Cooper pair splitting in a nanoSQUID geometry at high transparency

R. Jacquet,1 J. Rech,1 T. Jonckheere,1 A. Zazunov,2 and T. Martin1

1 Aix-Marseille Universite, Universite de Toulon, CPT, UMR 7332, France and2 Institut fur theoretische physik, Heinrich Heine Universitat, D-40225 Dusseldorf, Germany

(Dated: August 27, 2018)

We describe a Josephson device composed of two superconductors separated by two interactingquantum dots in parallel, as a probe for Cooper pair splitting. In addition to sequential tunneling ofelectrons through each dot, an additional transport channel exists in this system: crossed Andreevreflection, where a Cooper pair from the source is split between the two dots and recombined inthe drain superconductor. Unlike non-equilibrium scenarios for Cooper pair splitting which involvessuperconducting/normal metal “forks”, our proposal relies on an Aharonov-Bohm measurement ofthe DC Josephson current when a flux is inserted between the two dots. We provide a path integralapproach to treat arbitrary transparencies, and we explore all contributions for the individual phases(0 or π) of the quantum dots. We propose a definition of the Cooper pair splitting efficiency for arbi-trary transparencies, which allows us to find the phase associations which favor the crossed Andreevprocess. Possible applications to experiments using nanowires as quantum dots are discussed.

PACS numbers: 74.50.+r 85.25.Dq 73.21.La

I. INTRODUCTION

In superconductors, Cooper pairs provide a naturalsource of entanglement1 in both spin and momentumspace. If for some reason the constituent electrons ofthis pair are separated and are allowed to propagate intwo different metallic leads or nanodevices such as quan-tum dots, one expects that the entanglement is preservedbecause the tunneling processes are spin preserving. Thisprocess is called cross Andreev reflection (CAR), and hasbeen the focus of both theoretical and experimental in-vestigations in the last two decades. The initial man-ifestation of CAR was proposed theoretically for non-local current,2–7 as well as for non-equilibrium noisecross-correlations (current-current fluctuations betweenthe superconductor and the two outgoing devices).8–15

Indeed, the scattering formalism of electron and holetransport showed that noise cross-correlations could bepositive. Strictly speaking the positive cross-correlationsignal does not constitute a rigorous proof of electronsentanglement originating from superconductors, but itcertainly provides evidence of Cooper pair splitting. Al-ternative approaches exploiting electron energy filteringusing Coulomb blockade confirmed at the same time thenonlocal spin singlet nature of electron pairs in oppositequantum dots placed in the vicinity of a superconductor,exploiting T matrix calculations.6,16

The possibility of generating nonlocal entangled pairsof electrons in condensed matter settings bears funda-mental applications in the context of quantum informa-tion theory. Tests of quantum entanglement followedthese works, based on the idea that Bell inequality viola-tion measurements could be implemented via noise cross-correlations with the so-called superconducting Cooperpair splitter.17,18 Moreover, these ideas have been ap-plied to the paradigm of quantum teleportation,19,20 aswell as in other applications.21,22

On the experimental side, attention has mainly fo-

cused on nonlocal current measurement on the Cooperbeam splitter,23 a device where typically the supercon-ducting source of electrons is connected to two leads,sometimes via embedded quantum dots.24–27 Under spe-cific gate voltages imposed on such dots, it is possibleto trigger electronic transport in the two outgoing con-ductors. Only a single experiment managed to measurepositive noise cross-correlations when the source of elec-trons was rendered superconducting.28

The main challenge with these proposals resides in thefact that they rely on non-equilibrium measurements.Strictly speaking, these measurements, although to besaluted, constitute only an indirect evidence of Cooperpair splitting, while noise cross correlation measurementsrepresent a considerable ordeal due to the poor signal tonoise ratio, and no attempt has been tried so far to re-produce them.

A seminal theoretical work has been suggested early onto circumvent these difficulties by proposing a Joseph-son equilibrium current geometry to test Cooper pairsplitting.29 It describes two superconductors (with ap-plied phase difference) separated by two quantum dotsplaced in parallel. When a Cooper pair is transmittedfrom one superconductor to the other, the two electronscan either pass both through a given dot, or they cantransit through different dots (cf. Fig. 1). This indeedrealizes an Aharonov-Bohm (AB) experiment with su-perconductors as source and drain, driven by an appliedphase difference. The critical current as a function of theAB flux should be π periodic if electrons are not split be-tween the two dots, and 2π periodic if Cooper pair split-ting is effective. The clear originality of this proposalresides in the fact that unlike non-equilibrium noise se-tups, here Cooper pair splitting is uncovered using a cur-rent measurement at equilibrium albeit in a Josephsongeometry. The calculation was performed perturbativelyin the tunneling Hamiltonian, with infinite repulsion onthe dots. Dot gate voltages insured that on average each

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Figure 1. Illustration of the three different possibilities for thetransmission of a Cooper pair from the left superconductor tothe right one. The two electrons of the pair can either beboth transmitted through the upper dot (top left), throughthe lower dot (top right), or the Cooper pair can be split withone electron transmitted through each dot (bottom).

dot was occupied by a single electron. A complementarystudy appeared a decade later with the same setup30 andperturbative results for dot levels which were assumed tobe above the superconducting chemical potentials andwith a finite Coulomb repulsion were also presented.

First, so far no analysis of this superconductingAharonov-Bohm effect has allowed to go beyond lowestorder perturbation theory in the tunneling Hamiltonian.Advances in superconducting device fabrication31 seemto indicate that by burying nanowires underneath super-conductors, large transmission can be achieved betweenthe resulting quantum dot and the lead. Treating thetunnel coupling to all orders of perturbation theory thusconstitutes a first motivation of our study.

Secondly, it is established that when a quantum dot isembedded in a Josephson junction, away from the Kondoregime, the strength of the on-site repulsion, the couplingto the leads and the level position of the dot determinewhether it constitutes a 0 or a π junction (positive ornegative Josephson amplitude). This has been analyzedperturbatively,32 as well as with path integral formalism,followed by a saddle point approximation.33 This latterwork allows to distinguish between three phases of theJosephson junction: a) the π phase (where the dot issingly occupied); b) the 0(0) phase (where it is unoccu-pied); c) and the 0(2) phase (with double occupation).These predictions have been verified experimentally adecade ago.34 In the Josephson setup with two dots inparallel which is studied here, each dot can either be inthe 0(0), the 0(2) or the π phase. In the works of Ref. 29the two electrons are both in a π junction configurationwhile the work of Ref. 30 have also considered the casewhere both dots are in the 0(0) phase. However, there isto this date no systematic or comparative study specify-ing which combination of the dot phases may enhance orreduce the AB signal for maximal observation, even lessso for arbitrarily large transparencies.

The two above points constitute the main motivationsof the present paper. In this work, we employ the path

integral formalism to model the AB setup without anyrestrictions on the transmission properties of the sam-ple. Indeed, we provide results both in the tunneling andthe high transparency regimes, and we propose a way tomeasure the efficiency of CAR processes in both cases,by analyzing the AB signal. We find that the CAR pro-cesses are optimized when the two quantum dots are inthe same phase.

The paper is organized as follows. In Sec. II we pro-vide a description of the device and of the model withwhich we describe it, and we give an expression of thepartition function in terms of Grassmann variables. Thefree energy used to derive the Josephson current of thisnanoSQUID in a non-perturbative manner is presentedin Sec. III. In Sec. IV, we discuss the possible occupancystates of the dots. We propose the definition of a Cooperpair splitting efficiency in Sec. V which can be computedfor arbitrary transparencies of the studied junctions. TheAB signals are calculated first in the tunneling regimein Sec. VI and then in the non-perturbative regime inSec. VII and all possible phase associations for the dotsare considered. We discuss our results in Sec. VIII.

II. MODEL AND PARTITION FUNCTION

The device is illustrated in Fig. 2. For simplicity, thetwo leads consist of the same superconducting materialwith a controllable phase difference which can either beimposed by closing the device in a loop geometry or em-bedding the device in a macroscopic SQUID.29,30 Twoquantum dots are placed in parallel in the nanogap be-tween the two electrodes, and a magnetic flux (which is inprinciple independent from the one imposed to trigger aDC Josephson signal between the electrodes) threads thearea between the two dots. Electrons can tunnel from thesource electrode to the upper or lower dot, but on siteCoulomb repulsion favors zero, or single occupation onthe latter. In the presence of a magnetic flux between thetwo dots, because of the AB effect, different phase shiftsare expected between the two paths that an electron canfollow to reach the drain electrode. If the separation be-tween injection points is larger than the coherence length,Cooper pairs as a whole pass either by the upper or lowerpath, as in the work of Ref. 31. If the injection pointsare closer together than the superconducting coherencelength however, the two electrons of the pair can travelthrough opposite dots, realizing a CAR process, whichgives a third contribution to the Josephson current. Byadjusting the quantum dot energy levels and Coulombinteraction, we expect to filter the electrons and eventu-ally favor the CAR process. Granted, we cannot claim todirectly measure the degree of entanglement of this CARcontribution (like in a Bell inequalities measurement),nevertheless, the recombination of these two electrons onthe destination superconductor could not be possible ifthis outgoing state did not form a Cooper pair, i.e. anentangled state.

3

tL e-i(ϕ-α)/4 tR e-i(ϕ-α)/4

tL e-i(ϕ+α)/4 tR e-i(ϕ+α)/4

L R

U

D

Figure 2. Path dependent phase shifts.

We denote by d†aσ the creation operator for an electronwith spin σ =↑, ↓ on the quantum dot a = U,D and by

ψ†jkσ the creation operator for an electron with momen-tum k and spin σ =↑, ↓ in the superconductor j = L,R.It is convenient to introduce the Nambu spinors

da =

(da↑d†a↓

)and ψjk =

(ψjk,↑ψ†j(−k),↓

). (1)

σi (i = x, y, z) are the Pauli matrices that act in Nambuspace. The Hamiltonian of the double Josephson junctionreads

H =∑a=U,D

Ha +∑j=L,R

Hj +Ht. (2)

Ha is the Hamiltonian of the quantum dot a = U,D,characterized by its energy level εa and its on-siteCoulomb repulsion Ua:

Ha = εa∑σ=↑,↓

d†aσdaσ + Uana↑na↓ , (3)

with naσ = d†aσdaσ the dot occupation operator per spin.Hj is the Hamiltonian of the superconductor j = L,R,with gap ∆ and chemical potential µ:

Hj =∑k

ψ†jk (ξk σz + ∆σx) ψjk, ξk =k2

2m− µ. (4)

Here ∆ is real, the phase difference between electrodeshas been gauged out and instead appears in the tunnel-ing Hamiltonian. If we denote by rja the location of theinjection point from lead j to dot a, the tunneling Hamil-tonian Ht reads

Ht =∑jka

eik.rja ψ†jk Tja da + h.c. (5)

The tunneling matrices involved in Eq. (5) read

TLU = tL σz e+iφ−α4 σz , TRU = tR σz e−iφ−α

4 σz , (6a)

TLD = tL σz e+iφ+α4 σz , TRD = tR σz e−iφ+α

4 σz . (6b)

φ is the phase difference between the superconductorswhile α is related to the magnetic flux Φ inside the

SQUID loop: α = 2π ΦΦ0

where Φ0 = h/e is the flux quan-tum. For clarity, the phase shifts acquired by tunnelingelectrons are indicated in Fig. 2.

We employ a path integral approach in the Matsubaraformalism in order to compute the partition function ofthe device. We introduce then the eigenvalues of the

annihilation operators ψjkσ and daσ written as ψjkσ anddaσ respectively. These are Grassmann variables and weconsider also their conjugates ψjkσ and daσ as well as thecollections in Nambu spinors da and ψjk defined in thesame way as Eq. (1).

The partition function is given by a functional integra-tion over paths that are β antiperiodic:

Z =

∫da(β)=−da(0)ψjk(β)=−ψjk(0)

D[d, d, ψ, ψ

]exp

[−SE

(d, d, ψ, ψ

)]. (7)

The Euclidean action SE reads

SE(d, d, ψ, ψ

)=

β∫0

dτ

{H(d, d, ψ, ψ

)+∑a

da∂τda +∑jk

ψjk ∂τψjk

}(8)

where the matrix elements of the Hamiltonian can bewritten as

H(d, d, ψ, ψ

)=∑a

Ha

(da, da

)+∑jk

Hjk

(ψjk, ψjk

)+∑jka

Ht,jka

(da, da, ψjk, ψjk

). (9)

The expressions of Hjk and Ht,jka are readily obtainedfrom Eqs. (4)-(5) by substituting the annihilation opera-tors a by their eigenvalues a and the corresponding cre-ation operators a† by the conjugate Grassmann variablesa. For the quantum dots, we can also find an expressionin terms of Nambu spinors as follows

Ha

(da, da

)= εa + εa da σz da −

Ua2

(dada

)2, (10)

with εa = εa + Ua2 .

III. FREE ENERGY AND JOSEPHSONCURRENT

As the lead degrees of freedom are quadratic in theHamiltonian, they can be easily integrated out. The par-tition function is then expressed as a functional integralover the dot Grassmann variables:

Z = c1

∫da(β)=−da(0)

∏a

D[da, da

]exp

[−Seff

(d, d)]

, (11)

where c1 is the determinant arising from the integrationof the lead variables, which is independent of φ and α.The effective action in Eq. (11) reads

4

Seff

(d, d)

=∑a

β∫0

dτ

{εa + da(τ) [∂τ12 + εa σz] da(τ)− Ua

2

(da(τ) da(τ)

)2}−∑a,b

β∫0

dτ

β∫0

dτ ′ da(τ) Σab(τ − τ ′) db(τ ′)

(12)

where the self-energy term

Σab(τ) =∑jk

eik.(rjb−rja)T †jaGk(τ)Tjb (13)

involves the Green function of the leads Gk(τ) which ver-ifies

[∂τ12 + ξkσz + ∆σx]Gk(τ) = δ(τ)12. (14)

The quartic terms (dada)2 in Eq. (12) prohibit an exactcomputation of the partition function. As in Ref. 33,we use a Hubbard-Stratonovich transformation to treat

these terms and we neglect the temporal fluctuations ofthe two auxiliary fields XU and XD which are introduced:

eUa2

β∫0

dτ(dada)2

≈√

β

2πUa

∫ +∞

−∞dXa e

− β2Ua

X2a+Xa

β∫0

dτ dada.

(15)

Because both the Green functions Gk and the Nambuspinor components daσ are β antiperiodic, we use a Mat-subara series expansion F (τ) =

∑p∈Z e−iωpτF (ωp) over

the frequencies ωp =(p+ 1

2

)2π/β.

Rather than keeping track of a cumbersome device-specific position dependence of the self-energy, we choose tointroduce a phenomenological parameter η. It extrapolates between the two most relevant cases: η = 0 for infinitelydistant injection points (much more separated than the superconducting coherence length) and η = 1 for coincidinginjection points (much closer than the superconducting coherence length). The dots are now integrated out and thepartition function becomes

Zη(φ, α) = c1c2

∫ +∞

−∞dXU

∫ +∞

−∞dXD exp

[−SHS,η

eff (XU , XD, φ, α)]

(16)

where c2 is a fermionic determinant arising from the dot variables integration. The effective action reads

SHS,ηeff (XU , XD, φ, α) =

∑a

(β εa +

β

2UaX2a

)− 2

∑p∈N

ln(β4∣∣∣det

[Mη

p (XU , XD, φ, α)]∣∣∣), (17)

Mηp (XU , XD, φ, α) =

[− (iωp +XU )12 + εU σz −Ap(φ− α) −Bηp(φ,+α)

−Bηp(φ,−α) − (iωp +XD)12 + εD σz −Ap(φ+ α)

], (18)

Ap(φ) =Γ√

∆2 + ω2p

[iωp 12 −∆

(cos

φ

2σx + γ sin

φ

2σy

)], (19)

Bηp(φ, α) = ηΓ√

∆2 + ω2p

[iωp

(cos

α

212 + iγ sin

α

2σz

)−∆

(cos

φ

2σx + γ sin

φ

2σy

)]. (20)

The decay rate is defined as Γ = π ν(0)(t2L + t2R) where ν(0) is the density of states of the leads at the Fermi level.The contact asymmetry is given by γ = (t2L − t2R)/(t2L + t2R).

The CAR process is due to the off-diagonal terms Bpof the matrices Mp the determinants of which we haveto compute as a result of the Gaussian integrals overquantum dot degrees of freedom. In practice, these off-diagonal terms depend on the separation between injec-

tion points R ≡ |rjU − rjD|. They have an exponen-tial decay on the scale of the superconducting coherencelength. With the microscopic tunneling Hamiltonian for-mulation of Eq. (5), they also bear fast oscillations onthe Fermi wavelength, and possibly power law decay de-

5

pending on the dimensionality [η(R) = (sin kFR)/(kFR)for 3D superconductors29,35 and no power law decayfor quasi-one dimensional superconductors36]. ExistingCAR experiments24–27 based on nanowire quantum dotsembedded in the superconducting leads are typically per-formed for an injection separation which is less thanthe superconducting coherence length. Yet these experi-ments find no evidence of either Fermi wavelength oscil-lations or power law decay at all: the nonlocal signal isstrong and can be optimized by tuning the dot gate volt-ages. This can be attributed to the proximity effect fromthe bulk superconductors, which acts on the nanowiresused in the experiments. To keep our discussion moregeneral, and avoid such device-specific complications, inthis work, we use η as a phenomenological parameter, asin Refs. 30 and 37. Most of the results will be displayedfor the extreme values η = 0 and η = 1, but in orderto show the evolution of the AB signals, we sometimesallow it to vary smoothly between these two values.

To evaluate the partition function of Eq. (16),we use a saddle-point method.33 The effective action

SHS,ηeff (XU , XD, φ, α) is computed numerically by sum-

ming over Matsubara frequencies (up to a cut-off muchlarger than the superconducting gap). Its minimum, lo-cated in [X∗U (φ, α), X∗D(φ, α)], is obtained with a gradi-ent descent method for fixed φ and α (the number ofstarting points of the algorithm depends on the sym-metry of the function to be minimized). The freeenergy is then defined from this minimum value as

SHS,ηeff (X∗U (φ, α), X∗D(φ, α), φ, α) ≡ βF η(φ, α). The cur-

rent is finally obtained by differentiating the free energywith respect to the phase difference φ:

Jη(φ, α) = 2 ∂φFη(φ, α). (21)

The critical current, function of the flux α, is defined as

Jηc (α) = maxφ |Jη(φ, α)| . (22)

For η = 0, it is π periodic. Indeed, in this case, thereare no CAR processes and for a magnetic flux α, thecurrent characteristic (current as a function of the phasedifference φ) through the double junction is simply thesum of the current characteristics across the independentsingle junctions, one being shifted by −α, the other oneby +α. Shifting α by π results in adding a phase shift−π for one of the two junctions [Ap(φ − α) → Ap(φ −α − π) in the top left block of Eq. (18)] and a phaseshift +π for the other one [Ap(φ + α) → Ap(φ + α + π)in the bottom right block of Eq. (18)]. Using the 2πperiodicity in φ of the current through a single junction,and taking the maximum on φ shows immediately thatthe critical current in unchanged when α→ α+π. This isin contrast to the case with η = 1, i.e. in the presence ofCAR processes, where the off-diagonal terms of Eq. (18)implies that only the 2π periodicity of the critical currentcan be observed.

Figure 3. Mean occupation number diagram of the quantumdot of a single Josephson junction for symmetric couplings(tL = tR) and for Γ = ∆, β = 50∆−1.

IV. 0−π TRANSITION IN A SINGLEJOSEPHSON JUNCTION

As a first step, let us recall some known results con-cerning a single dot embedded in a Josephson junction.In this section, we summarize the properties of such asetup, and we determine under what condition the junc-tion is in the 0(0), the π or the 0(2) phase. The 0phase is characterized by a positive Josephson currentfor φ ∈ [0, π]. It can be further separated between a 0(0)

phase where the dot is almost empty and a 0(2) phasewhere the mean occupation number of the dot is almost2. The π phase is associated with a negative Joseph-son current for φ ∈ [0, π], and corresponds to a singlyoccupied dot.

In perturbative calculations, the Josephson currentflowing through the quantum dot is the result of thetunneling of Cooper pairs, which requires a fourth or-der perturbative expansion in the tunneling amplitudesbetween a superconductor and the quantum dot. At thisorder, the current can be written as J = J0 sinφ, whereφ is the phase difference between the superconductors,and the sign of J0 determines the 0 or π character of thejunction. Providing a single occupancy on the quantumdot (0 < −ε � U) J0 can be negative32 unlike the caseof an empty quantum dot.

This phenomenon has been investigated numerically33

for arbitrary transmissions between the dot and the su-perconducting leads. For a fixed quantum dot energylevel ε < 0 and a fixed decay rate Γ, the current as afunction of the phase difference φ between the super-conductors undergoes a discontinuity as one tunes theCoulomb interaction U across a critical value. Comput-ing the mean occupation number on the quantum dot inthe (−ε, U) plane reveals the presence of all three phases,as shown in Fig. 3, where the π phase, which lies aroundthe line 2ε+ U = 0, separates the 0(0) and 0(2) phases.

6

V. SPLITTING EFFICIENCY

In order to identify the optimal regime for observingsignatures of CAR processes in the AB signal, we needto define a splitting efficiency. We start by discussing thecase of low electron transmission, where intuition can begained from simple perturbation theory. We then aim atcomparing the results of our general approach for arbi-trary transmission to those perturbative results for theAB signal.

The fourth order perturbation expansion29,30 in thetunneling amplitudes tj allows us to write the Josephsoncurrent as three contributions, associated with the threedifferent processes illustrated in Fig. 1, so that

J(φ, α) = ID sin (φ+ α) + IU sin (φ− α) + ICAR sinφ.(23)

Indeed, in the presence of a magnetic flux α 6= 0, an ad-ditional phase shift is acquired, depending on the paththat each electron of the Cooper pair followed betweenthe two superconducting leads. When the Cooper pairtunnels through the U (resp. D) quantum dot, it ac-cumulates a phase shift −α (resp. +α) in addition tothe superconducting phase difference and contributes tothe Josephson current with an amplitude IU (resp. ID).However, when the Cooper pair is delocalized on the twoquantum dots (via the CAR process, thus contributingwith an amplitude ICAR), one electron gets a phase shift+α

2 while the other one gets a phase shift −α2 , the pairaccumulating as a result no additional phase shift.

The critical current associated with the Josephson cur-rent Eq. (23) then reads

Jc(α) = I0√

1 + a cosα+ b cos 2α (24)

with I0 =√I2U + I2

D + I2CAR, I2

0 a = 2ICAR (IU + ID)and I2

0 b = 2IDIU . For low enough values of Γ, we arein the tunneling regime and the approximation Eq. (23)for the Josephson current is justified. If we are able toextract the parameters IU , ID and ICAR, e.g. from a fitof the AB signal, we can calculate the quantity

rt =I2CAR

I2U + I2

D + I2CAR

(25)

which encodes the splitting efficiency of the doubleJosephson junction. It varies between 0 and 1. rt = 0corresponds to a low efficiency of the CAR process whilert = 1 is obtained when this process of spatial delocal-ization of the two electrons of a Cooper pair is muchmore important than the tunneling processes of a wholeCooper pair through a single quantum dot.

As already stressed out, the formalism developed inSecs. II-III is valid regardless of the strength of thecoupling between the superconductors and the quantumdots. However, the above definition of the splitting ef-ficiency relies on the expression Eq. (24) for the criticalcurrent which is no longer valid in the non-perturbativeregime. There, one needs an alternative diagnosis for the

detection of the CAR process. As it turns out, a relevantquantity can be extracted from the mean powers of thecritical current obtained for infinitely distant injectionpoints (η = 0) and for coinciding ones (η = 1). Indeed,

defining Pη =∫ 2π

0dα [Jηc (α)]

2, we compute the quantity

r =|Pη=1 − Pη=0|Pη=1

. (26)

This generalizes the concept of splitting efficiency to thecase of arbitrary transmission, and coincides with thedefinition of Eq. (25) in the tunneling regime.

VI. NANOSQUID IN THE TUNNELINGREGIME

We first focus on the tunneling regime, taking a lowvalue of the decay rate Γ = 0.01∆. We choose to exploreall the possible combinations for the phases of the twoquantum dots (see Figs. 4-5). The results were obtainedfor symmetric couplings γ = 0, at temperature β−1 =0.002∆. The energies of the quantum dots are: ε =−0.3∆ for the π phase, ε = 0.3∆ for the 0(0) phase andε = −0.9∆ for the 0(2) phase. The Coulomb interactionU is chosen to be the same for the two quantum dots,and within a specific range as staying in a given phase atfixed energy restricts the possible values for U .

The particle-hole symmetry ensures that the currentis invariant under the change (εU , εD) → (−εU ,−εD).For the values of energy ε mentioned above, this impliesthat the critical current is identical for 0(0) − 0(0) and0(2)− 0(2) phase associations with U = 0.6∆, for π− 0(0)

and π − 0(2) phase associations again with U = 0.6∆,and finally for 0(0) − 0(0) at U = 0.7∆ and 0(2) − 0(2) atU = 0.5∆.

We first consider the critical current curves of a SQUIDmade of two independent single Josephson junctions, i.e.

Figure 4. Critical current (in units of 104e∆/~) curves forsymmetric associations of dots in the tunneling regime: π−π(top panel), 0(0) − 0(0) (middle panel), 0(2) − 0(2) (bottompanel).

7

Figure 5. Critical current (in units of 104e∆/~) curves forasymmetric associations of dots in the tunneling regime: π−0(0) (top panel), π − 0(2) (middle panel), 0(0) − 0(2) (bottompanel).

for η = 0. In this particular case, ICAR = 0 and con-sequently a = 0 in Eq. (24), so that the extrema ofthe critical current are the zeros of sin 2α, and the totalJosephson current is given by Eq. (23) with ICAR = 0.From the results of Figs. 4 and 5, it clearly appears thatthere is a π/2 phase shift in the critical current betweenthe situation where the two dots are in the same phase(0 − 0 or π − π) and the one where they are in differ-ent phases (π − 0). More specifically, for two quantumdots in the same phase (IDIU > 0), there is no phaseshift between the currents of the two single Josephsonjunctions for α = 0, so that the maxima are added(|ID + IU | = |ID| + |IU |) and, as a result, the criticalcurrent is maximal for α = 0. However, for two quantumdots in different phases (IDIU < 0), the phase shift ofπ between the currents of the two single Josephson junc-

Figure 6. Influence of the parameter η on the critical current(in units of 104e∆/~) in the tunneling regime (U = 0.6∆).

tions for α = 0 is compensated by a phase shift α = +π/2for one of the currents and a phase shift −α = −π/2 forthe other one (|ID − IU | = |ID| + |IU |) resulting in amaximum of the critical current for α = π/2. Such abehavior has been observed experimentally.31

Comparing the left panels of Figs. 4-5 to the right ones(i.e. the case η = 0 to η = 1), we immediately obtainevidence of the cross-talk between the two single Joseph-son junctions: the period of the critical current doubles.This is a signature of the emergence of the CAR process.Note that the symmetric associations (cf. Fig. 4) differcompletely between η = 0 and η = 1: for the π − π as-sociation, the maximum at α = 0 for η = 0 becomes aminimum for η = 1, and for the 0(0)−0(0) and 0(2)−0(2)

associations, the maximum at α = π for η = 0 becomes azero for η = 1. Concerning the asymmetric associations(cf. Fig. 5), the critical current curves for η = 1 somehowlook like those obtained for η = 0: the positions of themaxima and minima are mostly preserved for all phaseassociations, only their local or global character changeswhen tuning η.

Increasing U results in a more pronounced filtering asthe processes where the quantum dots are doubly occu-pied are less favored. This explains the observed decreasein critical current for the 0(0) − 0(0) and π − π associa-tions (Fig. 4). The opposite behavior happens when thequantum dots are doubly occupied, i.e. for the 0(2)−0(2)

association. There, increasing U favors processes wherethe occupation of the dots is lowered, since approachingthe π transition results in the decrease of the mean occu-pation number on the quantum dot. Such opposite be-haviors while tuning U for 0(0)−0(0) and 0(2)−0(2) phaseassociations can be seen as a consequence of the alreadydiscussed particle-hole symmetry. Similarly, the π − 0(0)

and π − 0(2) combinations have opposite interaction de-pendence (see Fig. 5). Interestingly, the interaction hasno noticeable effect in the case of the 0(0) − 0(2) phaseassociation.

Figure 7. Splitting efficiency r given by Eq. (26) extractedfrom the curves of Figs. 4 and 5.

8

In order to monitor the emergence of CAR processes,we investigate in Fig. 6 intermediate regimes for the π−π,0(2) − 0(2), π − 0(2) and 0(0) − 0(2) phase associationsby turning on η progressively from 0 to 1. This is aphenomenological way to introduce more and more CARprocesses, as the injection points in the superconductorsare brought together, from η = 0 (no CAR effect) toη = 1 (maximal CAR effect). We observe a dramaticincrease of the critical current in the symmetric configu-rations (π − π and 0(2) − 0(2) phase associations), whilein the asymmetric configurations (π−0(2) and 0(0)−0(2)

phase associations), the modification of the AB signal isnoticeable, but not substantial.

We display the splitting efficiency r given by Eq. (26)for all phase associations in Fig. 7. In the studied domainfor Coulomb repulsion parameter U , we do not observenoticeable evolutions of r except for the 0(0)− π associa-tion for which we see a clear decrease. The symmetric as-sociations of phases (for which the critical current differsthe most between η = 0 and η = 1) lead to the highestvalues of r. The highest splitting efficiency is obtainedfor two quantum dots in the π phase: ensuring a meanoccupation number around 1 on each quantum dot favorsthe CAR process. One can get some insight concerningthe value of r for the associations 0(0)−0(0) and 0(2)−0(2)

from the perturbative calculation presented in the previ-ous section. Indeed, in the tunneling regime the splittingefficiency r matches the definition rt of Eq. (25) in termsof the amplitudes ID, IU and ICAR. For symmetric asso-ciations, we must add the constraint ID = IU ≡ I/2 tothe fit procedure, so that the critical current reads

Jc(α)

|I|=

∣∣∣∣ICAR

I+ cosα

∣∣∣∣ (27)

and since the critical current in symmetric associationsof 0 phases vanishes for α = π, we get ICAR/I = 1 andrt = 2/3.

VII. NANOSQUID IN THE HIGHTRANSPARENCY REGIME

The advantage of the formalism developed in Secs. II-III relies on the possibility to address high transparencyregimes of the double Josephson junction under study.The results presented in this Sec. are obtained for sym-metric couplings γ = 0, at temperature β−1 = 0.02∆and for a decay rate Γ = 2∆. The energies of the quan-tum dots are: ε = −4∆ for the π phase, ε = 4∆ for the0(0) phase and ε = −12∆ for the 0(2) phase. Particle-hole symmetry implies that the critical current is identi-cal for 0(0) − 0(0) and 0(2) − 0(2) phase associations withU = 8∆, for π−0(0) and π−0(2) phase associations againwith U = 8∆, and finally for 0(0) − 0(0) at U = 9∆ and0(2) − 0(2) at U = 7∆.

The strategy is again to investigate all the possiblecombinations of phases for the quantum dots (which cor-respond to different mean occupation numbers), in order

to reproduce the qualitative study of Sec. VI. Our goal isto determine what features are preserved and what haschanged, and to compute the splitting efficiency r definedby Eq. (26) in order to determine which associations ofphases favor nonlocal phenomena the most. The criticalcurrent curves are given in Figs. 8-9.

From the results of the π−π association for η = 1, it isclear that we are no longer in the tunneling regime as theflux dependence can not be fitted by Eq. (24) togetherwith the constraint that ID = IU . We can again notice,for η = 0, the π/2 phase shift in the critical currentbetween dots in the π− 0 phases (top and middle panelsof Fig. 9) and dots in the 0 − 0 or π − π phases (Fig. 8and bottom panel of Fig. 9).

The critical current in Figs. 8-9 again presents a dou-bling of its period when switching η from 0 to 1. This isdue to the emergence of nonlocal processes where bothquantum dots are involved. For the π − π association,during this switching, α = 0 remains a local maximum(contrary to the tunneling regime). For 0(0) − 0(0) and0(2) − 0(2) phase associations, the maximum at α = πfor η = 0 becomes a zero for η = 1 (as in the tunnelingregime). There is little influence (less than in the tunnel-ing regime) of η on the π− 0(2) and π− 0(0) associationswhereas, for the 0(0) − 0(2) association the maximum inα = 0 is considerably lowered (more than in the tunnelingregime) from η = 0 to η = 1.

For symmetric associations of phases, the evolution ofthe critical current with increasing U (Fig. 8) can be ex-plained following the same arguments as in the tunnelingregime. While the filtering of electrons tunneling through0(0) or π quantum dots is responsible for a decrease ofthe signal, favoring one-electron processes through a 0(2)

quantum dot results in an increase of the signal. Theopposite U -dependence for the 0(0) − 0(0) and 0(2) − 0(2)

phase associations is a consequence of particle-hole sym-

Figure 8. Critical current (in units of e∆/~) curves for sym-metric associations of dots in the high transparency regime:π−π (top panel), 0(0)−0(0) (middle panel), 0(2)−0(2) (bottompanel).

9

Figure 9. Critical current (in units of e∆/~) curves for asym-metric associations of dots in the high transparency regime:π−0(0) (top panel), π−0(2) (middle panel), 0(0)−0(2) (bottompanel).

metry. While there is still no noticeable effect of U onthe 0(0)− 0(2) association as in the tunneling regime, theπ− 0(2) association also shows little U -dependence. As aconsequence, we do not observe the opposite behavior asa function of U for the π− 0(0) and π− 0(2) associations(Fig. 9).

We introduce progressively nonlocal effects in Fig. 10by switching η from 0 to 1. For the symmetric associa-tions (π − π and 0(0) − 0(0)) as well as for the 0(0) − 0(2)

configuration (contrary to the tunneling regime), the ABsignal is dramatically increased when CAR processes areswitched on. On the contrary, the π − 0(2) phase asso-ciation is hardly influenced by the presence of nonlocalprocesses (even less than in the tunneling regime). Wecan quantify the importance of CAR processes comparedto the direct tunneling through a single quantum dot by

Figure 10. Influence of the parameter η on the critical current(in units of e∆/~) in the high transparency regime (U = 8∆).

calculating the splitting efficiency r given by Eq. (26)which we display in Fig. 11 for the different associa-tions of phases. As a function of U , the splitting effi-ciencies which are found are essentially constant over therange considered. As it turns out, although we work withspecific phases for the individual quantum dots, imply-ing specific populations, the quantization of the electroncharge on these dots is ineffective at high transparenciesbecause they constitute “open quantum dots”, involvinglarge fluctuations from their average population. Thisis consistent with the observation that the splitting effi-ciency varies little with U . As mentioned above, there islittle influence of η on the π−0(2) and π−0(0) associationsand this is why these are the lowest splitting efficiencieswe found. The splitting efficiencies of the associations of0 phases are around 0.55 and the highest splitting effi-ciency is still obtained for the π − π association.

Finally, we note that the energy levels of the dots canbe easily varied experimentally using electrostatic gates.Varying the position of the energy level ε of a quantumdot at fixed decay rate Γ and fixed Coulomb on-site re-pulsion U changes the effective transparency defined as

D(ε) =Γ2

Γ2 +(ε+ U

2

)2 . (28)

Thus, by varying the energy levels of the two dots, wecan optimize the splitting efficiency by reaching the π−πphase at high effective transparency. We show on the toppanel of Fig. 12 the splitting efficiency as a function of theenergy level of the two dots, taken to be identical. Theabrupt change of the efficiency near D(ε) = 0.4 showsthe crossover from the 0(0) − 0(0) to the π − π junction.Similarly, when tuning only one of the two energy levels,while maintaining the other dot in the π (middle panel)or the 0(0) phase (bottom panel), the splitting efficiencyshows a marked transition as a function of the effectivetransparency.

Figure 11. Splitting efficiency r given by Eq. (26) extractedfrom the curves of Figs. 8 and 9.

10

Figure 12. Evolution of the splitting efficiency as a functionof the effective transparency D(ε) [Eq. (28)], for fixed Γ =2∆ and U = 8∆. Top: The energy levels of both dots aretaken to be identical and varied simultaneously (εU = εD ≡ε). Middle: the D quantum dot is taken in the π phase(εD = −4∆) while the energy of the U dot is varied (εU ≡ ε).

Bottom: the D quantum dot is taken in the 0(0) phase (εD =4∆) while the energy of the U dot is varied (εU ≡ ε).

VIII. CONCLUSION

We have studied a double Josephson junction consist-ing of two quantum dots connected to two superconduc-tors as a tool to probe Cooper pair splitting. When thetwo single Josephson junctions which constitute our de-vice are coupled to each other via CAR processes, thedoubling of the period of the critical current measuredin a nanoSQUID experiment is an evidence for the emer-gence of nonlocal phenomena in the electronic transportof the double junction. This type of diagnosis may provemore convenient than non-equilibrium scenarios involv-ing a superconducting source of electrons and two nor-mal leads/dots where either nonlocal conductance signalor noise cross-correlations are measured.

While this device had been studied in the context ofperturbation theory in the tunneling Hamiltonian cou-pling the dot to the leads,29,30 no generalization to ar-bitrary transmission had been proposed, and no system-atic study for optimizing the CAR signal with respectto the phases (0(0), π, 0(2)) of the quantum dots hadbeen attempted so far. The path integral approach ofRef. 33, within reasonable approximations (saddle pointtreatment) allows precisely to meet these goals. One ofthe key results of the present work resides in defining thedegree of efficiency of Cooper pair splitting, and evaluat-ing it for the different dot configurations.

We first studied the tunneling regime where the usualperturbative expression for the Josephson signal allowsus to fit our numerical results and to introduce a nat-ural definition of the splitting efficiency. We were alsoable to generalize this quantity to arbitrary transparency,

providing a criterion for the efficiency of CAR processeswhich is based on an analysis of the AB signal of theJosephson critical current. We thus studied the promi-nence of nonlocal phenomena depending on the phasesof the quantum dots and found that the π − π associa-tion optimizes the splitting of the Cooper pairs that areemitted from one superconductor and recombined on theother one. Yet, our analysis shows that the 0(0) − 0(0)

and 0(2) − 0(2) combinations also provide robust Cooperpair splitting signals. Within each of these combinationsof phases, we see for the most part that variations of theCoulomb repulsion parameter has little influence on theCooper pair splitting efficiency.

The present treatment should only be valid ifthe superconductor–dots system is above the Kondoregime.38,39 For a single dot embedded between two su-perconductors, the Kondo effect manifests itself whenthe Kondo temperature is larger than the superconduct-ing gap, because quasiparticle excitations are requiredto trigger the spin flip between the leads and the dot.The expression of the Kondo temperature for a Hub-bard type Coulomb repulsion is given39,40 by TK =√

ΓU/2 eπε0(ε0+U)/(ΓU). Within the range of parameterschosen in our numerical study, we find that the Kondotemperature is at most 0.13∆, which gives us confidencein our working assumptions. At any rate, the Kondoregime could be avoided by working with the 0(0) − 0(0)

combination of phases, which according to Fig. 11 stillhas a sizable splitting efficiency.

Furthermore, we treated the CAR coupling parameterη in a phenomenological manner (varying it between 0and 1), justifiably so because so far no experimental in-vestigation of Cooper pair splitting in superconducting–normal metal “forks” out of equilibrium seems to find thepower law suppression which is attributed to bulk 3D su-perconductors. This may be due to the fact that micro-scopic models have to be revisited, taking into accountthat electron emission/absorption in the superconductorsis not point like, but should be averaged over some finitevolume, reducing the effect of oscillations over the Fermiwavelength of the CAR parameter.

Extensions of this work could be envisioned by goingbeyond the saddle point approximation of the HubbardStratonovich transformation. This transformation is ex-act in its functional integral form and neglecting the fluc-tuations of the auxiliary field and then using a method ofsteepest descent are approximations which were sufficientto exhibit the π-shift in a single junction33. However, thepossibility to go beyond this mean field type of approachcould be considered by looking at Gaussian fluctuationsaround the stationary value of the auxiliary field. Al-ternatively, a numerical renormalization group method38

could be employed in principle, but the proliferation ofcouplings and parameters is likely to render it cumber-some.

A limitation of the present Cooper pair diagnosis re-sides in the fact that the nanoSQUID requires reduceddimensions so that the injection points to the dots on

11

α↑

VU

VD

↓

Figure 13. Setup with nanowire/nanotube quantum dots.

both superconductors are separated within a distancesmaller than the coherence length. This imposes con-straints on the separation of the quantum dots, and asa consequence, the area where the AB flux is imposedbecomes reduced. Recall that in order to perform theAB diagnosis, a few flux quanta need to be introducedin this area, but if the imposed magnetic field needed toapply several flux quanta becomes larger than the criticalfield of the superconductors, the whole diagnosis breaksdown. In order to optimize the surface area encompassedbetween the dots, we believe the best choice would beto work with nanowire/nanotube quantum dots, whichhave a large aspect ratio, and which nevertheless canachieve large charging energies even though their lengthmay exceed several µm.41 This possibility is illustratedin Fig. 13. To work out the numbers, we find that im-

posing 2 magnetic flux quanta within a 1 µm2 area re-quires a magnetic field of 8 × 10−3 Tesla, which is stillsmaller than the critical field of superconducting mate-rials such as Aluminum (10−2 Teslas) or Niobium (0.2Teslas). Note that Aluminum could be quite adapteddue to its long coherence length (1.6 µm).

While this work came to completion, we became awareof a current biased measurement42 which studies pre-cisely the behavior of the double dot Josephson junction.There, a (remarkable) comparative study of the switch-ing current (the current required to transit to the dissi-pative regime with voltage bias) was performed for differ-ent dot phase configurations. This experiment seems toprovide realistic evidence of Cooper pair splitting, albeitindirectly, because the self organized quantum dots em-bedded in the Josephson junction are too close in orderto impose the necessary magnetic flux to observe the ABoscillations.

ACKNOWLEDGMENTS

We acknowledge the support of the French NationalResearch Agency, through the project ANR NanoQuar-tets (ANR-12-BS1000701). Part of this work has beencarried out in the framework of the Labex ArchimedeANR-11-LABX-0033 and of the A*MIDEX project ANR-11-IDEX-0001-02. The authors acknowledge discussionswith S. Tarucha and A. Oiwa about the nanoSQUID de-vice. Discussions with E. Paladino and G. Falci are alsogratefully acknowledged.

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