectrum that occurs upon changing the ratio of conductances of the inner and outer point contacts. At big values of the ratio (closed limit), the levels exhibit a generic behavior expected for the spectrum of a disordered system manifesting level repulsion and Brownian "motion" upon changing the phases. At small values of the ratio (open limit), the levels are squeezed into narrow bunches separated by wide smile gaps. Each bunch consists of almost degenerate ABS formed by Andreev reflection between two adjacent terminals. We study in detail the properties of the spectrum in the limit of a small ratio, paying special attention to the crossings of bunches. We distinguish two types of crossings: (i) with a regular phase dependence of the levels and (ii) crossings where the Brownian motion of the levels leads to an apparently irregular phase dependence. We work out a perturbation theory that explains the observations both at a detailed level of random scattering in the device and at a phenomenological level of positively defined, QN/Theoretical Physics, QN/Nazarov Group

It has been thought that the long chiral edge channels cannot support any supercurrent between the superconducting electrodes. We show theoretically that the supercurrent can be mediated by a nonlocal interaction that facilitates a long-distance information transfer in the direction opposite of electron flow. We compute the supercurrent for several interaction models, including that of an external circuit., QN/Nazarov Group

We address the statistics of continuous weak linear measurement on a few-state quantum system that is subject to a conditioned quantum evolution. For a conditioned evolution, both the initial and final states of the system are fixed: the latter is achieved by the postselection in the end of the evolution. The statistics may drastically differ from the nonconditioned case, and the interference between initial and final states can be observed in the probability distributions of measurement outcomes as well as in the average values exceeding the conventional range of nonconditioned averages. We develop a proper formalism to compute the distributions of measurement outcomes, and evaluate and discuss the distributions in experimentally relevant setups. We demonstrate the manifestations of the interference between initial and final states in various regimes. We consider analytically simple examples of nontrivial probability distributions. We reveal peaks (or dips) at half-quantized values of the measurement outputs. We discuss in detail the case of zero overlap between initial and final states demonstrating anomalously big average outputs and sudden jump in time-integrated output. We present and discuss the numerical evaluation of the probability distribution aiming at extending the analytical results and describing a realistic experimental situation of a qubit in the regime of resonant fluorescence., QN/Nazarov Group

Topological materials and their unusual transport properties are now at the focus of modern experimental and theoretical research. Their topological properties arise from the bandstructure determined by the atomic composition of a material and as such are difficult to tune and naturally restricted to ≤3 dimensions. Here we demonstrate that n-terminal Josephson junctions with conventional superconductors may provide novel realizations of topology in n-1 dimensions, which have similarities, but also marked differences with existing 2D or 3D topological materials. For n≥4, the Andreev subgap spectrum of the junction can accommodate Weyl singularities in the space of the n-1 independent superconducting phases, which play the role of bandstructure quasimomenta. The presence of these Weyl singularities enables topological transitions that are manifested experimentally as changes of the quantized transconductance between two voltage-biased leads, the quantization unit being 4e2/h, where e is the electric charge and h is the Planck constant., QN/Nazarov Group

Experimentally, the concentration of quasiparticles in gapped superconductors always largely exceeds the equilibrium one at low temperatures. Since these quasiparticles are detrimental for many applications, it is important to understand theoretically the origin of the excess. We demonstrate in detail that the dynamics of quasiparticles localized at spatial fluctuations of the gap edge becomes exponentially slow. This gives rise to the observed excess in the presence of a vanishingly weak nonequilibrium agent., QN/Nazarov Group

We study the density of states in disordered s-wave superconductors with a small gap anisotropy. We consider disorder in the form of common nonmagnetic scatterers and pairing-potential impurities, which interact with electrons via an electric potential and a local distortion of the superconducting gap. Using quasiclassical Green functions, we determine the bound-state spectrum at a single impurity and the density of states at a finite concentration of impurities. We show that, if the gap is isotropic, an isolated impurity with suppressed pairing supports an infinite number of Andreev states. With growing impurity concentration, the energy-dependent density of states evolves from a sharp gap edge with an impurity band below it to a smeared BCS singularity in the so-called universal limit. Within one spin sector, pairing-potential impurities and weak spin-polarized magnetic impurities have essentially the same effect on the density of states. We note that, if a gap anisotropy is present, the density of states becomes sensitive to ordinary potential disorder, and the existence of Andreev states localized at pairing-potential impurities requires special conditions. An unusual feature related to the anisotropy is a nonmonotonic dependence of the gap edge smearing on impurity concentration., QN/Nazarov Group

Topological materials and their unusual transport properties are now at the focus of modern experimental and theoretical research. Their topological properties arise from the bandstructure determined by the atomic composition of a material and as such are difficult to tune and naturally restricted to ≤3 dimensions. Here we demonstrate that n-terminal Josephson junctions with conventional superconductors may provide novel realizations of topology in n-1 dimensions, which have similarities, but also marked differences with existing 2D or 3D topological materials. For n≥4, the Andreev subgap spectrum of the junction can accommodate Weyl singularities in the space of the n-1 independent superconducting phases, which play the role of bandstructure quasimomenta. The presence of these Weyl singularities enables topological transitions that are manifested experimentally as changes of the quantized transconductance between two voltage-biased leads, the quantization unit being 4e2/h, where e is the electric charge and h is the Planck constant., QN/Nazarov Group

We study the density of states in disordered s-wave superconductors with a small gap anisotropy. We consider disorder in the form of common nonmagnetic scatterers and pairing-potential impurities, which interact with electrons via an electric potential and a local distortion of the superconducting gap. Using quasiclassical Green functions, we determine the bound-state spectrum at a single impurity and the density of states at a finite concentration of impurities. We show that, if the gap is isotropic, an isolated impurity with suppressed pairing supports an infinite number of Andreev states. With growing impurity concentration, the energy-dependent density of states evolves from a sharp gap edge with an impurity band below it to a smeared BCS singularity in the so-called universal limit. Within one spin sector, pairing-potential impurities and weak spin-polarized magnetic impurities have essentially the same effect on the density of states. We note that, if a gap anisotropy is present, the density of states becomes sensitive to ordinary potential disorder, and the existence of Andreev states localized at pairing-potential impurities requires special conditions. An unusual feature related to the anisotropy is a nonmonotonic dependence of the gap edge smearing on impurity concentration., QN/Nazarov Group

Experimentally, the concentration of quasiparticles in gapped superconductors always largely exceeds the equilibrium one at low temperatures. Since these quasiparticles are detrimental for many applications, it is important to understand theoretically the origin of the excess. We demonstrate in detail that the dynamics of quasiparticles localized at spatial fluctuations of the gap edge becomes exponentially slow. This gives rise to the observed excess in the presence of a vanishingly weak nonequilibrium agent., QN/Nazarov Group

We describe the proximity effect in a short disordered metallic junction between three superconducting leads. Andreev bound states in the multiterminal junction may cross the Fermi level. We reveal that for a quasicontinuous metallic density of states, crossings at the Fermi level manifest as a closing of the proximity-induced gap. We calculate the local density of states for a wide range of transport parameters using quantum circuit theory. The gap closes inside an area of the space spanned by the superconducting phase differences. We derive an approximate analytic expression for the boundary of the area and compare it to the full numerical solution. The size of the area increases with the transparency of the junction and is sensitive to asymmetry. The finite density of states at zero energy is unaffected by the electron-hole decoherence present in the junction, although decoherence is important at higher energies. Our predictions can be tested using tunneling transport spectroscopy. To encourage experiments, we calculate the current-voltage characteristic in a typical measurement setup. We show how the structure of the local density of states can be mapped out from the measurement., QN/Quantum Nanoscience, Applied Sciences

We present a universal relation between the flow of a Renyi entropy and the full counting statistics of energy transfers. We prove the exact relation for a flow to a system in thermal equilibrium that is weakly coupled to an arbitrary time-dependent and nonequilibrium system. The exact correspondence, given by this relation, provides a simple protocol to quantify the flows of Shannon and Renyi entropies from the measurements of energy transfer statistics., QN/Quantum Nanoscience, Applied Sciences

We study a short multichannel superconducting junction subject to dc and ac phase biases. The ac modulation changes the occupation of the Andreev bound states formed at the constriction by transitions between bound states and the continuum. In a short junction, the nonequilibrium Andreev bound-state population may relax through processes that conserve parity of the occupation number on the same bound state and processes that do not conserve it. We argue that the parity-conserving processes occur on a much faster time scale. In this case, even a weak driving may lead to a large nonequilibrium quasiparticle population scaling with the number of channels and results in a large deviation of the supercurrent from its equilibrium value. We show that this effect is accompanied by a quasiparticle current which may lead to a measurable charge imbalance in the vicinity of the junction. Furthermore, we study the time evolution of the supercurrent after switching off the ac drive. On a time scale where parity relaxation is negligible, the supercurrent relaxes to a stationary nonequilibrium state. Finally, we briefly outline the regime of ultraweak driving where the ac-induced processes occur on a time scale comparable to that of parity relaxation., QN/Quantum Nanoscience, Applied Sciences

We evaluate Rényi entropy flows from generic quantum heat engines (QHE) to a weakly coupled probe environment kept in thermal equilibrium. We show that the flows are determined not only by heat flow but also by a quantum coherent flow that can be separately measured in experiment apart from the heat flow measurement. The same pertains to Shannon entropy flow. This appeals for a revision of the concept of entropy flows in quantum nonequlibrium thermodynamics., QN/Quantum Nanoscience, Applied Sciences

We describe the proximity effect in a short disordered metallic junction between three superconducting leads. Andreev bound states in the multiterminal junction may cross the Fermi level. We reveal that for a quasicontinuous metallic density of states, crossings at the Fermi level manifest as a closing of the proximity-induced gap. We calculate the local density of states for a wide range of transport parameters using quantum circuit theory. The gap closes inside an area of the space spanned by the superconducting phase differences. We derive an approximate analytic expression for the boundary of the area and compare it to the full numerical solution. The size of the area increases with the transparency of the junction and is sensitive to asymmetry. The finite density of states at zero energy is unaffected by the electron-hole decoherence present in the junction, although decoherence is important at higher energies. Our predictions can be tested using tunneling transport spectroscopy. To encourage experiments, we calculate the current-voltage characteristic in a typical measurement setup. We show how the structure of the local density of states can be mapped out from the measurement., QN/Quantum Nanoscience, Applied Sciences

We present a universal relation between the flow of a Renyi entropy and the full counting statistics of energy transfers. We prove the exact relation for a flow to a system in thermal equilibrium that is weakly coupled to an arbitrary time-dependent and nonequilibrium system. The exact correspondence, given by this relation, provides a simple protocol to quantify the flows of Shannon and Renyi entropies from the measurements of energy transfer statistics., QN/Quantum Nanoscience, Applied Sciences

We study a short multichannel superconducting junction subject to dc and ac phase biases. The ac modulation changes the occupation of the Andreev bound states formed at the constriction by transitions between bound states and the continuum. In a short junction, the nonequilibrium Andreev bound-state population may relax through processes that conserve parity of the occupation number on the same bound state and processes that do not conserve it. We argue that the parity-conserving processes occur on a much faster time scale. In this case, even a weak driving may lead to a large nonequilibrium quasiparticle population scaling with the number of channels and results in a large deviation of the supercurrent from its equilibrium value. We show that this effect is accompanied by a quasiparticle current which may lead to a measurable charge imbalance in the vicinity of the junction. Furthermore, we study the time evolution of the supercurrent after switching off the ac drive. On a time scale where parity relaxation is negligible, the supercurrent relaxes to a stationary nonequilibrium state. Finally, we briefly outline the regime of ultraweak driving where the ac-induced processes occur on a time scale comparable to that of parity relaxation., QN/Quantum Nanoscience, Applied Sciences