Superconducting Phase Difference Research Articles

Practical Guide to Quantum Phase Transitions in Quantum-Dot-Based Tunable Josephson Junctions

Quantum dots attached to BCS superconducting leads exhibit a $0-\pi$ impurity quantum phase transition, which can be experimentally controlled either by the gate voltage or by the superconducting phase difference. For the pertinent superconducting single-impurity Anderson model, we newly present two simple analytical formulae describing the position of the phase boundary in parameter space for the weakly correlated and Kondo regime, respectively. Furthermore, we show that the two-level approximation provides an excellent description of the low temperature physics of superconducting quantum dots near the phase transition. We discuss reliability and mutual agreement of available finite temperature numerical methods (Numerical Renormalization Group and Quantum Monte Carlo) and suggest a novel approach for efficient determination of the quantum phase boundary from measured finite temperature data. Our results enable fast and efficient, yet reliable characterization and design of such nanoscopic tunable Josephson junction devices.

Transient dynamics of a quantum dot embedded between two superconducting leads and a metallic reservoir

Sub-gap transport properties of a quantum dot (QD) coupled to two superconducting and one metallic leads are studied theoretically, solving the time-dependent equation of motion by the Laplace transform technique. We focus on time-dependent response of the system induced by a sudden switching on the QD-leads couplings, studying the influence of initial conditions on the transient currents and the differential conductance. We derive analytical expressions for measurable quantities and find that they oscillate in time with the frequency governed by the QD-superconducting lead coupling and acquire damping, due to relaxation driven by the normal lead. Period of these oscillations increases with the superconducting phase difference $\phi$. In particular, for $\phi=\pi$ the QD occupancy and the normal current evolve monotonically (without any oscillations) to their stationary values. In such case the induced electron pairing vanishes and the superconducting current is completely blocked. We also analyze time-dependent development of the Andreev bound states. We show, that the measurable conductance peaks do not appear immediately after sudden switching of the QD coupling to external leads but it takes some finite time-interval for the system needs create these Andreev states. Such time-delay is mainly controlled by the QD-normal lead coupling.

Topological supercurrents interaction and fluctuations in the multiterminal Josephson effect

We study the Josephson effect in the multiterminal junction of topological superconductors. We use the symmetry-constrained scattering matrix approach to derive band dispersions of emergent sub-gap Andreev bound states in a multidimensional parameter space of superconducting phase differences. We find distinct topologically protected band crossings that serve as monopoles of finite Berry curvature. Particularly, in a four-terminal junction the admixture of $2\pi$ and $4\pi$ periodic levels leads to the appearance of finite energy Majorana-Weyl nodes. This topological regime in the junction can be characterized by a quantized nonlocal conductance that measures the Chern number of the corresponding bands. In addition, we calculate current-phase relations, variance, and cross-correlations of topological supercurrents in multiterminal contacts and discuss the universality of these transport characteristics. At the technical level these results are obtained by integrating over the group of a circular ensemble that describes the scattering matrix of the junction. We briefly discuss our results in the context of observed fluctuations of the gate dependence of the critical current in topological planar Josephson junctions and comment on the possibility of parity measurements from the switching current distributions in multiterminal Majorana junctions.

Josephson effects in one junction with one s-wave superconductor and two topological superconductors

We present an analysis of Josephson effects in one quantum-dot-embedded junction with one s-wave superconductor (SC) and two topological superconductors (TSs). It is found that when superconducting phase difference is applied between the two TSs, the dot-SC coupling can efficiently modulate the fractional Josephson effect, including the changes of the supercurrent oscillation direction and period. Even though the polarization directions of the TSs are opposite, the dot-SC coupling enables to drive the fractional Josephson effect between the TSs. Next, as the superconducting phase difference is introduced between the TSs and SC, only the normal Josephson effect takes place.

Braiding quantum circuit based on the 4π Josephson effect

We propose a topological qubit in which braiding and readout are mediated by the $4\pi$ Majorana-Josephson effect. The braidonium device consists of three Majorana nanowires that come together to make a tri-junction; in order to control the superconducting phase differences at the tri-junction the nanowires are enclosed in a ring made of a conventional superconductor; and in order to perform initialization/readout one of the nanowires is coupled to a fluxonium qubit through a topological Josephson junction. We analyze how flux-based control and readout protocols can be used to demonstrate braiding and qubit operation for realistic materials and circuit parameters.

Phase-dependent heat and charge transport through superconductor–quantum dot hybrids

We analyze heat and charge transport through a single-level quantum dot coupled to two BCS superconductors at different temperatures to first order in the tunnel coupling. In order to describe the system theoretically, we extend a real-time diagrammatic technique that allows us to capture the interplay between superconducting correlations, strong Coulomb interactions, and nonequilibrium physics. We find that a thermoelectric effect can arise due to the superconducting proximity effect on the dot. In the nonlinear regime, the thermoelectric current can also flow at the particle-hole symmetric point due to a level renormalization caused by virtual tunneling between the dot and the leads. The heat current through the quantum dot is sensitive to the superconducting phase difference. In the nonlinear regime, the system can act as a thermal diode.

Spin-Orbit Splitting of Andreev States Revealed by Microwave Spectroscopy

We have performed microwave spectroscopy of Andreev states in superconducting weak links tailored in an InAs-Al (core-full shell) epitaxially-grown nanowire. The spectra present distinctive features, with bundles of four lines crossing when the superconducting phase difference across the weak link is 0 or $\pi.$ We interpret these as arising from zero-field spin-split Andreev states. A simple analytical model, which takes into account the Rashba spin-orbit interaction in a nanowire containing several transverse subbands, explains these features and their evolution with magnetic field. Our results show that the spin degree of freedom is addressable in Josephson junctions, and constitute a first step towards its manipulation.

Magnetization Reversal by Tuning Rashba Spin–Orbit Interaction and Josephson Phase in a Ferromagnetic Josephson Junction

We theoretically investigate the magnetization inside a normal metal containing the Rashba spin-orbit interaction (RSOI) induced by the proximity effect in an s-wave superconductor/normal metal/ferromagnetic metal/s-wave superconductor (S/N/F/S) Josephson junction. By solving the linearized Usadel equation taking account of the RSOI,we find that the direction of the magnetization induced by the proximity effect in N can be reversed by tuning the RSOI.Moreover, we also find that the direction of the magnetization inside N can be reversed by changing the superconducting phase difference, i.e., Josephson phase. From these results, it is expected that the dependence of the magnetization on the RSOI and Josephson phase can be applied to superconducting spintronics.

Current noises in a topological Josephson junction

We study the transport properties of a superconductor-quantum spin Hall insulator-superconductor Josephson junction both in the absence and in the presence of a DC bias voltage. As the system is predicted to host Majorana fermions at its interfaces, the Andreev bond states are supposed to exhibit a distinct $4\pi$ periodicity in the superconducting phase difference, namely the fractional Josephson effect. Using the non-equilibrium Greens function method, we calculate the current and the related current noise based on a tight-binding Hamiltonian. Our direct results show that the fractional Josephson effect can not be seen in equilibrium junctions. While in non-equilibrium junctions, this effect can be confirmed by the multiple Andreev reflections induced peaks of the non-equilibrium noise, which appear at discrete frequencies $\omega=neV$ with $n$ being an integer number.

Landau-Zener-St\xfcckelberg Interferometry for Majorana Qubit

Stimulated by a recent experiment observing successfully two superconducting states with even- and odd-number of electrons in a nanowire topological superconductor as expected from the existence of two end Majorana quasiparticles (MQs) [Albrecht et al., Nature 531, 206 (2016)], we propose a way to manipulate Majorana qubit exploiting quantum tunneling effects. The prototype setup consists of two one-dimensional (1D) topological superconductors coupled by a tunneling junction which can be controlled by gate voltage. We show that the time evolution of superconducting phase difference at the junction under a voltage bias induces an oscillation in energy levels of the Majorana parity states, whereas the level-crossing is avoided by a small coupling energy of MQs in the individual 1D superconductors. This results in a Landau-Zener-Stückelberg (LZS) interference between the Majorana parity states. Adjusting pulses of bias voltage and gate voltage, one can construct a LZS interferometry which provides an arbitrary manipulation of the Majorana qubit.

Andreev spectrum and supercurrents in nanowire-based SNS junctions containing Majorana bound states.

Hybrid superconductor–semiconductor nanowires with Rashba spin–orbit coupling are arguably becoming the leading platform for the search of Majorana bound states (MBSs) in engineered topological superconductors. We perform a systematic numerical study of the low-energy Andreev spectrum and supercurrents in short and long superconductor–normal–superconductor junctions made of nanowires with strong Rashba spin–orbit coupling, where an external Zeeman field is applied perpendicular to the spin–orbit axis. In particular, we investigate the detailed evolution of the Andreev bound states from the trivial into the topological phase and their relation with the emergence of MBSs. Due to the finite length, the system hosts four MBSs, two at the inner part of the junction and two at the outer one. They hybridize and give rise to a finite energy splitting at a superconducting phase difference of π, a well-visible effect that can be traced back to the evolution of the energy spectrum with the Zeeman field: from the trivial phase with Andreev bound states into the topological phase with MBSs. Similarly, we carry out a detailed study of supercurrents for short and long junctions from the trivial to the topological phases. The supercurrent, calculated from the Andreev spectrum, is 2π-periodic in the trivial and topological phases. In the latter it exhibits a clear sawtooth profile at a phase difference of π when the energy splitting is negligible, signalling a strong dependence of current–phase curves on the length of the superconducting regions. Effects of temperature, scalar disorder and reduction of normal transmission on supercurrents are also discussed. Further, we identify the individual contribution of MBSs. In short junctions the MBSs determine the current–phase curves, while in long junctions the spectrum above the gap (quasi-continuum) introduces an important contribution.

Quantum transport properties in one mesoscopic circuit with an embedded fractional Josephson junction

We investigate the quantum transport properties in one mesoscopic circuit which is embedded by a fractional Josephson junction. The calculation results show that the quantum interference takes nontrivial effects to the quantum transport through this system. This is manifested as the contributions of the superconducting phase difference and the coupling manners between the Majorana zero modes and leads to the conductance and Fano factor, respectively. The results in this work provide helpful information for understanding the transport behavior assisted by Majorana zero modes.

Manipulatable Andreev reflection due to the interplay between the DIII-class topological and s-wave superconductors

We construct one mesoscopic circuit in which one quantum dot couples to one DIII-class topological superconductor and one s-wave superconductor, in addition to its connection with the metallic lead. And then, the Andreev reflection current in the metallic lead is evaluated. It is found that the two kinds of superconductors drive the Andreev reflection in the constructive manner. Next as finite superconducting phase difference is taken into account, the Andreev reflection oscillates in period π2, and it can be suppressed in the low-energy region if the superconducting phase difference is (n+12)π2 (n ∈ Integer). Such a result is almost independent of the increase of the intradot Coulomb interaction. Therefore, this structure can assist to realize the manipulation of the Andreev reflection. Also, the result in this work provides useful information for understanding the property of the DIII-class topological superconductor.

Z4 parafermions in an interacting quantum spin Hall Josephson junction coupled to an impurity spin

$\mathbb{Z}_4$ parafermions can be realized in a strongly interacting quantum spin Hall Josephson junction or in a spin Hall Josephson junction strongly coupled to an impurity spin. In this paper we study a system that has both features, but with weak (repulsive) interactions and a weakly coupled spin. We show that for a strongly anisotropic exchange interaction, at low temperatures the system enters a strong coupling limit in which it hosts two $\mathbb{Z}_4$ parafermions, characterizing a fourfold degeneracy of the ground state. We construct the parafermion operators explicitly, and show that they facilitate fractional $e/2$ charge tunneling across the junction. The dependence of the effective low-energy spectrum on the superconducting phase difference reveals an $8\pi$ periodicity of the supercurrent.

Tunability of Andreev levels via spin-orbit coupling in Zeeman-split Josephson junctions

We study Andreev reflection and Andreev levels $\ensuremath{\varepsilon}$ in Zeeman-split superconductor/Rashba wire/Zeeman split superconductor junctions by solving the Bogoliubov--de Gennes equation. We theoretically demonstrate that the Andreev levels $\ensuremath{\varepsilon}$ can be controlled by tuning either the strength of Rashba spin-orbit interaction or the relative direction of the Rashba spin-orbit interaction and the Zeeman field. In particular, it is found that the magnitude of the band splitting is tunable by the strength of the Rashba spin-orbit interaction and the length of the wire, which can be interpreted by a spin precession in the Rashba wire. We also find that if the Zeeman field in the superconductor has the component parallel to the direction of the junction, the $\ensuremath{\varepsilon}\ensuremath{-}\ensuremath{\phi}$ curve becomes asymmetric with respect to the superconducting phase difference $\ensuremath{\phi}$. Whereas the Andreev reflection processes associated with each pseudospin band are sensitive to the relative orientation of the spin-orbit field and the exchange field, the total electric conductance interestingly remains invariant.

Theoretical Study of Magnetism Induced by Proximity Effect in a Ferromagnetic Josephson Junction with a Normal Metal

We theoretically study the magnetism induced by the proximity effect in the normal metal of ferromagnetic Josephson junction composed of two $s$-wave superconductors separated by ferromagnetic metal/normal metal/ferromagnetic metal junction (${S}/{F}/{N}/{F}/{S}$ junction). We calculate the magnetization in the $N$ by solving the Eilenberger equation. We show that the magnetization arises in the ${N}$ when the product of anomalous Green's functions of the spin-triplet even-frequency odd-parity Cooper pair and spin-singlet odd-frequency odd-parity Cooper pair in the ${N}$ has a finite value. The induced magnetization $M(d,\theta)$ can be decomposed into two parts, $M(d,\theta)=M^{\rm I}(d)+M^{\rm II}(d,\theta)$, where $d$ is the thickness of $N$ and $\theta$ is superconducting phase difference between two ${S}$s. Therefore, $\theta$ dependence of $M(d,\theta)$ allows us to control the amplitude of magnetization by changing $\theta$. The variation of $M(d,\theta)$ with $\theta$ is indeed the good evidence of the magnetization induced by the proximity effect, since some methods of magnetization measurement pick up total magnetization in the ${S}/{F}/{N}/{F}/{S}$ junction.

Josephson flux-flow oscillator: The microscopic tunneling approach

We elaborate a theoretical description of large Josephson junctions which is based on the Werthamer's microscopic tunneling theory. The model naturally incorporates coupling of electromagnetic radiation to the tunnel currents and, therefore, is particularly suitable for description of the self-coupling effect in Josephson junction. In our numerical calculations we treat the arising integro-differential equation, which describes temporal evolution of the superconducting phase difference coupled to the electromagnetic field, by the Odintsov-Semenov-Zorin algorithm. This allows us to avoid evaluation of the time integrals at each time step while taking into account all the memory effects. To validate the obtained microscopic model of large Josephson junction we focus our attention on the Josephson flux flow oscillator. The proposed microscopic model of flux flow oscillator does not involve the phenomenological damping parameter, rather, the damping is taken into account naturally in the tunnel current amplitudes calculated at a given temperature. The theoretically calculated current-voltage characteristics is compared to our experimental results obtained for a set of fabricated flux flow oscillators of different lengths. Our theoretical calculation agrees well with the obtained experimental results, and, to our knowledge, is the first where theoretical description of Josephson flux flow oscillator is brought beyond the perturbed sine-Gordon equation.

Ballistic edge states in Bismuth nanowires revealed by SQUID interferometry

The protection against backscattering provided by topology is a striking property. In two-dimensional insulators, a consequence of this topological protection is the ballistic nature of the one-dimensional helical edge states. One demonstration of ballisticity is the quantized Hall conductance. Here we provide another demonstration of ballistic transport, in the way the edge states carry a supercurrent. The system we have investigated is a micrometre-long monocrystalline bismuth nanowire with topological surfaces, that we connect to two superconducting electrodes. We have measured the relation between the Josephson current flowing through the nanowire and the superconducting phase difference at its ends, the current–phase relation. The sharp sawtooth-shaped phase-modulated current–phase relation we find demonstrates that transport occurs selectively along two ballistic edges of the nanowire. In addition, we show that a magnetic field induces 0–π transitions and ϕ0-junction behaviour, providing a way to manipulate the phase of the supercurrent-carrying edge states and generate spin supercurrents.

Simultaneously scanning two connected tips in a scanning tunneling microscope

We have modified a dual-tip scanning tunneling microscope (STM) by electrically connecting the tips together with a short (3 mm) strip of flexible 25 μm thick Nb foil. For simultaneous topographic imaging with both tips, we moved each tip to within tunneling distance z of a surface and modulated one tip's z-piezo at 5 kHz and the other at 10 kHz. The resulting combined tunneling current has modulation at both frequencies which we detect using individual lock-in amplifiers. Each lock-in output is fed back to its corresponding tip's individual STM z-position controller to maintain a stable current in both junctions. During the tests at room temperature, simultaneous imaging was performed with both tips made of Pt-Ir on Au/mica and highly oriented pyrolytic graphite (HOPG) samples, where a small tip-to-tip mechanical coupling was observed. We describe the system's performance, show results from simultaneous imaging, and discuss the potential application of the system to imaging superconducting phase differences.

Quasiclassical theory for the superconducting proximity effect in Dirac materials

We derive the quasiclassical non-equilibrium Eilenberger and Usadel equations to first order in quantities small compared to the Fermi energy, valid for Dirac edge and surface electrons with spin-momentum locking, as relevant for topological insulators. We discuss in detail several of the key technical points and assumptions of the derivation, and provide a Riccati-parametrization of the equations. Solving first the equilibrium equations for S/N and S/F bilayers and Josephson junctions, we study the superconducting proximity effect in Dirac materials. Similarly to related works, we find that the effect of an exchange field depends strongly on the direction of the field. Only components normal to the transport direction lead to attenuation of the Cooper pair wavefunction inside the F. Fields parallel to the transport direction lead to phase-shifts in the dependence on the superconducting phase difference for both the charge current and density of states in an S/F/S-junction. Moreover, we compute the differential conductance in S/N and S/F bilayers with an applied voltage bias, and determine the dependence on the length of the N and F regions and the exchange field.