In a Josephson junction (JJ), Cooper pairs are transported via Andreev bound states (ABSs) between superconductors. The ABSs in the weak link of multi-terminal (MT) JJs can coherently hybridize two Cooper pairs among different superconducting electrodes, resulting in the Cooper quartet (CQ) involving four fermions entanglement. The energy spectrum of these CQ-ABS can be controlled by biasing MT-JJs due to the AC Josephson effect. Here, using gate tunable four-terminal graphene JJs complemented with a flux loop, we construct CQs with a tunable spectrum. The critical quartet supercurrent exhibits magneto-oscillation associated with a charge of 4e; thereby presenting the evidence for interference between entangled CQ-ABS. At a finite bias voltage, we find the DC quartet supercurrent shows non-monotonic bias dependent behavior, attributed to Landau-Zener transitions between different Floquet bands. Our experimental demonstration of coherent non-equilibrium CQ-ABS sets a path for design of artificial topological materials based on MT-JJs.
We report an observation of a new, non dissipative and non local supercurrent, carried by quartets; each consisting of four entangled electrons. The supercurrent is a result of a novel Andreev bound state (ABS), formed among three superconducting terminals. While in a two-terminal Josephson junction the usual ABS, and thus the DC Josephson current, exist only in equilibrium, in the present realization the ABS exists also in the strongly nonlinear regime (biased terminals). The presence of supercurrent carried by quartets was established by performing non-local conductance and cross-correlation…
Bijunctions are three-terminal Josephson junctions where three superconductors are connected by a single weak link made of a metallic region or of quantum dots. Biasing two of the superconductors with commensurate voltages yields Andreev resonances that produce d.c. Josephson currents made of correlated Cooper pairs. For instance with applied voltages (0, V,-V), quartets formed by two entangled Cooper pairs are emitted by one reservoir towards the two others. Theory involving non-equilibrium Green's functions reveal the microsopic mechanism at play, e.g. multiple coherent Andreev reflections t…
A three-terminal Josephson junction biased at opposite voltages can sustain a phase-sensitive dc-current carrying three-body static phase coherence, known as the "quartet current". We calculate the zero-frequency current noise cross-correlations and answer the question of whether this current is noisy (like a normal current in response to a voltage drop) or noiseless (like an equilibrium supercurrent in response to a phase drop). A quantum dot with a level at energy $ε_0$ is connected to three superconductors $S_a$, $S_b$ and $S_c$ with gap $Δ$, biased at $V_a=V$, $V_b=-V$ and $V_c=0$, and wit…
In connection with the recent Harvard group experiment on graphene-based four-terminal Josephson junctions containing a grounded loop, we consider biasing at opposite voltages on the quartet line and establish lowest-order perturbation theory in the tunnel amplitudes between a two-dimensional (2D) metal and four superconducting leads in the dirty limit. We present in addition general nonperturbative and nonadiabatic results. The critical current on the quartet line $I_c(Φ/Φ_0)$ depends on the reduced flux $Φ/Φ_0$ via interference between the three-terminal quartets (3TQ) and the nonstandard fo…
Cooper quartets and fractional vortices in frustrated Josephson junction dice arrays
University of Cologne（科隆大学）；Forschungszentrum Jülich GmbH, Institute of Quantum Control, Peter Grünberg Institut (PGI-8)（于利希研究中心有限公司，量子控制研究所，彼得·格林贝格研究所（PGI-8））；University of Oslo（奥斯陆大学）；Università di Pisa（比萨大学）；INFN, Sezione di Pisa（意大利国家核物理研究所，比萨分部）
Multiterminal Josephson junctions have aroused considerable theoretical interest recently and numerous works aim at putting the predictions of correlations among Coopers (i.e. the so-called quartets) and simulation of topological matter to the test of experiments. The present paper is motivated by recent experimental investigation from the Harvard group reporting $h/4e$-periodic quartet signal in a four-terminal configuration containing a loop pierced by magnetic flux, together with inversion controlled by the bias voltage, i.e. the quartet signal can be larger at half-flux quantum than in zer…
We theoretically study the ground-state properties and the condensations of exciton-like Cooper pairs and biexciton-like Cooper quartets in an electron-hole system. Applying the variational approach associated based on the quartet Bardeen-Cooper-Schrieffer (BCS) model to the four-component fermionic system consisting of spin-1/2 electrons and spin-1/2 holes, we show how Cooper pairs and quartet correlations appear in the equation of state at the thermodynamic limit. The biexciton-like four-body correlations survive even at the high-density regime as a many-body BCS-like state of Cooper quartet…
We have investigated the Josephson inductance $L_J$ of a Superconducting Cooper Pair Transistor (SCPT). We traced the inductance using microwave reflection measurements on a tuned resonance circuit in which a SCPT was mounted in parallel to a $ \sim 200$ pH strip line inductance. When the inverse of the Josephson inductance, determined on the charge-phase bias plane for a SCPT with a Josephson to Coulomb energy ratio of $E_J/E_C =1.75$, is integrated over the phase, we obtain a current-phase relation, which is strongly non-sinusoidal near the charge degeneracy point.
In a transparent three-terminal Josephson junction, modeling nonequilibrium transport is numerically challenging, owing to the interplay between multiple Andreev reflection (MAR) thresholds and multipair resonances in the pair current. An approximate method, coined as "partially resummed perturbation theory in the number of nonlocal Green's functions", is presented that can be operational on a standard computer and demonstrates compatibility with results existing in the literature. In a linear structure made of two neighboring interfaces (with intermediate transparency) connected by a central …
We have developed a quantitative theory of Cooper pair pumping in gated one-dimensional arrays of Josephson junctions. The pumping accuracy is limited by quantum tunneling of Cooper pairs out of the propagating potential well and by direct supercurrent flow through the array. Both corrections decrease exponentially with the number N of junctions in the array, but give a serious limitation of accuracy for any practical array. The supercurrent at resonant gate voltages decreases with N only as sin(v/N)/N, where v is the Josephson phase difference across the array.
We investigate the photon-mediated heat flow between two Josephson-coupled Bardeen-Cooper-Schrieffer (BCS) superconductors. We demonstrate that in standard low temperature experiments involving temperature-biased superconducting quantum interference devices (SQUIDs), this radiative contribution is negligible if compared to the direct galvanic one, but it largely exceeds the heat exchanged between electrons and the lattice phonons. The corresponding thermal conductance is found to be several orders of magnitude smaller, for real experiments setup parameters, than the universal quantum of therma…
We study the dynamics of a quantum superconducting circuit which consists of a Josephson charge qubit, coupled capacitively to a current biased Josephson junction. Under certain conditions, the eigenstates of the qubit and the junction become entangled. We obtain the time evolution of these states in the limit of weak coupling. Rabi oscillations occur, as a result of the spontaneous emission and re-absorption of a single oscillation quantum in the junction. We discuss a possible way to experimentally determine the quantum state of the junction and hence observe the Rabi oscillations.
The advent of quantum optical techniques based on superconducting circuits has opened new regimes in the study of the non-linear interaction of light with matter. Of particular interest has been the creation of non-classical states of light, which are essential for continuous-variable quantum information processing, and could enable quantum-enhanced measurement sensitivity. Here we demonstrate a device consisting of a superconducting artificial atom, the Cooper pair transistor, embedded in a superconducting microwave cavity that may offer a path toward simple, continual production of non-class…
We consider a finite-size topological Josephson junction formed at the edge of a two-dimensional topological insulator in proximity to conventional superconductors and study the impact of Cooper pair symmetries on the electron transport. We find that, due to the finite junction size, electron transport is highly tunable by the superconducting phase difference $ϕ$ across the junction. At zero frequency and $ϕ=π$, the setup exhibits vanishing local Andreev reflection and perfect normal transmission due to the interplay of finite junction size and formation of topological Andreev bound states in …
The equilibrium current-phase structure of a tri-terminal superconducting Josephson junction (bijunction) is analyzed as a function of the two relevant phases. The bijunction is made of two noninteracting quantum dots, each one carrying a single level. Nonlocal processes coupling the three terminals are described in terms of quartet tunneling and pair cotunneling. These couplings are due to nonlocal Andreev and cotunneling processes through the central superconductor $S_0$, as well as direct interdot coupling. In some cases, two degenerate midgap Andreev states appear, symmetric with respect t…
We study the critical Josephson current flowing through a double quantum dot weakly coupled to two superconducting leads. We use analytical as well as numerical methods to investigate this setup in the limit of small and large bandwidth leads in all possible charging states, where we account for on-site interactions exactly. Our results provide clear signatures of nonlocal spin-entangled pairs, which support interpretations of recent experiments [Deacon, R. S. et al., Nat. Commun. 6, 7446 (2015)]. In addition, we find that the ground state with one electron on each quantum dot can undergo a tu…
An all-superconducting bijunction consists of a central superconductor contacted to two lateral superconductors, such that non-local crossed Andreev reflection is operating. Then new correlated transport channels for the Cooper pairs appear in addition to those of separated conventional Joseph- son junctions. We study this system in a configuration where the superconductors are connected through gate-controllable quantum dots. Multipair phase-coherent resonances and phase-dependent multiple Andreev reflections are both obtained when the voltages of the lateral superconductors are commensurate,…
The dc transport properties of long arrays of small Al Josephson junctions, biased through on-chip Cr resistors, are studied. The IV-characteristics show a large Coulomb threshold for current as well as negative-slope regions indicating the regime of autonomous Bloch oscillations up to rather high frequencies of f = I/2e ~1 GHz, comparable to those reported by other groups for single junctions. On the other hand, a small depth of the back-bending implies a low duty cycle and a broad spectrum of the oscillations, which we attribute to the insufficiently high impedance of the bias resistors. A s…
Probing Cooper pair momentum by quasiparticle steering with planar Josephson junctions
QuTech, Delft University of Technology（代尔夫特理工大学 QuTech）；Kavli Institute of Nanoscience, Delft University of Technology（代尔夫特理工大学卡弗里纳米科学研究所）；Donostia International Physics Center（圣塞巴斯蒂安国际物理中心）；Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen（哥本哈根大学尼尔斯·玻尔研究所量子器件中心）
We examine decoherence effects in the Josephson current of a Cooper pair shuttle. Dephasing due to gate voltage fluctuations can either suppress or enhance the critical current and also change its sign. The current noise spectrum displays a peak at the Josephson coupling energy and shows a phase dependence.
The supercurrent of a Josephson junction is reduced by phase diffusion. For ultrasmall capacitance junctions the current may be further decreased by Coulomb blockade effects. We calculate the Cooper pair current by means of time-dependent perturbation theory to all orders in the Josephson coupling energy and obtain the current-voltage characteristic in closed form in a range of parameters of experimental interest. The results comprehend phase diffusion of the coherent Josephson current in the classical regime as well as the supercurrent peak due to incoherent Cooper pair tunneling in the stron…
We consider microscopically low-temperature transport in weakly disordered arrays of Josephson junctions in the Coulomb blockade regime. We demonstrate that at sufficiently low temperatures the main contribution to the dc conductivity comes from the motion of single-Cooper-pair excitations, scattered by irregularities in the array. Being proportional to the concentration of the excitations, the conductivity is exponentially small in temperature with the activation energy close to the charging energy of a Cooper pair on a superconductive island. Applying a diagrammatic approach to treat the dis…
We measure the high-frequency emission of a single Cooper pair transistor(SCPT) in the regime where transport is only due to tunneling of Cooper pairs. This is achieved by coupling on-chip the SCPT to a superconductor-insulator-superconductor junction and by measuring the photon assisted tunneling current of quasiparticles across the junction. This technique allows a direct detection of the AC Josephson effect of the SCPT and provides evidence of Landau-Zener transitions for proper gate voltage. The emission in the regime of resonant Cooper pair tunneling is also investigated. It is interprete…
We study correlated transport in a Josephson junction array for small Josephson energies. In this regime transport is dominated by Cooper-pair hopping, although we observe that quasiparticles can not be neglected. We assume that the energy dissipated by a Cooper-pair is absorbed by the intrinsic impedance of the array. This allows us to formulate explicit Cooper-pair hopping rates without adding any parameters to the system. We show that the current is correlated and crucially, these correlations rely fundamentally on the interplay between the Cooper-pairs and equilibrium quasiparticles.
Three-terminal superconductor (S) - normal metal (N) - superconductor (S) Josephson junctions are investigated. In a geometry where a T-shape normal metal is connected to three superconducting reservoirs, new sub-gap structures appear in the differential resistance for specific combinations of the superconductor chemical potentials. Those correspond to a correlated motion of Cooper pairs within the device that persist well above the Thouless energy and is consistent with the prediction of quartets formed by two entangled Cooper pairs. A simplified nonequilibrium Keldysh Green's function calcul…
The production of entangled pairs of electrons in ferromagnet-superconductor-ferromagnet or normal metal-superconductor-normal metal three-terminal structures has aroused considerable interest in the last twenty years. In these studies, the distance between the contacts is limited by the zero-energy superconducting coherence length. Here, we demonstrate nonlocality and quantum correlations in voltage-biased three-terminal Josephson junctions over the ultralong distance that exceeds the superconducting coherence length by orders of magnitude. The effect relies on} the interplay between the time…
Local spins coupled to superconductors give rise to several emerging phenomena directly linked to the competition between Cooper pair formation and magnetic exchange. These effects are generally scrutinized using a spectroscopic approach which relies on detecting the in-gap bound modes arising from Cooper pair breaking, the so-called Yu-Shiba-Rusinov (YSR) states. However, the impact of local magnetic impurities on the superconducting order parameter remains largely unexplored. Here, we use scanning Josephson spectroscopy to directly visualize the effect of magnetic perturbations on Cooper pai…
We study a one-dimensional Josephson junction chain embedded in a magnetic field. We show that when the magnetic flux per elementary loop equals half the superconducting flux quantum $ϕ_0=h/2e$, a local $\nbZ_2$ symmetry arises. This symmetry is responsible for a nematic Luttinger liquid state associated to bound states of Cooper pairs. We analyze the phase diagram and we discuss some experimental possibilities to observe this exotic phase.
We have measured the current-voltage characteristics of small-capacitance single Josephson junctions at low temperatures (T < 0.04 K), where the strength of the coupling between the single junction and the electromagnetic environment was controlled with one-dimensional arrays of dc SQUIDs. We have clearly observed Coulomb blockade of Cooper-pair tunneling and even a region of negative differential resistance, when the zero-bias resistance of the SQUID arrays is much higher than the quantum resistance h/e^2 = 26 kohm. The negative differential resistance is evidence of coherent single-Cooper-pa…
An all-superconductor charge qubit enabling a radio-frequency readout of its quantum state is described. The core element of the setup is a superconducting loop which includes the single-Cooper-pair (Bloch) transistor. This circuit has two functions: First, it operates as a charge qubit with magnetic control of Josephson coupling and electrostatic control of the charge on the transistor island. Secondly, it acts as the transducer of the rf electrometer, which probes the qubit state by measuring the Josephson inductance of the transistor. The evaluation of the basic parameters of this device sh…
When two superconductors are connected across a ferromagnet, the spin configuration of the transferred Cooper pairs can be modulated due to magnetic exchange interaction. The resulting supercurrent can reverse its sign across the Josephson junction (JJ) [1-4]. Here we demonstrate Josephson phase modulation in van der Waals heterostructures when Cooper pairs from superconducting NbSe$_2$ tunnel through atomically thin magnetic insulator (MI) Cr$_2$Ge$_2$Te$_6$. Employing a superconducting quantum interference device based on MI JJs, we probe a doubly degenerate non-trivial JJ phase ($ϕ$) origin…
We study inhomogeneous Cooper pairs distribution and localization effects in tree-like networks of superconducting islands coupled via Josephson weak links. Using a generalized Feynman's approach, reminiscent of the Bose-Hubbard model, we demonstrate that the Cooper pairs fraction which localizes on a specific network's island is limited by the network topology and, if present, by the repulsive interaction. These findings contribute to clarify the interplay between confinement effects induced by the network's topology and interaction and shed some light on recent experiments dealing with netwo…
Superconducting circuits incorporating Josephson tunnel junctions are widely used for fundamental research as well as for applications in fields such as quantum information and magnetometry. The quantum coherent nature of Josephson junctions makes them especially suitable for metrology applications. Josephson junctions suffice to form two sides of the quantum metrology triangle, relating frequency to either voltage or current, but not its base, which directly links voltage to current. We propose a five Josephson tunnel junction circuit in which simultaneous pumping of flux and charge results i…
The Hamiltonian operator for an unbiased array of Josephson junctions with gate voltages is constructed when only Cooper pair tunnelling and charging effects are taken into account. The supercurrent through the system and the pumped current induced by changing the gate voltages periodically are discussed with an emphasis on the inaccuracies in the Cooper pair pumping. Renormalisation of the Hamiltonian operator is used in order to reliably parametrise the effects due to inhomogeneity in the array and non-ideal gating sequences. The relatively simple model yields an explicit, testable predictio…
The anomalous proximity effect in dirty superconducting junctions is one of most striking phenomena highlighting the profound nature of Majorana bound states and odd-frequency Cooper pairs in topological superconductors. Motivated by the recent experimental realization of planar topological Josephson junctions, we describe the anomalous proximity effect in a superconductor/semiconductor hybrid, where an additional dirty normal-metal segment is extended from a topological Josephson junction. The topological phase transition in the topological Josephson junction is accompanied by a drastic chang…
We have fabricated a Cooper-pair transistor (CPT) with parameters such that for appropriate voltage biases, the sub-gap charge transport takes place via slow tunneling of quasiparticles that link two Josephson-coupled charge manifolds. In between the quasiparticle tunneling events, the CPT behaves essentially like a single Cooper-pair box (SCB). The effective capacitance of a SCB can be defined as the derivative of the induced charge with respect to gate voltage. This capacitance has two parts, the geometric capacitance, C_geom, and the quantum capacitance C_Q. The latter is due to the level a…
We discuss dephasing by the dissipative electromagnetic environment and by measurement in circuits consisting of small Josephson junctions. We present quantitative estimates and determine in which case the circuit might qualify as a quantum bit. Specifically, we analyse a three junction Cooper pair pump and propose a measurement to determine the decoherence time $τ_ϕ$.
Motivated by recent experiments, we analyze transport of Cooper pairs through a double-island Josephson qubit. At low bias in a certain range of gate voltages coherent superpositions of charge states play a crucial role. Analysis of the evolution of the density matrix allows us to cover a wide range of parameters, incl. situations with degenerate levels, when dissipation strongly affects the coherent eigenstates. At high noise levels the so-called Zeno effect can be observed, which slows down the transport. Our analysis explains certain features of the I-V curves, in particular the visibility …
Over the years, supersymmetric quantum mechanics has evolved from a toy model of high energy physics to a field of its own. Although various examples of supersymmetric quantum mechanics have been found, systems that have a natural realization are scarce. Here, we show that the extension of the conventional Cooper-pair box by a 4pi-periodic Majorana-Josephson coupling realizes supersymmetry for certain values of the ratio between the conventional Josephson and the Majorana- Josephson coupling strength. The supersymmetry we find is a "hidden" minimally bosonized supersymmetry that provides a non…
We analyze tunneling of Cooper pairs across voltage biased asymmetric single-Cooper-pair transistors. Also tunneling of Cooper pairs across two capacitively coupled Cooper-pair boxes is considered, when the capacitive coupling and Cooper pair tunneling are provided by a small Josephson junction between the islands. The theoretical analysis is done at subgap voltages, where the current-voltage characteristics depend strongly on the macroscopic eigenstates of the island(s) and their coupling to the dissipative environment. As the environment we use an impedance which satisfies Re[Z]<<R_Q and a f…
The Bloch-oscillating transistor (BOT) is an amplifier that utilizes semiclassical dynamics of states in energy bands under traveling quasimomentum. In a BOT, a single quasiparticle tunneling across a base tunnel junction switches the state of a superconducting tunnel junction to a lower Bloch band, triggering a series of resonant Cooper-pair tunnelings through an emitter Josephson junction (Bloch oscillations). Here, we investigate experimentally and theoretically an alternative realization of this device, based only on superconducting tunnel junctions. We discover new amplification schemes, …
We propose a scheme for enhancing the optomechanical coupling between microwave and mechanical resonators by up to seven orders of magnitude to the ultrastrong coupling limit in a circuit optomechanical setting. The tripartite system considered here consists of a Josephson junction Cooper-pair box that mediates the coupling between the microwave cavity and the mechanical resonator. The optomechanical coupling can be modified by tuning the gate charge and the magnetic flux bias of the Cooper-pair box which in turn affect the Josephson capacitance of the Cooper-pair box. We additionally show tha…
Not so long ago, thermoelectricity in superconductors was believed to be possible only by breaking explicitly the particle-hole symmetry. Recently, it has been theoretically predicted that a superconducting tunnel junction can develop bipolar thermoelectric phenomena in the presence of a large thermal gradient owing to non-equilibrium spontaneous PH symmetry breaking. The experimental realization of the first thermoelectric Josephson engine then followed. Here, we give a more extended discussion and focus on the impact of the Josephson contribution on thermoelectricity modulating the Cooper pa…
There exists a large number of experimental and theoretical results supporting the picture of "macroscopic qubits" implemented by nanoscopic Josephson junctions. On the other hand the standard model of such systems given in terms of a single degree of freedom suggests their semiclassical behavior due to a localization mechanism caused by a strong coupling to an environment. Indeed, such a mechanism is observed in an atomic Bose-Einstein condensate (BEC) placed in a double-well potential - a system mathematically equivalent to a Josephson junction. In this note it is shown, on the example of a …
We study theoretically electromagnetic radiation emitted by inelastic Cooper-pair tunneling. We consider a dc-voltage-biased superconducting transmission line terminated by a Josephson junction. We show that the generated continuous-mode electromagnetic field can be expressed as a function of the time-dependent current across the Josephson junction. The leading-order expansion in the tunneling coupling, similar to the "$P(E)$-theory", has previously been used to investigate the photon emission statistics in the limit of sequential (independent) Cooper-pair tunneling. By explicitly evaluating t…
We theoretically study the magnetization inside a normal metal induced in an s-wave superconductor/ferromagnetic metal/normal metal/ferromagnetic metal/s-wave superconductor (S/F1/N/F2/S) Josephson junction. Using quasiclassical Green's function method, we show that the magnetization becomes finite inside N. The origin of this magnetization is due to odd-frequency spin-triplet Cooper pairs formed by electrons of equal and opposite spins, which are induced by proximity effect in the S/F1/N/F2/S junction. We find that the magnetization M(d,q) in N can be decomposed into two parts, M(d,q)=MI(d)+M…
In this work we investigate the supercurrent in a hybrid topological Josephson junction consisting of two planes of topological insulator (TI) in a specific configuration, which allows both local (LAR) and crossed (CAR) Andreev processes at the interfaces with two conventional s-wave superconductors. We describe the effects of gate voltage and magnetic flux controls applied to the edge states of each TI. In particular, we demonstrate that the voltage gating allows the manipulation of the entaglement symmetry of non-local Cooper pairs associated to the CAR process. We establish a connection bet…
We discuss the properties of devices of small Josephson junctions in the light of the phase fluctuation theory and the energy band structure, which arises from the delocalization of the phase variable. The theory is applied in the realization of a mesoscopic amplifier, the Bloch oscillating transistor. The device characteristics and comparison with theory and simulations are discussed. The current gain of the device in a stable operating mode has been measured to be as high as 30. Measurements on input impedance and the power gain show that the BOT is an amplifier designed for middle-range imp…
