Non-Equilibrium Josephson and Andreev Current through Interacting Quantum Dots

We present a theory of transport through interacting quantum dots coupled to normal and superconducting leads in the limit of weak tunnel coupling. A Josephson current between two superconducting leads, carried by first-order tunnel processes, can be established by non-equilibrium proximity effect. Both Andreev and Josephson current is suppressed for bias voltages below a threshold set by the Coulomb charging energy. A $\pi$-transition of the supercurrent can be driven by tuning gate or bias voltages.


Introduction
Non-equilibrium transport through superconducting systems attracted much interest since the demonstration of a Superconductor-Normal-Superconductor (SNS) transistor [1]. In such a device, supercurrent suppression and its sign reversal (π-transition) are achieved by driving the quasi-particle distribution out of equilibrium by means of applied voltages [2,3,4,5]. Another interesting issue in mesoscopic physics is transport through quantum dots attached to superconducting leads. For DC transport through quantum dots coupled to a normal and a superconducting lead, subgap transport is due to Andreev reflection [6,7,8,9,10,11]. Also transport between two superconductors through a quantum dot has been studied extensively. The limit of a non-interacting dot has been investigated in [12]. Several authors considered the regime of weak tunnel coupling where the electrons forming a Cooper pair tunnel one by one via virtual states [13,14,15]. The Kondo regime was also addressed [13,16,17,18,19]. Multiple Andreev reflection through localized levels was investigated in [20,21]. Numerical approaches based on the non-crossing approximation [22], the numerical renormalization group [23] and Monte Carlo [24] have also been used. The authors of [25] compare different approximation schemes, such as mean field and second-order perturbation in the Coulomb interaction. In double-dot systems the Josephson current has been shown to depend on the spin state of the double dot [26]. Experimentally, the supercurrent through a quantum dot has been measured through dots realized in carbon nanotubes [27] and in indium arsenide nanowires [28].
In this Letter we study the transport properties of a system composed of an interacting single-level quantum dot between two equilibrium superconductors where a third, normal lead is used to drive the dot out of equilibrium. A Josephson coupling in SNS heterostructures can be mediated by proximity-induced superconducting correlations in the normal region. In case of a single-level quantum dot, superconducting correlations are indicated by the correlator d ↓ (0)d ↑ (t) , where d σ is the annihilation operator of the dot level with spin σ. To obtain a large pair amplitude, i.e. the equaltime correlator d ↓ d ↑ , at least two conditions need to be fulfilled: (i) the states of an empty and a doubly-occupied dot should be nearly energetically degenerate and (ii) the overall probability of occupying the dot with an even number of electrons should be finite. For a non-interacting quantum dot, i.e. vanishing charging energy U for double occupancy, this can be achieved by tuning the level position ǫ in resonance with the Fermi energy of the leads, ǫ = 0 [12]. In this case, the Josephson current can be viewed as transfers of Cooper pairs between dot and leads and the expression of the current starts in first order in the tunnel coupling strength Γ.
The presence of a large charging energy U ≫ k B T, Γ destroys this mechanism since the degeneracy condition 2ǫ+U ≈ 0 is incompatible with a finite equilibrium probability to occupy the dot with an even number of electrons. Nevertheless, a Josephson current can be established by higher-order tunnelling processes (see, for example, [13,14,15]), associated with a finite superconducting correlator d ↓ (0)d ↑ (t) at different times. The amplitude of the Josephson coupling is, however, reduced by a factor Γ/∆, i.e., the current starts only in second order in Γ, and the virtual generation of quasiparticles in the leads suppresses the Josephson current for large superconducting gaps ∆. In particular, it vanishes for ∆ → ∞.
The main purpose of the present paper is to propose a new mechanism that circumvents the above-stated hindrance to achieve a finite pair amplitude in an interacting quantum dot, and, thus, restores a Josephson current carried by first-order tunnel processes that survives in the limit ∆ → ∞. For this aim, we attach a third, normal, lead to the dot that drives the latter out of equilibrium by applying a bias voltage, so that condition of occupying the dot with an even number of electrons is fulfilled even for 2ǫ + U ≈ 0.
We relate the current flowing into the superconductors to the nonequilibrium Green's functions of the dot. In the limit of a large superconducting gap, ∆ → ∞, the current is only related to the pair amplitude. The latter is calculated by means of a kinetic equation derived from a systematic perturbation expansion within realtime diagrammatic technique that is suitable for dealing with both strong Coulomb interaction and nonequilibrium at the same time. H D + η=N,S L ,S R (H η + H tunn,η ). The quantum dot is described by the Anderson model

Model
is the number operator for spin σ =↑, ↓, ǫ is the energy level, and U is the charging energy for double occupation. The leads, labeled by η = N, S L , S R , are modeled by Here, V η are the spin-and wavevector-independent tunnel matrix elements, and c ηkσ (c † ηkσ ) and d σ (d † σ ) represent the annihilation (creation) operators for the leads and dot, respectively. The tunnel-coupling strengths are characterized by Γ η = 2π|V η | 2 k δ(ω − ǫ k ).

Current formula
We start with deriving a general formula for the charge current in lead η by using the approach of Ref. [29] generalized to superconducting leads. Similar formulae that relate the charge current to the Green's function of the dot in the presence of superconducting leads have been derived in previous works, in particular for equilibrium situations, see e.g. Refs. [16,22]. The formula derived below is quite general, as it allows for arbitrary bias and gate voltages, temperatures, and superconducting order parameters for a quantum dot coupled to an arbitrary number of normal and superconducting leads. For this, it is convenient to use the operators 3 indicate the Pauli matrices in Nambu space and e > 0 the electron charge. Evaluating the commutator leads to with V η = Diag(V η , −V * η ) and the lead-dot lesser Green's functions (G < D,ηk (ω)) m,n that are the Fourier transforms of i ψ † ηkn (0)φ m (t) . In the following, we assume the tunnelling matrix elements V η to be real (any phase of V η can be gauged away by substituting ∆ η → ∆ η exp(−2i arg V η )). The Green's function G < D,ηk is related to the full dot Green's functions and the lead Green's functions by a Dyson equation in Keldysh formalism: is the retarded (lesser) dot Green's function, and and g A(<) ηk (ω) the lead advanced (lesser) Green's function. Using this relation and assuming energy-independent tunnel rates Γ η , we obtain for the current where The two weighting functions D η (ω) and S η (ω) are given by The terms J 1η and J 2η involve excitation energies ω above and below the superconducting gap, respectively. For η = N, only the part of J 1η that involves normal (diagonal) components of the Green's functions contributes, and the current reduces to the result presented in [29]. For superconducting leads, this part describes quasiparticle transport that is independent of the superconducting phase difference. The other part of J 1η involves anomalous (off-diagonal) components of the Green's functions and is, in general, phase dependent. The contribution to the Josephson current stemming from this term is the dominant one in the regime considered in [13,14,15]. The excitation energies above the gap are only accessible either for transport voltages exceeding the gap or by including higher-order tunnelling, involving virtual states with quasiparticles in the leads, and, therefore, J 1η vanishes for large |∆ η |. In this case J 2η , that involves only anomalous Green's functions with excitation energies below the gap, dominates transport. It is, in general, phase dependent, and describes both Josephson as well as Andreev tunnelling.
In the following we consider the limit |∆ η | → ∞, where the current simplifies to with Φ η being the phase of ∆ η and d ↓ d ↑ = | d ↓ d ↑ | exp(iΨ) the pair amplitude of the dot that has to be determined in the presence of Coulomb interaction, coupling to all (normal and superconducting) leads and in non-equilibrium due to finite bias voltage. We now consider a symmetric three-terminal setup with Γ S L = Γ S R = Γ S , ∆ S L = |∆| exp(iΦ/2) and ∆ S R = |∆| exp(−iΦ/2), and µ S L = µ S R = 0. The quantities of interest are the the current that flows between the two superconductors (Josephson current) J jos = (J S L − J S R )/2 and the current in the normal lead (Andreev current) Furthermore, we focus on the limit of weak tunnel coupling, Γ S < k B T . In this regime, an Josephson current through the dot in equilibrium would be suppressed even in the absence of Coulomb interaction, U = 0, since the influence of the superconductors on the quantum-dot spectrum could not be resolved for the resonance condition ǫ ≈ 0. This can, e.g., be seen in the exactly-solvable limit of U = 0 together with Γ N = 0, where the Josephson current is J jos = (e/2h) S cos 2 (Φ/2). This provides an additional motivation to look for a non-equilibrium mechanism to proximize the quantum dot.

Kinetic equations for quantum-dot degrees of freedom
The Hilbert space of the dot is four dimensional: the dot can be empty, singly occupied with spin up or down, or doubly occupied, denoted by |χ ∈ {|0 , | ↑ , For convenience we define the detuning as δ = E D − E 0 = 2ǫ + U. The dot dynamics is fully described by its reduced density matrix ρ D , with matrix elements P χ 1 χ 2 ≡ (ρ D ) χ 2 χ 1 . The dot pair amplitude d ↓ d ↑ is given by the off-diagonal matrix element P 0 D . The time evolution of the reduced density matrix is described by the kinetic equations We define the generalized transition rates by W , which are the only quantities to be evaluated in the stationary limit. Together with the normalization condition χ P χ = 1, (5) determines the matrix elements of ρ D . Furthermore, in (5) we retain only linear terms in the tunnel strengths Γ η and the detuning δ. Hence, we calculate the rates W to the lowest (first) order in Γ η for δ = 0. This is justified in the transport regime Γ S , Γ N , δ < k B T .
The rates are evaluated by means of a real-time diagrammatic technique [30], that we generalize to include superconducting leads. This technique provides a convenient tool to perform a systematic perturbation expansion of the transport properties in powers of the tunnel-coupling strength. In the following, we concentrate on transport processes to first order in tunnelling (a generalization to higher orders is straightforward). This includes the transfer of charges through the tunnelling barriers as well as energy-renormalization terms that give rise to nontrivial dynamics of the quantum-dot degrees of freedom.
We find for the (first-order) diagonal rates W χ 1 χ 2 ≡ W χ 1 χ 2 The N lead also contributes to the rates W DD 00 = (W 00 , with µ N being the chemical potential of the normal lead and ψ(z) the Digamma function. Notice that B vanishes when µ N = 0 or U = 0. The superconducting leads do not enter here due to the gap in the quasi-particle density of states. These leads, though, contribute to the off-diagonal rates W 00 For an intuitive representation of the system dynamics we define, in analogy to [31], a dot isospin by From (5), we find that in the stationary limit the isospin dynamics can be separated into three parts, 0 = dI/dt = (dI/dt) acc + (dI/dt) rel + (dI/dt) rot , with whereê z is the z-direction and B eff = {2Γ S cos(Φ/2), 0, −Γ N B − 2ǫ − U} is an effective magnetic field in the isospin space. The accumulation term (7) builds up a finite isospin, while the relaxation term (8) decreases it. Finally, (9) describes a rotation of the isospin direction.

Non-equilibrium Josephson current
In the isospin language the current in the superconducting leads is where the upper(lower) sign refers to the left(right) superconducting lead. The I y component contributes to the Andreev current, while I x is responsible for the Josephson current. To obtain subgap transport, we first need to build up a finite isospin component along the z-direction, i.e. we need a population imbalance between the empty and doubly occupied dot [(this is generated by the accumulation term in (7)]; second, we need a finite B eff which rotates the isospin so that it acquires an inplane component. In order to have a finite Josephson current (I x = 0), we need the z-component, −Γ N B − 2ǫ − U, of the effective magnetic field producing the rotation to be non zero. The Josephson current and the Andreev current read These results take into account only first-order tunnel processes, i.e. the rates W The considerations above establish the importance of the non-equilibrium voltage to induce and control proximity effect in the interacting quantum dot. In figure 2 we show in a density plot (a) J jos and (b) J and for Φ = π/2 as a function of the voltage µ N and the level position ǫ. Both the control of proximity effect by the chemical potential µ N and the voltage driven π-transition are clearly visible. If the detuning is too large, |δ + Γ N B| > Γ 2 N + 4Γ 2 S cos 2 (Φ/2), it becomes difficult to build a superposition of the states |0 and |D , which is necessary to establish proximity. As a consequence, the Josephson and the Andreev current are algebraically suppressed by δ −1 and δ −2 , respectively. Figure 3 shows the Josephson current as a function of δ = 2ǫ + U. The fact that the Josephson current is non zero for δ = 0 is due to the term Γ N B, i.e. of the interaction induced contribution to the z-component of the effective field B eff acting on the isospin. The term |B| has a maximum at µ N = U/2, which causes this effect to be more pronounced at the onset of transport. The fact that the value of the Josephson current varies on a scale smaller than temperature indicates its nonequilibrium nature.
A π-transition of the Josephson current can also be achieved by changing the sign of δ +Γ N B, as shown in figure 4 where J jos is plotted as a function of the phase difference Φ for different values of the level position. Notice that the current for δ = 0 (ǫ = −U/2) is different from zero only due to the presence of the term Γ N B acting on the isospin.

Conclusion
In conclusion, we have studied non-equilibrium proximity effect in an interacting single-level quantum dot weakly coupled to two superconducting and one normal lead. We propose a new mechanism for a Josephson coupling between the leads that is qualitatively different from earlier proposals based on higher-order tunnelling processes via virtual states. Our proposal relies on generating a finite non-equilibrium pair amplitude on the dot by applying a bias voltage between normal and superconducting leads. The charging energy of the quantum dot defines a threshold bias voltage above which the non-equilibrium proximity effect allows for a Josephson current carried by first-order tunnelling processes, that is not suppressed in the limit of a large superconducting gap. Both the magnitude and the sign of the Josephson current are sensitive to the energy difference between empty and doubly-occupied dot. A πtransition can be driven by either bias or gate voltage. In addition to defining a threshold bias voltage, the charging energy induces many-body correlations that affect the dot's pair amplitude, visible in a bias-voltage-dependent shift of the π-transition as a function of the gate voltage.