Adiabatic pumping in a double-dot Cooper-pair beam splitter
Bastian Hiltscher, Michele Governale, Janine Splettstoesser, Jürgen König
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Adiabatic pumping in a double-dot Cooper-pair beam splitter
Bastian Hiltscher , Michele Governale , Janine Splettstoesser , and J¨urgen K¨onig Theoretische Physik, Universit¨at Duisburg-Essen and CeNIDE, D-47048 Duisburg, Germany School of Chemical and Physical Sciences and MacDiarmid Institute for Advanced Materials and Nanotechnology,Victoria University of Wellington, P.O. Box 600, Wellington 6140, New Zealand Institut f¨ur Theorie der Statistischen Physik, RWTH Aachen University,D-52056 Aachen, and JARA-Fundamentals of Future Information Technology, Germany (Dated: August 8, 2018)We study adiabatic pumping through a double quantum dot coupled to normal and supercon-ducting leads. For this purpose a perturbation expansion in the tunnel coupling between the dotsand the normal leads is performed and processes underlying the pumping current are discussed.Features of crossed Andreev reflection are investigated in the average pumped charge and related tolocal Andreev reflection in a single quantum dot. In order to distinguish Cooper-pair splitting fromquasi-particle pumping, we compare the properties of Cooper-pair pumping with single-electronpumping in a system with only normal leads. The dependence on the average dot level and the cou-pling asymmetry turn out to be the main distinguishing features. This is contrasted with the linearconductance for which it is more difficult to distinguish single-particle from Cooper-pair transport.
PACS numbers: 74.45.+c,73.23.Hk,72.10.Bg
I. INTRODUCTION
Charge transport through interfaces between super-conductors and normal conductors takes place by dif-ferent processes. Above the energy gap of the super-conductor’s density of states, mainly single electrons aretransferred, while subgap transport is sustained by An-dreev reflection (AR). In an AR process, an electronin the normal conductor that impinges on the interfaceis retroreflected as a hole while a Cooper-pair is trans-ferred into the superconductor. In junctions with morethan one normal conductor also crossed Andreev reflec-tion (CAR) may occur, that is, the two electrons formingthe Cooper-pair stem from different normal conductors(or tunnel into different normal contacts in the oppositetransport process). This nonlocal transport mechanismhas been extensively studied both theoretically and ex-perimentally. In recent years advancements in nanofabrication havemade it possible to contact quantum dots (QDs) with su-perconducting leads.
Such QD-superconductor devicesare of great relevance, because they enable the investi-gation of the interplay between superconducting corre-lations and Coulomb repulsion in nonequilibrium situa-tions. Andreev reflection as well as crossed Andreevreflection through quantum dots have been the fo-cus of many theoretical works. Recently, CAR throughQDs has also been observed in experiment.
The setupconsists of a superconducting lead tunnel coupled to twoparallel quantum dots realized in an InAs semiconductingnanowire and a carbon nanotube, respectively. Each ofthe two quantum dots is additionally coupled to separatenormal reservoirs (see Fig. 1). As a result the depen-dence of the current in one arm of the beam splitter onthe parameters of the other arm indicates the occurrenceof CAR.In the examples mentioned above, a bias voltage is ap- Γ S,R Γ S,L ε L ( t ) ε R ( t ) N L N R S Γ L Γ R FIG. 1: (Color online) NDSDN setup: two quantum dotscoupled to the same superconductor and each dot coupled toa normal conductor. plied to generate dc transport. In this paper we considera different transport mechanism: adiabatic pumping . Theprinciple of pumping is to transport electrons in the ab-sence of a bias voltage by varying certain system param-eters periodically in time. Pumping is therefore a mech-anism converting an ac into a dc signal, which has beenexperimentally realized in different systems. In the adi-abatic regime the pumping period is large compared toother characteristic time scales of the system. It wasshown that adiabatic pumping reveals features whichare not visible in stationary transport. Here, the mainmotivation of our work is to use adiabatic pumping in or-der to investigate features of CAR. We therefore considera system consisting of two quantum dots coupled both tonormal and superconducting leads as shown in Fig. 1. Inthe experiments performed so far, the CAR and AR sig-nals coexist. This happens even though strong Coulombinteraction within each dot tends to suppress AR, there-fore enhancing the visibility of CAR. Adiabatic pumpingrequires two out-of-phase time-dependent parameters inorder to obtain a finite dc current. Choosing gates ap-plied to the two dots, belonging to two different arms ofthe beam splitter, as pumping parameters, only transportmechanisms relying on nonlocal correlations between thetwo arms contribute to the pumped charge. Therefore,such a type of pumping cycle has the advantage with re-spect to biased transport that it singles out CAR, whilelocal effects do not yield any finite dc current.Theoretically, the dynamical scattering approach pro-vides a general framework for pumping as long as theCoulomb interaction is weak. In noninteracting sys-tems, the influence of the superconducting proximityeffect on pumping was studied in Refs. 18. However,Coulomb interaction cannot be neglected in the setupconsidered here. In recent years much effort has beenput on the treatment of pumping through strongly inter-acting systems such as quantum dots. While pumpingthrough a single quantum dot with a superconductinglead was studied in the limit of zero temperature andinfinitely strong Coulomb interaction, in this paper,we are interested in systems in which Coulomb interac-tion can be arbitrary and where coupling to the leadsis weak. To this purpose we use an adiabatic extensionof a generalized master equation approach. Inthe stationary limit the generalized master equation ap-proach has been applied to hybrid quantum dot sys-tems before. The motivation of this paper is to identify and un-derstand CAR in adiabatic pumping. To this purposewe investigate two quantum dots, with infinite intra-dot Coulomb repulsion, tunnel coupled to the same su-perconductor and each of them to a normal conductor(NDSDN) (see Fig. 1). Pumping is realized by apply-ing time-dependent potentials, namely one to each of thequantum dots, via gates with a phase-difference in thedriving. This gives us the possibility to identify uniquefeatures of crossed Andreev reflection in adiabatic pump-ing which rely on the nonlocality of the effect and can -as we show by a comparison with a setup with the super-conductor replaced by a normal lead (NDNDN) - not bereproduced by other parasitic nonlocal effects mediatedby quasiparticles.However, the complexity of this setup makes it difficultto obtain compact analytic formulae. Therefore, we addi-tionally consider a quantum dot with Zeeman-split levels,tunnel coupled to a ferromagnetic and a superconductinglead (FDS). In hybrid systems containing ferromagnets,superconductors, and quantum dots the influence of spinasymmetry on Andreev reflections has been investigatedbefore.
In the present work our motivation of con-sidering the FDS setup is to get a better understandingof the transport processes in the NDSDN system becausewe can relate the CAR in the NDSDN setup to AR in theFDS setup. The Zeeman splitting and the polarizationin the FDS setup corresponds to a difference of the twodot levels and an asymmetry of the coupling to the twonormal conductors of the NDSDN system, respectively.From a theoretical point of view the main difference be- tween the two setups is the existence of triplet statesin the NDSDN system. Experimentally, although hy-brid systems containing ferromagnets and superconduc-tors are realizable, the time dependence of the trans-port channels through the dot are easier to control in theNDSDN setup.The paper is structured as follows. In Sec. II we presentthe models of the considered setups. The technique usedto compute the pumping current into the superconduc-tor is described in Sec. III. The results divide in threedifferent parts. In Secs. IV A and IV B the results for lo-cal Andreev reflections and crossed Andreev reflections,respectively, are given. The features of CAR and single-particle transport are compared in Sec. IV C. Finally,conclusions are drawn in Sec. V. II. MODEL
The systems we consider are generally described by aHamiltonian for a hybrid system composed by multiplequantum dots tunnel coupled to both normal and super-conducting leads. Each individual dot, labeled by the in-dex r , is described by the Anderson-impurity model withan onsite interaction U intra and the level energy ε rσ . Theinteraction between electrons in different dots is charac-terized by the inter-dot repulsion U inter . The quantumdots are described by the Hamiltonian H dot = X rσ ε rσ ( t ) n rσ + U intra X r n r ↑ n r ↓ + 12 U inter X r = r ′ σσ ′ n rσ n r ′ σ ′ , (1)where n rσ = d † rσ d rσ is the number operator for electronsin the dot r with spin σ and d rσ ( d † rσ ) is the correspond-ing annihilation (creation) operator. Here we explicitlyintroduce the time-dependence of the dot levels, whichis used to realize the pumping cycles. The leads are de-scribed by the Hamiltonian H η = X kσ ε ηk c † ηkσ c ηkσ − δ ηS X k (∆ c η − k ↓ c ηk ↑ + h . c . ) (2)where the different reservoirs are identified by the label η . The operator c ηkσ ( c † ηkσ ) annihilates (creates) an elec-tron with momentum k and spin σ in lead η . The secondterm is only present for the superconducting leads and itis simply the attractive potential of the mean-field BCSHamiltonian. Without loss of generality the pair poten-tial ∆ can be chosen to be real, because we consider onlyone superconductor. Finally, the dots are coupled to thedifferent leads by means of the tunneling Hamiltonian H tunn = X ηrkσ t ηr c † ηkσ d rσ + h . c . . (3)Both the tunnel matrix elements and the density of statesof the leads ρ η are chosen to be energy independentin the window relevant for transport. Tunnel-couplingstrengths are then defined as Γ η,r,σ = 2 π | t η,r | ρ η,σ . No-tice that no inter-dot tunneling is included in the model.Finally, the total Hamiltonian for this type of hybrid sys-tem can be written as H = H dot + H tunn + P η H η . Weset in the following ~ = 1. A. Double-dot device
The main focus of this paper is on the parallel double-dot device shown in Fig. 1, that is ideal for studyingCooper-pair splitting. It is composed of two quantumdots which are tunnel-coupled to different normal con-ductors but the same superconducting lead. We will re-fer to it as to the NDSDN system, where N indicates anormal lead, S a superconducting lead and D a quantumdot. The Hamiltonian of the NDSDN system is obtainedfrom the general Hamiltonian of the previous subsectionby having r ∈ { L, R } , η ∈ N L , N R , S and Γ N L ≡ Γ N L ,L ,Γ N R ≡ Γ N R ,R , Γ S,r as spin-independent tunnel-couplingstrengths. With this we define Γ N ≡ Γ N L + Γ N R . For thedouble-dot system we assume the dots’ levels to be spindegenerate, that is ε r ↑ = ε r ↓ = ε r , the Coulomb repul-sion within one dot to be infinite U intra → ∞ , and a finiteinter-dot interaction U inter ≡ U . The limit U intra → ∞ excludes the possibility of double occupation of the samedot and, therefore, only CAR and no local AR appears.As independent pumping parameters we choose the twospin-degenerate dot levels, { ε L , ε R } , which can be var-ied by means of gate voltages. This system will be con-trasted to the system with the lead S in its normal state,which is referred to as NDNDN and in which we take η ∈ N L , N R , N c . B. Single-dot device
In order to identify the processes relevant for pumping,we consider a single-level quantum dot tunnel coupled toa ferromagnet and a superconductor (FDS), which havinga smaller Hilbert space allows for a simpler analysis. TheHamiltonian of the single-dot system is obtained from thegeneral Hamiltonian considering only one dot (we consis-tently drop the index r ) and two leads: η ∈ F, S . The fer-romagnet is described by the Stoner model which inducesΓ ↑ = Γ ↓ . The nonvanishing tunnel-coupling strengthsare: Γ F and Γ S . The pumping cycle in this case is re-alized by varying independently the two spin-split levels ε ↑ , ε ↓ . This can be done by means of a time-dependentgate voltage and magnetic field. C. Large- ∆ limit In the ∆ → ∞ limit quasi-particle transport in thesuperconducting lead is suppressed and an effective de-scription of the dot that takes into account Andreev tun- neling can be obtained by integrating out the supercon-ducting degrees of freedom.
Here we will discussthe resulting effective Hamiltonian only for the NDSDNsystem. The one for the FDS system is completely analo-gous. The effective Hamiltonian in the limit U intra → ∞ reads H eff = X rσ ε r n rσ + U X σσ ′ n Lσ n Rσ ′ + 12 Γ S (cid:16) d † R ↑ d † L ↓ − d † R ↓ d † L ↑ + h . c . (cid:17) (4)with Γ S = √ Γ SL Γ SR being the effec-tive coupling. The eigenstates are | χ i ∈{| + i , |−i , | σ, i , | , σ i , | T − i , | T i , | T i} , where | σ, i ( | , σ i ) corresponds to the left (right) dot be-ing singly occupied with spin σ and the right (left)dot being empty. The triplet states are | T − i = |↓ , ↓i , | T i = ( |↓ , ↑i + |↑ , ↓i ) / √ | T i = |↑ , ↑i . Thetunnel-coupling to the superconductor leads to eigen-states that are coherent superpositions of the statewith both dots empty | , i and the singlet state | S i = ( |↓ , ↑i − |↑ , ↓i ) / √ |±i = 1 √ r ∓ δ ε A | i ∓ √ r ± δ ε A | S i , (5)where δ ≡ ε L + ε R + U is the detuning between theempty state and the singlet and 2 ε A ≡ p δ + 2Γ S is theenergy splitting between the | + i and |−i states. Thecorresponding eigenenergies are E ± = δ ± ε A , E ( σ, = ε L , E (0 ,σ ) = ε R , and E T − = E T = E T = ε L + ε R + U .In the FDS setup the eigenenergies and eigenstates arethe same except that L and R are replaced by ↑ and ↓ ,respectively, the triplet state does not exist, the singletstate | S i is replaced by a double occupation | d i = d †↑ d †↓ | i of the dot, and 2 ε A ≡ p δ + Γ S . III. METHODA. Generalized master equation
The system, which is described by the Hamiltoniangiven in the previous section, can be subdivided into twodifferent subsystems, the (proximized) quantum dots andthe normal conducting leads. Since we are not interestedin the dynamics of the leads’ degrees of freedom, we cantrace them out. This leads to an effective description ofthe quantum dots in terms of the reduced density ma-trix ρ red . The elements of this reduced density matrixare denoted by p χ χ = h χ | ρ red | χ i , where χ and χ are states of the dots. The diagonal elements p χ ≡ p χχ give the probability to find the dots in state χ . We in-troduce the vector π = ( p χ , ..., p χ m , ..., p χ i χ j , ... ) T , with i = j , where the first m components are the diagonal ele-ments of the reduced density matrix of an m -dimensionalHilbert space followed by the off-diagonal elements. Thedynamics of the reduced density matrix is governed by ageneralized master equation (in matrix notation) ddt π ( t ) = − i E ( t ) π ( t ) + t Z −∞ dt ′ W ( t, t ′ ) π ( t ′ ) . (6)The matrix elements W χ χ ′′ χ ′ χ ′′′ ( t, t ′ ) of the kernel describetransitions from an initial state at time t ′ described by p χ ′′ χ ′′′ to a final state at time t described by p χχ ′ . In thesystems which we consider, consisting of a single dot ortwo dots coupled in parallel, the matrix elements of E ( t )are given by E χχ ′′ χ ′ χ ′′′ ( t ) = δ χχ ′′ δ χ ′ χ ′′′ ( E χ ( t ) − E χ ′ ( t )).We study transport relying on the periodic variationof a set of pumping parameters { X i ( t ) } . Assuming theparameter modulation to be slow, that is the pumpingfrequency Ω to be small compared to all other energiesof the system, we can perform an adiabatic expansionof Eq. (6) following the lines of Ref. 15. Within theadiabatic expansion with respect to a reference time t ,the reduced density matrix is written as the sum of aninstantaneous contribution and its adiabatic correction, π ( t ) → π ( i ) t + π ( a ) t . The instantaneous contribution re-sults from freezing all parameters to their value at time t and yields a contribution in zeroth order in Ω / Γ N , in-dicated by the superscript ( i ). The fact that the actualstate of the system always slightly lags behind the pa-rameter modulation is captured in the adiabatic term offirst order in Ω / Γ N , indicated by ( a ).On top of the adiabatic expansion we perform a system-atic expansion in the weak tunnel-coupling strengths be-tween normal conductor and leads, Γ N < k B T , of the ker-nel and the reduced density matrix, taking into accounttunneling processes up to first order in Γ N . Orders in theperturbation expansion in the tunneling coupling are de-noted by numbers in the superscript. The instantaneouscontribution to the reduced density matrix is determinedby 0 = (cid:16) − i E ( t ) + W ( i, t (cid:17) π ( i, t (7)together with the normalization condition nπ (i , t = 1with n = (1 , ... , , ..., m compo-nents of n are 1 and the other components are 0. TheLaplace transform of the Kernel at zero frequency, withall parameters frozen to the time t is given by W ( i ) t ≡ lim z → + R t −∞ dt ′ e − z ( t − t ′ ) W ( i ) t ( t − t ′ ), where here we consideronly the first order in Γ N (if not specified otherwise). Theadiabatic correction to the reduced density matrix turnsout to have a contribution in minus first order in Γ N , which due to the adiabaticity condition Ω / Γ N ≪ ddt π ( i, t = (cid:16) − i E ( t ) + W ( i, t (cid:17) π ( a, − t (8)with nπ ( a, − = 0. The rates W ( i, t between diagonalelements of the reduced density matrix can be obtained by means of Fermi’s Golden Rule. Solely for the ones con-necting off-diagonal elements this is not sufficient and onehas to resort to a diagrammatic method which has beendeveloped in Refs. 23. In general, offdiagonal elements ofthe reduced density matrix, p χχ ′ , enter Eqs. (7) and (8).However, we assume weak coupling to the normal con-ductors Γ N ≪ k B T, ε A , where for the FDS as well as theNDSDN setup the offdiagonal elements of the reduceddensity matrix are decoupled from the diagonal ones. As we are interested in the diagonal elements, needed forthe computation of the current, we can therefore disre-gard the offdiagonal ones. Solely in the NDNDN setupthe dynamics of the offdiagonal elements p ( σ, ,σ ) and p (0 ,σ )( σ, is coupled with the dynamics of the occupation proba-bilities. In the NDNDN system, where also offdiagonalelements of the reduced density matrix contribute, we as-sume ∆ ε = ε L − ε R ≈ Γ N and E and W ( i, t have to beof the same order in the small parameter Γ N ≃ ∆ ε . In a similar way, one can write rate equations for theexpectation value of the current into lead η . The instan-taneous contribution to the current is I ( i ) η ( t ) = e nW η, ( i ) t π ( i ) t , (9)which we consider in first order in the tunnel cou-pling, only. The current rates W ηt take into accountthe number of electrons transferred to lead η . FromEq. (9), we derive the conductance, which is given by G = ( dI ( i, /dV ) | V =0 , with V being the bias voltage.The instantaneous current vanishes exactly in the ab-sence of an applied bias. The adiabatic correction to thecurrent is then the dominant one and it is given by I ( a, η ( t ) = e nW η, ( i, t π ( a, − t . (10)We are in the following interested in the charge trans-ferred into lead η per cycle of the parameter variation.This is found by integrating the current over one period Q ηX ,X = Z π/ Ω0 dtI ( a, η ( t ) . (11)In the following we consequently drop the index η ifthe pumped charge corresponds to the superconductor, Q X ,X ≡ Q SX ,X . Two time-dependent parameters arenecessary to create a nonvanishing pumped charge; we in-dicate the parameter choice in the subscript. The pump-ing parameters can be written as X i ( t ) = X i + δX i ( t ),where X i is the mean value and δX i ( t ) the oscillatingcomponent. We concentrate on the limit of weak pump-ing, that is, the oscillating component is small comparedto the tunnel coupling δX i ( t ) ≪ Γ N . Therefore, weonly account for terms up to bilinear order in δX i ( t )and the pumped charge is proportional to A X ,X = R π/ Ω0 dtδX ( t ) ddt δX ( t ). IV. RESULTS
Using the effective Hamiltonian and performing theperturbation expansion as presented in the previous sec-tion we calculate the pumped charge in lowest order inΓ N or Γ F , respectively. Close to the dot levels being atresonance, the lowest order processes are the dominantones and cotunneling processes can safely be neglected.Before tackling the more complicated problem of CAR,we will first study the FDS system, in order to under-stand the features of local AR in adiabatic pumping andto identify the different transport processes occurring inthis simple setup. For this setup, we also examine the in-fluence of cotunneling processes on the pumped chargefar from resonance (Coulomb-blockade regime), whichare important when the interaction U becomes muchlarger than the temperature. In Sec. IV B, we discusshow adiabatic pumping provides the possibility to studyCAR. To this end, we finally compare the NDSDN setupwith the NDNDN setup. A. Local Andreev reflection
In this section we consider adiabatic pumping throughthe FDS setup. We choose the dot-level positions forelectrons with different spins ε ↑ ( t ) and ε ↓ ( t ) to be thepumping parameters. Such a situation can be realizedby a time-dependent gate voltage and a time-dependentmagnetic field, the latter introducing a time-dependentZeeman splitting. This choice of pumping parameters isconvenient here as it allows for a direct comparison witha double dot in the absence of a magnetic field, in whichgate voltages applied to the two dots are independentlymodulated. Pumping is possible whenever the polariza-tion of the leads or the average level splitting ∆ ε ≡ ε ↑ − ε ↓ are nonvanishing. To get a better understanding of thetransport properties we first focus on two different limits:a vanishing polarization ( p = 0) and a vanishing averagelevel splitting ( ε ↑ = ε ↓ ). We start with the case of avanishing polarization and a finite level splitting.For the pumped charge we find Q ε ↑ ,ε ↓ ( p = 0) ≈ − eA ε ↑ ,ε ↓ Γ S [Γ S + ( U + ε ↑ + ε ↓ ) ] f ( E − − ε ↑ ) f ′ ( E − − ε ↓ ) − f ( E − − ε ↓ ) f ′ ( E − − ε ↑ ) (cid:2) f ( E − − ε ↑ ) + f ( E − − ε ↓ ) − f ( E − − ε ↑ ) f ( E − − ε ↓ ) (cid:3) (12)with f ′ ( x ) = ddx f ( x ) being the derivative of the Fermifunction. We made use of the approximation f ( E + − ε ↑ ) ≈ f ( E + − ε ↓ ) ≈ f ( ε ↑ − E + ) ≈ f ( ε ↓ − E + ) ≈ S > k B T . Equation (12) showsthat the pumped charge vanishes for an average Zeemansplitting equal to zero, that is ε ↑ = ε ↓ .In Fig. 2(a), we show the pumped charge Q ε ↑ ε ↓ , with-out the approximation on the Fermi functions used towrite Eq. (12), as function of the average value of themean dot level ε ≡ ( ε ↑ + ε ↓ ) /
2. The pumped chargeexhibits a three-peak structure. The two external peaksare observed when the dot is in resonance with the nor-mal lead, that is when the addition energy for a singleelectron equals the chemical potential. This is realizedfor E − − ε σ = 0. Since we consider Zeeman splitting ∆ ε being larger than k B T (with ε ↓ being the level with thelower energy), only the resonance E − − ε ↓ = 0 is accessi-ble due to Coulomb blockade. The other Andreev boundstate, with energy E + , is only accessible in the high-biasor high temperature regime. The two resonances associ-ated to the condition E − − ε ↓ = 0 are at the two posi-tions, ε max , ± ≈ ( − U ± [( U + | ∆ ε | ) − Γ S ] / ), and areenhanced for an increased average Zeeman splitting.The central peak appears when the dot is in resonancewith the superconductor, that is the average dot level is ε max , ≈ − U/ δ < Γ N . In this casethe dot undergoes fast oscillations between the emptyand doubly-occupied state due to coherent Cooper-pairtransfer. In particular these oscillations are much faster than tunneling events of single particles between the nor-mal conductor and the dot. However, transport requiresexchange of charge both with the normal and the su-perconducting leads. Therefore, increasing the Coulombrepulsion U leads to an overall suppression of the pumpedcharge. The three peaks are not suppressed in the samemanner. The side peaks are suppressed by the factor (cid:2) U + ε ↑ + ε ↓ ) / Γ S (cid:3) − , appearing in Eq. (12). In-stead the central peak is suppressed by the combinationof Fermi functions in Eq. (12).We now focus on the limit of a vanishing average levelsplitting (∆ ε = 0) and a finite polarization. The pumpedcharge is then given by Q ε ↑ ,ε ↓ (∆ ε = 0) ≈ eA ε ↑ ,ε ↓ p (cid:0) − p (cid:1) Γ S δk B T [Γ S + (1 − p ) δ ] · − f ( E − − ε )2 − f ( E − − ε )(13)approximating the Fermi functions as done above. Wefind that the pumped charge is an odd function of δ ,therefore vanishing at the electron-hole symmetric point.The full result for the pumped charge at zero averagedetuning, ∆ ε = 0, is shown in Fig. 2(b). As shown inEq. (13), the pumped charge vanishes at ε = − U/ ε ≈ − U/
2, with opposite sign. Asargued above this relies on fast Cooper-pair oscillation.The amplitude of the pumped charge is much larger than -10 0 ! / k B T Q / Q "! = k B T "! = 2k B T "! = 3k B T ( a ) -15 -10 -5 0 5 ! / k B T -0.06-0.04-0.0200.020.040.06 Q / Q p = -0.4p = 0.1p = 0.4 -15 -10 -5 0 5 ! / k B T -0.04-0.0200.020.04 Q / Q p = -0.4p = 0.1p = 0.4 ( b ) (c) p = 0 FDS ∆ ε = k B T ∆ ε = 0 FIG. 2: (Color online) Pumped charge Q ≡ Q ε ↑ ,ε ↓ in unitsof Q = eA ε ↑ ,ε ↓ ( k B T ) as a function of the average dot level ε . Theparameters in all figures are Γ S = 4 k B T and U = 10 k B T . in Fig. 2(a) and strongly depends on the polarization ofthe leads: the stronger the polarization the larger the am-plitude. Furthermore, the pumped charge, in the vicinityof the electron-hole symmetric point is not suppressedby the strong Coulomb repulsion. We will address this,when discussing the cotunneling regime.Instead of giving the lengthy expression of the pumpedcharge for a finite average level splitting, ∆ ε = 0, and a finite polarization, p = 0, we show it in Fig. 2(c) as afunction of the average dot level. The shape is a com-bination of the two structures shown in Figs. 2(a) and2(b). We find that the effect for the finite polarizationdominates. Therefore, the peaks around ε ≈ − U/ U ≫ k B T and δ ≡ ε ↑ + ε ↓ + U ≈ k B T , when thesequential tunneling rates to reach an empty or doubly-occupied dot are exponentially small, higher-order pro-cesses such as cotunneling need to be taken into account.To compare with the result presented in Eq. (13), we an-alyze the pumped charge in the cotunneling regime. Forthis, we first of all note that Eq. (8) looses its validityin the Coulomb-blockade regime, since the rates W ( i, get exponentially suppressed, while - in contrast to situ-ations where the magnetic field is constant - the time-derivative of the instantaneous occupation probabilities, ddt p ( i, t,σ , of single occupation with spin σ do not. Thetime-evolution of the probabilities of single occupationis then governed by spin-flip processes in second order inthe tunneling, W ( i, t, ↓↑ , entering Eqs. (8) and (10) togetherwith adiabatic corrections to the probability in minus sec-ond order in Γ, p ( a, − t,σ . However, since U ≫ k B T and δ ≈ k B T results in an exponential suppression of ddt p ( i, t, ± ,also the elements p ( a, − t, ± are suppressed and will not enterthe current in the Coulomb blockade regime. For the cal-culation of the cotunneling rates we follow the procedureintroduced in Refs. 28 for metallic islands and appliedfor single-level quantum dots, for example, in Ref. 29. Incontrast to Eq. (10) in the cotunneling regime the currentis then I ( a, F ( t ) = e h W F, ( i, t, ↓↑ p ( a, − t, ↑ + W F, ( i, t, ↑↓ p ( a, − t, ↓ i , (14)which is nonvanishing due to p ( a, − t, ↑ = − p ( a, − t, ↓ and W F, ( i, t, ↓↑ = − W F, ( i, t, ↑↓ . Due to charge conservation, Q ε ↑ ,ε ↓ = − Q Fε ↑ ,ε ↓ , the charge pumped into the super-conductor is found as Q ε ↑ ,ε ↓ ≈− eπ k B T A ε ↑ ,ε ↓ Γ S p (cid:0) − p (cid:1) δ (cid:0) Γ S + δ − U (cid:1)h (1 + p ) π ( k B T ) Γ S + (1 − p ) (Γ S + δ − U ) i , (15)where we used ∆ ε/U ≪
1. The qualitative behavior ofthe pumped charge in the cotunneling regime stronglydiffers from the sequential tunneling regime. For strongCoulomb interaction, in the cotunneling regime transportis suppressed with 1 /U . To find a possible explanationfor this suppression we focus on the transport processesduring one pumping cycle. Consider the following pro-cess where a net transport is obtained in the cotunnelingas well as in the sequential tunneling regimes: An elec-tron tunnels from the ferromagnet onto a singly occupieddot. The dot is then, for example, in state |−i . To obtaina net transport another electron has to tunnel from theferromagnet onto the quantum dot bringing it back intosingle occupation which is possible due to Cooper-pairoscillations. A comparison of the system’s time scalesfor the two regimes might shed light on the origin ofthe suppression of the pumped charge. In the sequentialtunneling regime the time between two single-electrontransport processes scales with 1 / Γ N . In the cotunnel-ing regime the intermediate state can only be virtuallyoccupied due to energy conservation and hence the timebetween two tunneling events scales with 1 /U . In theconsidered limit of large U ≫ k B T and small Γ N ≪ k B T ,Cooper-pair oscillations are fast compared to the time be-tween two tunneling events in the sequential but slow inthe cotunneling regime. This gives an interpretation ofthe suppression of the pumped charge in the cotunnelingregime. B. Crossed Andreev reflection
We now consider a system made out of two quantumdots each coupled to one normal conducting lead. Thetwo QDs are then coupled to each other via a commonsuperconducting lead (see Fig. 1). We take the pairpotential in the superconducting lead to be the largestenergy scale (∆ → ∞ ), such that single-particle trans-port between superconductor and QDs is suppressed.Furthermore, we take the intra-dot Coulomb repulsion( U intra → ∞ ) to be large excluding double occupation ofeach of the single dots, as discussed in Sec. II C. In thisregime, only nonlocal effects enable transport betweenthe superconductor and the dots, that is a Cooper-pairhas to be split into two electrons occupying different dotsor electrons from different dots enter the superconductorto form a Cooper-pair.We now calculate the charge pumped through the systemdue to the periodic modulation of the dot levels ε L ( t )and ε R ( t ), which can be achieved by two time-dependentgate voltages. We are interested in the charge, Q ε L ,ε R ,pumped into the superconducting lead, which due tocharge conservation and to the fact that only CAR isallowed is twice the charge pumped out of each normallead.In Figs. 3(a) and 3(b) we show Q ε L ,ε R as a functionof ε for different values of ∆ ε and for different cou-pling asymmetries with the normal conducting leads, λ = (Γ N L − Γ N R ) / Γ N , respectively. Features appearat the resonance condition with the normal and super-conducting leads, that are equivalent to the one in theFDS case with Zeeman splitting replaced by the differ- -15 -10 -5 0 5 ! / k B T Q / Q "! !"! k B T "! = 2k B T "! = 3k B T ( a ) -15 -10 -5 0 5 ! / k B T -0.02-0.0100.010.02 Q / Q " !"! " !"! " !"! ( b ) NDSDN λ = 0 ∆ ε = k B T FIG. 3: (Color online) Pumped charge Q ≡ Q ε L ,ε R in unitsof Q = eA εL,εR ( k B T ) as a function of the average dot level ε . Theparameters are Γ S = 3 k B T and U = 10 k B T . -15 -10 -5 0 5 ! / k B T G / G " !"! " !"! " !"! linear conductance NDSDN
FIG. 4: (Color online) Linear conductance as a function ofthe average dot level ε . for different coupling asymmetries λ .The parameters are Γ S = 3 k B T , Γ N = k B T , U = 10 k B T ,and ∆ ε = k B T . ence of the energy levels of the left and right dots and thepolarization p replaced by coupling asymmetry λ . If thecouplings to the normal leads are symmetric, λ = 0, thecharge as a function of the average mean dot level po-sition ε , shows three peaks similarly to the FDS case.In this respect, CAR exhibits similar features to ARthrough the single dot. The main difference between thetwo is the asymmetry in the heights of the external peakswhich can be attributed to the triplet blockade discussedin Ref. 13. Since the proximization by the superconduc-tor solely causes a coupling between the empty and thesinglet state, Cooper-pair tunneling is blocked wheneverthe dot is in the triplet state. In the FDS setup thesymmetry of the two external peaks can be related toparticle-hole symmetry which is broken by this tripletblockade in the NDSDN structure.As in the FDS with finite polarization, also inthe NDSDN the scenario changes completely in theasymmetric-coupling case ( λ = 0). In this case the peakat ε = − U/ λ → − λ ).However, in the linear conductance, the coupling asym-metry does not introduce any new feature, as shown inFig. 4, where for different coupling asymmetries only theweight of the three peaks is influenced and not their po-larity. Furthermore, the central peak is strongly sup-pressed. That means that the characteristic features ofCAR in adiabatic pumping are not present in the linearconductance. As we will see in the next section these fea-tures are fundamental to distinguish single-particle trans-port from CAR. C. Single-particle transport
A finite pumped charge can be obtained by varying intime the properties of the two spatially-separated dotsexclusively by nonlocal correlations. CAR has such anonlocal character. However, there may be other nonlo-cal effects that can produce a finite pumped charge and,thus, mask the signal from CAR. In order to distinguishCAR from other nonlocal transport processes, we investi-gate single-particle transport in a NDNDN setup, wherethe superconductor in the NDSDN setup is replaced by anormal conductor. While in the NDSDN setup the non-locality arises from CAR, in the NDNDN setup pumpingis possible due to the formation of a coherent superposi-tion of states with one electron either in the left or theright dot. This superposition is generated by the tun-nel coupling to the common normal lead. In contrast tothe NDSDN setup, the coherent superposition is stronglysuppressed if the difference of the two dot levels is largecompared to temperature ( | ∆ ε | ≫ k B T ).Furthermore, in the NDSDN setup pumping cannotlead to an average charge transfer from the left intothe right normal lead (and vice versa) because transportthrough the superconductor always involves CAR in theinfinite-∆ limit. Instead, in the NDNDN setup, chargecan also be transferred from the left lead N L to the rightlead N R . Therefore an asymmetry of transport into lead N L and into lead N R is one possible indication for single- -20 -15 -10 -5 0 5 10 ! / k B T -0.02-0.015-0.01-0.00500.005 Q / Q " !"! " !"! " !"! NDNDN pumped charge
FIG. 5: (Color online) Pumped charge Q ≡ Q N c ε L ,ε R in unitsof Q = eA εL,εR ( k B T ) as a function of the average dot level ε for different coupling asymmetries λ . The other parametersare Γ N c , L = 0 . k B T , Γ N c , R = 0 . k B T , Γ N = 0 . k B T , U =10 k B T , and ∆ ε = k B T . particle transport.The motivation of this work is the identification ofCAR with respect to quasi-particle transport in form ofan easily detectable signature in the pumped charge. Wefind this to be the peak-trough structure at ε = − U/ λ = 0) CAR can be distinguished from single-particletransport by the presence of the peak at ε = − U/ λ as well as itsbehavior around ε ≈ − U/ V. CONCLUSIONS
We have investigated adiabatic pumping through twoquantum dots tunnel coupled to the same superconduc-tor and additionally coupled to different normal conduc-tors. For an infinite intra-dot Coulomb repulsion in thissetup pumping relies on CAR. In order to understandthe underlying transport processes we mapped the setupto the simpler setup of a quantum dot tunnel coupledto a ferromagnet and a superconductor where only ARappears. We found that most of the features of pump-ing including CAR are also present in pumping with lo-cal AR. The main difference are asymmetries due to thepresence of the triplet state. To distinguish CAR fromsingle-electron tunneling, which does not appear in ourmodel but might be relevant in experiments, we com-pare transport through the double-dot setup containinga superconductor with a setup where the superconductoris replaced by a normal conductor. 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