Critical current 0-pi transition in designed Josephson Quantum Dot junctions

Supercurrent in the Presence of Direct Transmission and a Resonant Localized State

We study the current-phase relation (CPR) of an InSb-Al nanowire Josephson junction in parallel magnetic fields up to 700 mT. At high magnetic fields and in narrow voltage intervals of a gate under the junction, the CPR exhibits π shifts. The supercurrent declines within these gate intervals and shows asymmetric gate voltage dependence above and below them. We detect these features sometimes also at zero magnetic field. The observed CPR properties are reproduced by a theoretical model of supercurrent transport via interference between direct transmission and a resonant localized state.

Josephson Junction π-0 Transition Induced by Orbital Hybridization in a Double Quantum Dot

In this Letter, we manipulate the phase shift of a Josephson junction using a parallel double quantum dot (QD). By employing a superconducting quantum interference device, we determine how orbital hybridization and detuning affect the current-phase relation in the Coulomb blockade regime. For weak hybridization between the QDs, we find π junction characteristics if at least one QD has an unpaired electron. Notably the critical current is higher when both QDs have an odd electron occupation. By increasing the inter-QD hybridization the critical current is reduced, until eventually a π-0 transition occurs. A similar transition appears when detuning the QD levels at finite hybridization. Based on a zero-bandwidth model, we argue that both cases of phase-shift transitions can be understood considering an increased weight of states with a double occupancy in the ground state and with the Cooper pair transport dominated by local Andreev reflection.

Microwave Susceptibility Observation of Interacting Many-Body Andreev States

Electrostatic charging affects the many-body spectrum of Andreev states, yet its influence on their microwave properties has not been elucidated. We developed a circuit quantum electrodynamics probe that, in addition to transition spectroscopy, measures the microwave susceptibility of different states of a semiconductor nanowire weak link with a single dominant (spin-degenerate) Andreev level. We found that the microwave susceptibility does not exhibit a particle-hole symmetry, which we qualitatively explain as an influence of Coulomb interaction. Moreover, our state-selective measurement reveals a large, π-phase shifted contribution to the response common to all many-body states which can be interpreted as arising from a phase-dependent continuum in the superconducting density of states.

Proposal for Detection of the 0^{'} and π^{'} Phases in Quantum-Dot Josephson Junctions

The competition between the Kondo correlation and superconductivity in quantum-dot Josephson junctions (QDJJs) has been known to drive a quantum phase transition between 0 and π junctions. Theoretical studies so far have predicted that under strong Coulomb correlations the 0-π transition should go through intermediate states, 0^{'} and π^{'} phases. By combining a nonperturbative numerical method and the resistively shunted junction model, we investigated the magnetic-field-driven phase transition of the QDJJs in the Kondo regime and found that the low-field magnetotransport exhibits a unique feature which can be used to distinguish the intermediate phases. In particular, the magnetic-field driven π^{'}-π transition is found to lead to the enhancement of the supercurrent which is strongly related to the Kondo effect.

Superconducting Proximity in Intrinsic Magnetic Topological Insulator MnBi2Te4-NbN Hybrid Device Modulated by Coulomb Blockade Effect

The combination of nontrivial topology, magnetism, and superconductivity could offer the potential to realize exotic excitations of quasiparticles. MnBi₂Te₄, as an intrinsic magnetic topological insulator, may be a good platform to create Majorana fermions if coupled to an s-wave superconductor. Here, we report the transport properties of a MnBi₂Te₄-NbN hybrid device. This device exhibits clear Coulomb blockade oscillations. We observe a large zero-bias conductance peak that exists over considerable changes in gate voltage, magnetic field, and temperature, which is interpreted as a not fully developed supercurrent. The zero-bias peak shows a nonmonotonic evolution with a magnetic field and an abrupt π phase shift with changing temperature. Zero-energy bound states and a topological phase transition may exist in this hybrid system. Our results provide the first experimental investigation into the properties of the intrinsic magnetic topological insulator/superconductor hybrid structures modulated by the Coulomb blockade effect.

Signatures of Interactions in the Andreev Spectrum of Nanowire Josephson Junctions

We performed microwave spectroscopy of an InAs nanowire between superconducting contacts implementing a finite-length, multichannel Josephson weak link. Certain features in the spectra, such as the splitting by spin-orbit interactions of the transition lines among Andreev states, have been already understood in terms of noninteracting models. However, we identify here additional transitions, which evidence the presence of Coulomb interactions. By combining experimental measurements and model calculations, we reach a qualitative understanding of these very rich Andreev spectra.

Supercurrent parity meter in a nanowire Cooper pair transistor

We study a Cooper pair transistor realized by two Josephson weak links that enclose a superconducting island in an InSb-Al hybrid nanowire. When the nanowire is subject to a magnetic field, isolated subgap levels arise in the superconducting island and, because of the Coulomb blockade, mediate a supercurrent by coherent cotunneling of Cooper pairs. We show that the supercurrent resulting from such cotunneling events exhibits, for low to moderate magnetic fields, a phase offset that discriminates even and odd charge ground states on the superconducting island. Notably, this phase offset persists when a subgap state approaches zero energy and, based on theoretical considerations, permits parity measurements of subgap states by supercurrent interferometry. Such supercurrent parity measurements could, in a series of experiments, provide an alternative approach for manipulating and protecting quantum information stored in the isolated subgap levels of superconducting islands.

Collapse of the Josephson Emission in a Carbon Nanotube Junction in the Kondo Regime

We probe the high frequency emission of a carbon nanotube based Josephson junction and compare it to its dc Josephson current. The ac emission is probed by coupling the carbon nanotube to an on-chip detector (a superconductor-insulator-superconductor junction), via a coplanar waveguide resonator. The measurement of the photoassisted current of the detector gives direct access to the signal emitted by the carbon nanotube. We focus on the gate regions that exhibit Kondo features in the normal state and demonstrate that when the dc supercurrent is enhanced by the Kondo effect, the ac Josephson effect is strongly reduced. This result is compared to numerical renormalization group theory and is attributed to a transition between the singlet ground state and the doublet excited state which is enabled only when the junction is driven out-of-equilibrium by a voltage bias.

Quantum Dot Parity Effects in Trivial and Topological Josephson Junctions

An odd-occupied quantum dot in a Josephson junction can flip transmission phase, creating a π junction. When the junction couples topological superconductors, no phase flip is expected. We investigate this and related effects in a full-shell hybrid interferometer, using gate voltage to control dot-junction parity and axial magnetic flux to control the transition from trivial to topological superconductivity. Enhanced zero-bias conductance and critical current for odd parity in the topological phase reflects hybridization of the confined spin with zero-energy modes in the leads.

Charge localization and reentrant superconductivity in a quasi-ballistic InAs nanowire coupled to superconductors

A semiconductor nanowire with strong spin-orbit coupling in proximity to a superconductor is predicted to display Majorana edge states emerging under a properly oriented magnetic field. The experimental investigation of these exotic states requires assessing the one-dimensional (1D) character of the nanowire and understanding the superconducting proximity effect in the presence of a magnetic field. Here, we explore the quasi-ballistic 1D transport regime of an InAs nanowire with Ta contacts. Fine-tuned by means of local gates, the observed plateaus of approximately quantized conductance hide the presence of a localized electron, giving rise to a lurking Coulomb blockade effect and Kondo physics. When Ta becomes superconducting, this local charge causes an unusual, reentrant magnetic field dependence of the supercurrent, which we ascribe to a 0 - π transition. Our results underline the relevant role of unintentional charge localization in the few-channel regime where helical subbands and Majorana quasi-particles are expected to arise.

The Anderson-Josephson quantum dot-a theory perspective

Recent progress in nanoscale manufacturing has allowed to experimentally investigate quantum dots coupled to two superconducting leads in controlled and tunable setups. The equilibrium Josephson current was measured in on-chip superconducting quantum interference devices, and subgap states were investigated using weakly coupled metallic leads for spectroscopy. This has reinstated two 'classic' problems on the agenda of theoretical condensed matter physics: (1) the Josephson effect and (2) quantum spins in superconductors. The relevance of the former is obvious as the barrier, which separates the two superconductors in a standard Josephson junction, is merely replaced by the quantum dot with well separated energy levels. For odd filling of the dot it acts as a quantum mechanical spin-1/2 and thereby the relevance of the latter becomes apparent also. For normal conducting leads and at odd dot filling the Kondo effect strongly modifies the transport properties as can, e.g. be studied within the Anderson model. One can expect the same for superconducting leads, and in certain parameter regimes remnants of Kondo physics, i.e. strong electronic correlations, will affect the Josephson current. In this topical review, we discuss the status of the theoretical understanding of the Anderson-Josephson quantum dot in equilibrium, mainly focusing on the Josephson current. We introduce a minimal model consisting of a dot which can only host a single spin-up and a single spin-down electron repelling each other by a local Coulomb interaction. The dot is tunnel-coupled to two superconducting leads as described by the Bardeen-Cooper-Schrieffer Hamiltonian. This model was investigated using a variety of methods, some capturing aspects of Kondo physics, while others failing in this respect. We briefly review this. The model shows a first order level-crossing quantum phase transition when varying any parameter, provided that the others are within appropriate ranges. At vanishing temperature it leads to a jump of the Josephson current. To study the qualitative behavior of the phase diagram, or the Josephson current, several of the methods can be used. However, for a quantitative description, elaborate quantum many-body methods must be employed. We show that a quantitative agreement between accurate results obtained for the simple model and measurements of the current can be reached. This confirms that the experiments reveal the finite temperature signatures of the zero temperature transition. In addition, we consider two examples of more complex dot geometries, which might be experimentally realized in the near future. The first is characterized by the interplay of the above level-crossing physics and the Fano effect, and the second by the interplay of superconductivity and almost degenerate singlet and triplet two-body states.

Supercurrent in a Double Quantum Dot

We demonstrate the Josephson effect in a serial double quantum dot defined in a nanowire with epitaxial superconducting leads. The supercurrent stability diagram adopts a honeycomb pattern. We observe sharp discontinuities in the magnitude of the critical current, I_{c}, as a function of dot occupation, related to doublet to singlet ground state transitions. Detuning of the energy levels offers a tuning knob for I_{c}, which attains a maximum at zero detuning. The consistency between experiment and theory indicates that our device is a faithful realization of the two-impurity Anderson model.

Enhanced Shot Noise of Multiple Andreev Reflections in a Carbon Nanotube Quantum Dot in SU(2) and SU(4) Kondo regimes

The sensitivity of shot noise to the interplay between Kondo correlations and superconductivity is investigated in a carbon nanotube quantum dot connected to superconducting electrodes. Depending on the gate voltage, the SU(2) and SU(4) Kondo unitary regimes can be clearly identified. We observe enhancement of the shot noise via the Fano factor in the superconducting state. Its divergence at low bias voltage, which is more pronounced in the SU(4) regime than in the SU(2) one, is larger than what is expected from proliferation of multiple Andreev reflections predicted by the existing theories. Our result suggests that the Kondo effect is responsible for this strong enhancement.

Fermi-Liquid Approach for Superconducting Kondo Problems

We present a Fermi liquid approach to superconducting Kondo problems applicable when the Kondo temperature is large compared to the superconducting gap. To illustrate the theory, we study the current-phase relation and the Andreev level spectrum for an Anderson impurity between two s-wave superconductors. In the particle-hole symmetric Kondo limit, we find a 4π periodic Andreev spectrum. The 4π periodicity persists under a small voltage bias which however causes an asymmetric distortion of Andreev levels. The latter distinguishes the present 4π effect from the one in topological Majorana junctions.

Josephson effect in junctions of conventional and topological superconductors

We present a theoretical analysis of the equilibrium Josephson current-phase relation in hybrid devices made of conventional s-wave spin-singlet superconductors (S) and topological superconductor (TS) wires featuring Majorana end states. Using Green's function techniques, the topological superconductor is alternatively described by the low-energy continuum limit of a Kitaev chain or by a more microscopic spinful nanowire model. We show that for the simplest S-TS tunnel junction, only the s-wave pairing correlations in a spinful TS nanowire model can generate a Josephson effect. The critical current is much smaller in the topological regime and exhibits a kink-like dependence on the Zeeman field along the wire. When a correlated quantum dot (QD) in the magnetic regime is present in the junction region, however, the Josephson current becomes finite also in the deep topological phase as shown for the cotunneling regime and by a mean-field analysis. Remarkably, we find that the S-QD-TS setup can support φ0-junction behavior, where a finite supercurrent flows at vanishing phase difference. Finally, we also address a multi-terminal S-TS-S geometry, where the TS wire acts as tunable parity switch on the Andreev bound states in a superconducting atomic contact.

Shiba Bound States across the Mobility Edge in Doped InAs Nanowires

We present a study of Andreev quantum dots fabricated with small-diameter (30 nm) Si-doped InAs nanowires where the Fermi level can be tuned across a mobility edge separating localized states from delocalized states. The transition to the insulating phase is identified by a drop in the amplitude and width of the excited levels and is found to have remarkable consequences on the spectrum of superconducting subgap resonances. While at deeply localized levels only quasiparticle cotunneling is observed, for slightly delocalized levels Shiba bound states form and a parity-changing quantum phase transition is identified by a crossing of the bound states at zero energy. Finally, in the metallic regime, single Andreev resonances are observed.

Transport Signatures of Quasiparticle Poisoning in a Majorana Island

We investigate effects of quasiparticle poisoning in a Majorana island with strong tunnel coupling to normal-metal leads. In addition to the main Coulomb blockade diamonds, "shadow" diamonds appear, shifted by 1e in gate voltage, consistent with transport through an excited (poisoned) state of the island. Comparison to a simple model yields an estimate of parity lifetime for the strongly coupled island (∼1  μs) and sets a bound for a weakly coupled island (>10  μs). Fluctuations in the gate-voltage spacing of Coulomb peaks at high field, reflecting Majorana hybridization, are enhanced by the reduced lever arm at strong coupling. When converted from gate voltage to energy units, fluctuations are consistent with previous measurements.

6π Josephson Effect in Majorana Box Devices

We study Majorana devices featuring a competition between superconductivity and multichannel Kondo physics. Our proposal extends previous work on single-channel Kondo systems to a topologically nontrivial setting of a non-Fermi liquid type, where topological superconductor wires (with gap Δ) represent leads tunnel coupled to a Coulomb-blockaded Majorana box. On the box, a spin degree of freedom with Kondo temperature T_{K} is nonlocally defined in terms of Majorana states. For Δ≫T_{K}, the destruction of Kondo screening by superconductivity implies a 4π-periodic Josephson current-phase relation. Using a strong-coupling analysis in the opposite regime Δ≪T_{K}, we find a 6π-periodic Josephson relation for three leads, with critical current I_{c}≈eΔ^{2}/ℏT_{K}, corresponding to the transfer of fractionalized charges e^{*}=2e/3.

Perturbation theory of a superconducting 0 - π impurity quantum phase transition

A single-level quantum dot with Coulomb repulsion attached to two superconducting leads is studied via the perturbation expansion in the interaction strength. We use the Nambu formalism and the standard many-body diagrammatic representation of the impurity Green functions to formulate the Matsubara self-consistent perturbation expansion. We show that at zero temperature second order of the expansion in its spin-symmetric version yields a nearly perfect agreement with the numerically exact calculations for the position of the 0 − π phase boundary at which the Andreev bound states reach the Fermi energy as well as for the values of single-particle quantities in the 0-phase. We present results for phase diagrams, level occupation, induced local superconducting gap, Josephson current, and energy of the Andreev bound states with the precision surpassing any (semi)analytical approaches employed thus far.

Andreev bound-state dynamics in quantum-dot Josephson junctions: a washing out of the 0-π transition

We consider a Josephson junction formed by a quantum dot connected to two bulk superconductors in the presence of Coulomb interaction and coupling to both an electromagnetic environment and a finite density of electronic quasiparticles. In the limit of a large superconducting gap we obtain a Born-Markov description of the relevant Andreev bound-states dynamics. We calculate the current-phase relation and we find that the experimentally unavoidable presence of quasiparticles can dramatically modify the 0-π standard transition picture. We show that photon-assisted quasiparticle absorption allows the dynamic switching from the 0 to the π state and vice versa, washing out the 0-π transition predicted by purely thermodynamic arguments. Spectroscopic signatures of Andreev bound-states broadening are investigated by considering microwave irradiation.

Spin-resolved Andreev levels and parity crossings in hybrid superconductor-semiconductor nanostructures

The physics and operating principles of hybrid superconductor-semiconductor devices rest ultimately on the magnetic properties of their elementary subgap excitations, usually called Andreev levels. Here we report a direct measurement of the Zeeman effect on the Andreev levels of a semiconductor quantum dot with large electron g-factor, strongly coupled to a conventional superconductor with a large critical magnetic field. This material combination allows spin degeneracy to be lifted without destroying superconductivity. We show that a spin-split Andreev level crossing the Fermi energy results in a quantum phase transition to a spin-polarized state, which implies a change in the fermionic parity of the system. This crossing manifests itself as a zero-bias conductance anomaly at finite magnetic field with properties that resemble those expected for Majorana modes in a topological superconductor. Although this resemblance is understood without evoking topological superconductivity, the observed parity transitions could be regarded as precursors of Majorana modes in the long-wire limit.

Tunneling spectroscopy of quasiparticle bound states in a spinful Josephson junction

The spectrum of a segment of InAs nanowire, confined between two superconducting leads, was measured as function of gate voltage and superconducting phase difference using a third normal-metal tunnel probe. Subgap resonances for odd electron occupancy-interpreted as bound states involving a confined electron and a quasiparticle from the superconducting leads, reminiscent of Yu-Shiba-Rusinov states-evolve into Kondo-related resonances at higher magnetic fields. An additional zero-bias peak of unknown origin is observed to coexist with the quasiparticle bound states.

Transport measurement of Andreev bound states in a Kondo-correlated quantum dot

We report nonequilibrium transport measurements of gate-tunable Andreev bound states in a carbon nanotube quantum dot coupled to two superconducting leads. In particular, we observe clear features of two types of Kondo ridges, which can be understood in terms of the interplay between the Kondo effect and superconductivity. In the first type (type I), the coupling is strong and the Kondo effect is dominant. Levels of the Andreev bound states display anticrossing in the middle of the ridge. On the other hand, crossing of the two Andreev bound states is shown in the second type (type II) together with the 0-π transition of the Josephson junction. Our scenario is well understood in terms of only a single dimensionless parameter, k(B)T(K)(min)/Δ, where T(K)(min) and Δ are the minimum Kondo temperature of a ridge and the superconducting order parameter, respectively. Our observation is consistent with measurements of the critical current, and is supported by numerical renormalization group calculations.

Josephson current through interacting double quantum dots with spin-orbit coupling

We study the effect of Rashba spin-orbit interaction on the Josephson current through a double quantum dot in the presence of Coulomb repulsion. In particular, we describe the characteristic effects on the magnetic field-induced singlet-triplet transition in the molecular regime. Exploring the whole parameter space, we analyze the effects of the device asymmetry, the orientation of the applied magnetic field with respect to the spin-orbit interaction, and finite temperatures. We find that at finite temperatures the orthogonal component of the spin-orbit interaction exhibits a similar effect to the Coulomb interaction inducing the occurrence of a π-phase at particle-hole symmetry. This provides a new route to the experimental observability of the π-phase in multi-level quantum dots.

Thermoelectric transport in the three terminal quantum dot

The thermoelectric transport in the system composed of a quantum dot in contact with superconducting, ferromagnetic and normal metal electrodes has been studied. Such a system can support pure spin current in the normal electrode. In the limit of a large superconducting gap and weak coupling between the dot and the electrodes we investigate the sub-gap charge and spin transport via Andreev mechanism using the standard master equation technique, which is known to be valid in the sequential tunnelling regime. The Zeeman splitting of the dot level induces pure spin current in the ferromagnetic electrode under an appropriate bias. This opens a novel possibility to switch the spin current between two electrodes by electric means. The calculated spin and charge thermopower coefficients attain very large values, of the order of a few hundreds μV K(-1), and show similar dependences on the position of the on-dot energy level and temperature.

Understanding the Josephson current through a Kondo-correlated quantum dot

We study the Josephson current 0-π transition of a quantum dot tuned to the Kondo regime. The physics can be quantitatively captured by the numerically exact continuous time quantum Monte Carlo method applied to the single-impurity Anderson model with Bardeen-Cooper-Schrieffer superconducting leads. For a comparison to an experiment, the tunnel couplings are determined by fitting the normal-state linear conductance. Excellent agreement for the dependence of the critical Josephson current on the level energy is achieved. For increased tunnel couplings the Kondo scale becomes comparable to the superconducting gap, and the regime of the strongest competition between superconductivity and Kondo correlations is reached; we predict the gate voltage dependence of the critical current in this regime.

Magnetic-field dependence of tunnel couplings in carbon nanotube quantum dots

By means of sequential and cotunneling spectroscopy, we study the tunnel couplings between metallic leads and individual levels in a carbon nanotube quantum dot. The levels are ordered in shells consisting of two doublets with strong- and weak-tunnel couplings, leading to gate-dependent level renormalization. By comparison to a one- and two-shell model, this is shown to be a consequence of disorder-induced valley mixing in the nanotube. Moreover, a parallel magnetic field is shown to reduce this mixing and thus suppress the effects of tunnel renormalization.

Hybrid superconductor-quantum dot devices

Advances in nanofabrication techniques have made it possible to make devices in which superconducting electrodes are connected to non-superconducting nanostructures such as quantum dots. The properties of these hybrid devices result from a combination of a macroscopic quantum phenomenon involving large numbers of electrons (superconductivity) and the ability to control single electrons, offered by quantum dots. Here we review research into electron transport and other fundamental processes that have been studied in these devices. We also describe potential applications, such as a transistor in which the direction of a supercurrent can be reversed by adding just one electron to a quantum dot.

Josephson current in strongly correlated double quantum dots

We study the Josephson current through a serial double quantum dot and the associated 0-π transitions which result from the subtle interplay between the superconductivity, the Kondo physics, and the interdot superexchange interaction. The competition between them is examined by tuning the relative strength Δ/T(K) of the superconducting gap and the Kondo temperature, for different strengths of the superexchange coupling determined by the interdot tunneling t relative to the level broadening Γ. We find strong renormalization of t, a significant role of the superexchange coupling J, and a rich phase diagram of the 0 and π-junction regimes. In particular, when both the superconductivity and the exchange interaction compete with the Kondo physics (Δ∼J∼T(K)), there appears an island of π' phase at large values of the superconducting phase difference.

Tunneling spectroscopy of Andreev energy levels in a quantum dot coupled to a superconductor

The coupling of a quantum dot with a BCS-type superconducting reservoir results in an intriguing system where low energy physics is governed by the interplay of two distinct phases, singlet and doublet. In this Letter we show that the spectrum of Andreev energy levels, which capture the properties of the two phases, can be detected in transport measurements with a quantum dot strongly coupled to a superconducting lead and weakly coupled to a normal metal lead. We observe phase transitions between BCS singlet and degenerate magnetic doublet states when the quantum dot chemical potential is tuned with an electrostatic gate, in good qualitative agreement with numerical renormalization group calculations.

Gate-tunable dissipation and "superconductor-insulator" transition in carbon nanotube josephson junctions

Dissipation is ubiquitous in quantum systems, and its interplay with fluctuations is critical to maintaining quantum coherence. We experimentally investigate the dissipation dynamics in single-walled carbon nanotubes coupled to superconductors. The voltage-current characteristics display gate-tunable hysteresis, with sizes that perfectly correlate with the normal state resistance R_{N}, indicating the junction undergoes a periodic modulation between the underdamped and overdamped regimes. Surprisingly, when a device's Fermi level is tuned through a local conductance minimum, we observe a gate-controlled transition from superconductinglike to insulatinglike states, with a "critical" R_{N} value approximately 8-20 kOmega.