Dynamic Cooper Pair Splitter
DDynamic Cooper Pair Splitter
Fredrik Brange, Kacper Prech,
1, 2 and Christian Flindt Department of Applied Physics, Aalto University, 00076 Aalto, Finland School of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK
Cooper pair splitters are promising candidates for generating spin-entangled electrons. However,the splitting of Cooper pairs is a random and noisy process, which hinders further synchronizedoperations on the entangled electrons. To circumvent this problem, we here propose and analyze adynamic Cooper pair splitter that produces a noiseless and regular flow of spin-entangled electrons.The Cooper pair splitter is based on a superconductor coupled to quantum dots, whose energy levelsare tuned in and out of resonance to control the splitting process. We identify the optimal operatingconditions for which exactly one Cooper pair is split per period of the external drive and the flow ofentangled electrons becomes noiseless. To characterize the regularity of the Cooper pair splitter inthe time domain, we analyze the g (2) -function of the output currents and the distribution of waitingtimes between split Cooper pairs. Our proposal is feasible using current technology, and it pavesthe way for dynamic quantum information processing with spin-entangled electrons. Introduction. — Superconductors are natural sources ofentangled particles [1]. By splitting the Cooper pairs ina supercondutor into different normal-state leads, spin-entanglement between spatially separated electrons canbe achieved [2, 3]. Cooper pair splitters have been re-alized in several types of solid-state architectures [4–19],for instance using quantum dots [6, 9, 11], carbon nan-otubes [7], or graphene [15, 16, 18]. Experimentally, thesplitting process has been observed by measuring thenon-local conductance or the noise [8, 12] and recentlyusing single-electron detectors [19]. However, with staticvoltages, the generation of spin-entangled electrons is arandom and noisy process, which offers little control overthe regularity and the timing of the Cooper pair splitting.In parallel with these developments, dynamic single-electron emitters have emerged as accurate sources ofnoiseless currents [20–28]. By applying periodic gate orbias voltages to a nano-scale structure, such as a quan-tum dot [24, 25, 27, 28], a mesocopic capacitor [20, 22],or an ohmic contact [23, 26], single electrons can be peri-odically emitted into a ballistic conductor, leading to anelectric current which is given simply by the driving fre-quency times the charge of an electron [29]. So far, theseefforts have mainly focused on dynamic sources that emita single electron per cycle. However, one may envisionother types of sources that emit more complex quantumstates, for example with several entangled particles.In this Letter, we propose and analyze a dynamicCooper pair splitter that can deliver a noiseless and reg-ular stream of spin-entangled electrons. Specifically, weshow how the splitting of Cooper pairs can be controlledby applying time-dependent gate voltages to two quan-tum dots that are connected to a superconductor. Weevaluate the average current and the fluctuations in theoutput leads and identify the optimal operating condi-tions for the dynamic Cooper pair splitter to produce anoiseless and regular current. Our proposal seems feasi-ble in the light of recent experimental advances, and itmay be realized using current technology.
Dynamic Cooper pair splitter. — Figure 1(a) shows ourdynamic Cooper pair splitter consisting of a supercon-ductor coupled to two single-level quantum dots. Cooperpairs from the superconductor are split between the dotsdue to strong on-site Coulomb interactions, which pre-vent each dot from being doubly occupied. Electronson the dots are collected in separate normal-metal drain T (b) (c)(a) Γ T V g ( t ) γ Γ t V g ( t )
010 1 2 3 γ T / π (d) Average current (1/ T ) Γ T = 5 Γ T = 1 Average current (1/ T ) γ T / π T FIG. 1. Dynamic Cooper pair splitter. (a) The Cooper pairsplitter consists of two quantum dots (light green) coupled toa superconductor (blue) and two normal-metal drains (green).The splitting of Cooper pairs is controlled with the time-dependent gate voltage, V g ( t ). (b) In phase 1 ○ of the periodicdriving protocol, Cooper pair splitting is tuned into resonancefor the time T , so that a split Cooper pair tunnels into thedots (see insets). In phase 2 ○ , Cooper pair splitting is off res-onance for the time T , and the electrons may escape via thedrains. (c,d) Average current in the drains as a function of T and T for Γ = 0 . γ , κ = γ , δ = 100 κ , with ε = 0 in phase 1 ○ and ε = 100 γ in phase 2 ○ (see main text for definitions). Forthe sweet-spot conditions, γT = π/
2, and Γ T = 5, shownwith a red dot, one Cooper pair is split per period of the drive. a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b electrodes. Importantly for our proposal, we dynami-cally control the splitting of Cooper pairs using a time-dependent gate voltage V g ( t ) as we explain below.For a large superconducting gap, the coherent dynam-ics of the dots due to the coupling to the superconductorcan be described by the effective Hamiltonian [30–33]ˆ H ( t ) = (cid:88) (cid:96)σ (cid:15) (cid:96) ( t ) ˆ d † (cid:96)σ ˆ d (cid:96)σ − (cid:18) γ ˆ d † S + (cid:88) σ κ ˆ d † Lσ ˆ d Rσ +H.c. (cid:19) , (1)where ˆ d † (cid:96)σ creates an electron with spin σ = ↑ , ↓ in dot (cid:96) = L, R , while ˆ d † S ≡ ( ˆ d † L ↓ ˆ d † R ↑ − ˆ d † L ↑ ˆ d † R ↓ ) / √ γ and κ , respectively, and (cid:15) (cid:96) ( t ) is the time-dependent energy level of each dot, which we tune byexternal gates to control the splitting of Cooper pairsand elastic cotunneling between the dots. Elastic cotun-neling occurs mainly, when the dot levels are aligned andthe detuning δ = (cid:15) L − (cid:15) R vanishes. Similarly, Cooperpair splitting is on resonance, when the doubly occu-pied dots have the same energy as the empty dots andthe sum ε = (cid:15) L + (cid:15) R vanishes. In general, Cooper pairsplitting and elastic cotunneling lead to coherent oscil-lations with angular frequencies ω γ = (cid:112) γ + ε and ω κ = √ κ + δ , respectively, and both processes aresuppressed as γ / ( γ + ε /
4) and κ / ( κ + δ /
4) as wemove away from the resonances. These suppression fac-tors provide us with efficient experimental knobs to con-trol the two types of processes. Thus, in the following,we consider the periodic driving protocol in Fig. 1(b),where Cooper pair splitting is tuned in ( ε = 0) and out( ε (cid:29) γ ) of resonance, while elastic cotunneling is keptoff resonance ( δ (cid:29) κ ). The duration of each phase isdenoted by T j , j = 1 ,
2, with ˆ H j being the correspondingHamiltonian, and T = T + T is the period of the drive.With large negative voltages on the drains, the time-evolution is governed by the Lindblad equation [34, 35] ddt ˆ ρ ( t ) = L j ˆ ρ ( t ) = − i (cid:126) [ ˆ H j , ˆ ρ ( t )] + D ˆ ρ ( t ) (2)for each of the phases with Liouvillian L j , j = 1 ,
2, andˆ ρ ( t ) is the density matrix of the dots, while the dissipator D ˆ ρ ( t ) = Γ (cid:88) σ,(cid:96) = L,R (cid:18) J (cid:96)σ ˆ ρ ( t ) − { ˆ ρ ( t ) , ˆ d † (cid:96)σ ˆ d (cid:96)σ } (cid:19) (3)describes the coupling to the drains. The jump opera-tors J (cid:96)σ ˆ ρ ( t ) ≡ ˆ d (cid:96)σ ˆ ρ ( t ) ˆ d † (cid:96)σ describe the tunneling of singleelectrons to the drains with the same rate Γ to keep thediscussion simple, and from hereon we take (cid:126) , e = 1. Driving protocol. — We first consider our driving pro-tocol in the weak coupling limit, Γ (cid:28) γ, κ , suppress-ing elastic cotunneling using the off-resonance condition δ (cid:29) κ . The protocol is designed as follows: In the first phase ( 1 ○ ), Cooper pair splitting is tuned into resonance( ε = 0) for the time T . Ideally, during this phase, aCooper pair is split between the dots, and we refer to T as the loading time. In the second phase ( 2 ○ ), Cooperpair splitting is turned off resonance ( ε (cid:29) γ ) for the un-loading time T , allowing the split Cooper pair to leavethe dots via the drains. We now consider the current inthe drains to optimize the loading and unloading times.Figure 1(c,d) shows the average current for the peri-odic driving protocol obtained using a method describedbelow. The current oscillates as a function of the load-ing time T due to the coherent oscillations induced byCooper pair splitting in the first phase. By contrast,the current increases monotonously as a function of T until it saturates for Γ T (cid:38)
5, reflecting that the unload-ing time is sufficiently long for the electrons to leave thedots. In this regime, the average current [the blue curvein Fig. 1(d)] can be captured by the simple expression I = 1 T (cid:18) sin ( γT ) + 12 (cid:2) Γ T + (cid:0) − e − Γ T (cid:1) cos (2 γT ) (cid:3)(cid:19) , (4)where the first term stems from the coherent oscillationsfor very small drain couplings, Γ T (cid:28)
1. Correspondingto this term, exactly one Cooper pair is split per period,if γT = π (1 / m ) for integer m . For finite couplings,the second term describes an unwanted leakage currentduring the first phase, which can be minimized by choos-ing a short loading time, Γ T (cid:28)
1. Thus, we find thesweet-spot conditions γT = π/ (cid:29) Γ T and Γ T (cid:39) Fluctuations. — We now proceed with a refined analy-sis of the Cooper pair splitter by investigating the autoand cross-correlations of the output currents. To thisend, we use techniques from full counting statistics anddecompose the density matrix as ˆ ρ ( t ) = (cid:80) n ˆ ρ ( n , t ),so that P ( n , t ) = tr { ˆ ρ ( n , t ) } is the joint probabilitythat n = ( n L , n R ) electrons have been collected in thedrains during the time span [0 , t ] [36, 37]. The Lind-blad equation renders a hierarchy of coupled equationsfor ˆ ρ ( n , t ), which are decoupled by introducing count-ing fields χ = ( χ L , χ R ) via the transformation ˆ ρ ( χ , t ) = (cid:80) n ˆ ρ ( n , t ) e i n · χ . We thereby obtain a generalized masterequation for ˆ ρ ( χ , t ) with χ -dependent Liouvillians L j ( χ ),obtained by substituting J (cid:96)σ → e iχ (cid:96) J (cid:96)σ in Eq. (3) [33].The moment generating function for the number ofemitted electrons after N periods now reads [38–40] M ( χ , N ) = tr { ˆ ρ ( χ , N T ) } = tr (cid:8) [ U ( χ , T, N ˆ ρ C (0) (cid:9) , (5)where the evolution operator for a time-dependent Li-ouvillian is given by a time-ordered exponential as U ( χ , t, t ) = T { exp[ (cid:82) tt L ( χ , t (cid:48) ) dt (cid:48) ] } , which we can explic-itly evaluate for our piecewise constant protocol, and thecyclic state is determined by the eigenproblem U ( , T + t, t )ˆ ρ C ( t ) = ˆ ρ C ( t ). The cumulants of the currents are (a) (b) γ T / π γ T / π δ / κ Γ T = 1 Γ T = 2 Γ T = 5 F FF FF (c)
FIG. 2. Noise and spin correlations. (a) Fano factor as a function of the loading time T for Γ = 0 . γ , γ = κ , δ = 100 κ ,with ε = 0 in phase 1 ○ , and ε = 100 γ in phase 2 ○ . (b) Fano factors of the spin cross-correlations as a function of T for thesame settings as in panel (a). (c) Fano factors of the spin-correlations as functions of δ with γT = π/ then given by derivatives of the cumulant generatingfunction F ( χ ) = lim N →∞ ln[ M ( χ , N )] /N T with respectto the counting fields as (cid:104)(cid:104) I kL I lR (cid:105)(cid:105) = ∂ kiχ L ∂ liχ R F ( χ ) | χ =0 .With these definitions at hand, we can calculate the av-erage currents and their correlations using the methodsdeveloped in Refs. [41–43] and in some regimes obtainsimple expressions as those in Eq. (4) and Eq. (6) below. Noise and Fano factor. — Figure 2(a) shows the Fanofactor, F (cid:96) = (cid:104)(cid:104) I (cid:96) (cid:105)(cid:105) / (cid:104)(cid:104) I (cid:96) (cid:105)(cid:105) , (cid:96) = L, R , of the noise inthe drain electrodes, which for regular transport is sup-pressed below the Poisson value of one. Here, the Fanofactor oscillates as a function of the loading time T , sim-ilarly to the average current in Fig. 1(c,d), and for longunloading times, Γ T (cid:29)
1, we find the simple expression F = 1 − T I + 2Γ T (1 + Γ T ) + e − T − T I , (6)corresponding to the dashed line in Fig. 2(a). For weakdrain couplings, Γ T (cid:28)
1, the Fano factor reduces to F = cos ( γT ), reflecting the coherent oscillations in thefirst phase. For larger couplings, the leakage current inthe loading phase generates noise since more than oneCooper pair may be split during each period. Thus, wemay minimize the noise by choosing γT = π/ T (cid:29)
1, corresponding tothe red dot in Fig. 2(a). (For short unloading times,the dots may not be emptied after each period, leadingto cycle-missing events that generate noise.) Thus, thedevice produces N split Cooper pairs after N periods,when operated at the sweet spot in Figs. 1 and 2, and itis noiseless in contrast to a static Cooper pair splitter. Spin-current correlations. — Next, we consider thecross-correlations between the drain currents and, in par-ticular, between the spin currents in each drain, whichplay an important role for detecting the entanglementof the split Cooper pairs [44–47]. It it straightforwardto include spin-dependent counting fields in Eq. (5), andin Fig. 2(b) we show the resulting cross-correlations ofthe spin-currents, F σσ (cid:48) = (cid:104)(cid:104) I Lσ I Rσ (cid:48) (cid:105)(cid:105) / (cid:112) (cid:104)(cid:104) I Lσ (cid:105)(cid:105)(cid:104)(cid:104) I Rσ (cid:48) (cid:105)(cid:105) , as functions of the loading time T . The anti-parallel spincurrents are positively correlated, while parallel spins ex-hibit negative correlations, as expected for a split Cooperpair in a spin-singlet state. For long unloading times,Γ T (cid:29)
1, we find that the spin correlations Fig. 2(b) canbe related to the Fano factor of the charge currents as F ↑↓ = 1 + F ↑↑ = 12 ( F + 1) , (7)making it possible to determine the spin correlations fromnoise measurements of the charge currents in the drains.In Fig. 2(c), we consider the spin-correlations as wetune elastic cotunneling into resonance. The negativecorrelations for parallel spins are essentially unchangedas elastic cotunneling is included. On the other hand,the positive correlations for anti-parallel spins are grad-ually washed out by elastic cotunneling. To better un-derstand the effect of cotunneling, we consider a shortloading time, Γ T (cid:28)
1, so that the leakage current isnegligible during the first phase, and a long unloadingtime, Γ T (cid:29)
1, ensuring that at most one Cooper pair issplit per period. Each uncorrelated period may then pro-duce one of five outcomes; either no Cooper pair is splitand no electrons reach the drain, or one Cooper pair issplit and the two electrons tunnel into either of the twodrains. We then approximate the generating function as M ( χ , N ) (cid:39) (cid:20) − p + pq (cid:16) e i ( χ L ↑ + χ L ↓ ) + e i ( χ R ↑ + χ R ↓ ) (cid:17) + p (1 − q )2 (cid:16) e i ( χ L ↑ + χ R ↓ ) + e i ( χ R ↑ + χ L ↓ ) (cid:17)(cid:21) N , (8)where p = sin ( ω γ T / γ / ( γ + ε /
4) is the probabilitythat a Cooper pair is split in the first phase, and q = κ / [ κ +(Γ + δ ) /
4] is the probability that the electronsleave via the same drain in the second phase due to elasticcotunneling. In Fig. 2(c), we show calculations of thespin-correlations based on this approximation and findgood agreement with the exact results. Furthermore, we (a) (b) (c) x x g ( ) Γ T = 1 Γ T = 5 Γ T = 10 -1 -0.5 0 0.5 1 0 1 2 0 1 2 /T /T T T /T g (2) g (2) Γ T = 10 FIG. 3. Time-domain observables. (a) Auto and cross-correlations ( g (2) x ) of the drain currents. (b) Distributions of waitingtimes between electrons tunneling into the same drain. (c) Distributions of waiting times between electrons tunneling intodifferent drains. Parameters are γT = π/
2, Γ = 0 . γ , κ = γ , δ = 20 κ with ε = 0 in phase 1 ○ and ε = 20 γ in phase 2 ○ . find F ↑↑ = − p/ F ↑↓ = 1 − p/ − q , which shows that,in this regime, F ↑↑ and F ↑↓ are sufficient to determine p and q and thus fully characterize the noise statistics. Time-domain analysis. — Above, we focused on con-ventional low-frequency measurements of the current andthe noise. However, as in the recent experiment ofRef. [19], additional information can be obtained by ana-lyzing the fluctuations in the time-domain. (For example,a device that splits two Cooper pairs every second periodproduces the same low-frequency noise as one that splitsone Cooper pair per period.) To this end, we considerthe g (2) -function of the drain currents [48, 49], defined as g (2) (cid:96)(cid:96) (cid:48) ( τ ) = (cid:90) T dt (cid:104)(cid:104)J (cid:96) U ( , t + τ, t ) J (cid:96) (cid:48) (cid:105)(cid:105) t (cid:104)(cid:104)J (cid:96) (cid:105)(cid:105) t + τ (cid:104)(cid:104)J (cid:96) (cid:48) (cid:105)(cid:105) t P (cid:96) (cid:48) ( t ) (9)for a periodic drive. Here, the probability density forthe time that a tunneling event occurs reads P (cid:96) ( t ) = (cid:104)(cid:104)J (cid:96) (cid:105)(cid:105) t / (cid:82) T dτ (cid:104)(cid:104)J (cid:96) (cid:105)(cid:105) τ , and (cid:104)(cid:104)A(cid:105)(cid:105) t = tr {A ˆ ρ C ( t ) } is the ex-pectation value of A . In this definition, the correlationsdue to the periodic drive have been factored out.Figure 3(a) shows the g (2) -functions for the autoand the cross-correlations ( g (2) x ) of the drain currents.The auto-correlation function shows a clear suppressionaround τ = 0, corresponding to the anti-bunching of elec-trons due to the strong Coulomb interactions on the dots,which prevent the simultaneous tunneling of electronsinto the same lead. The cross-correlations, by contrast,exhibit a peak well above the uncorrelated value of one,showing how Cooper pair splitting leads to nearly simul-taneous tunneling of electrons into different leads.Information about the regularity of the dynamicCooper pair splitter can be obtained from the waitingtimes between the tunneling events [19, 50]. The distri-bution of waiting times can be expressed as [51–55] W (cid:96)(cid:96) (cid:48) ( τ ) = (cid:90) T dt (cid:104)(cid:104)J (cid:96) U (cid:96) ( t + τ, t ) J (cid:96) (cid:48) (cid:105)(cid:105) t (cid:104)(cid:104)J (cid:96) (cid:48) (cid:105)(cid:105) t P (cid:96) (cid:48) ( t ) , (10)where U (cid:96) ( t, t ) = T exp[ (cid:82) tt ( L ( , t (cid:48) ) − (cid:80) σ J (cid:96)σ ) dt (cid:48) ] is thetime-evolution operator excluding electron tunneling into drain (cid:96) . For (cid:96) = (cid:96) (cid:48) , we obtain the distribution of waitingtimes between electrons tunneling into the same lead,while for (cid:96) (cid:54) = (cid:96) (cid:48) , the distribution characterizes the waitingtime between electrons tunneling into different leads.Figure 3(b,c) shows both types of distributions. Witha long unloading time, Γ T (cid:29)
1, electrons tunneling intothe same drain tend to be separated by the period of thedrive, as seen in Fig. 3(b), reflecting the regular split-ting of Cooper pairs. For shorter unloading times, thedots are not always emptied in the second phase, and theCooper pair splitting becomes less regular. This pictureis corroborated by the distributions for tunneling intodifferent leads shown in Fig. 3(c). Here, the first peak atshort waiting times corresponds to the tunneling of elec-trons from the same split Cooper pair into different leads.In addition, the second peak corresponds to the waitingtime between the last electron from one split Cooper pairand the first electron from the next split pair, and thoseare spaced by an interval, which is slightly shorter thanthe period of the drive. Again, we observe that a shortunloading time reduces the regularity of the splitting.Based on our analysis of the low-frequency noise andthe time-domain statistics, we see that the dynamicCooper pair splitter can deliver a noiseless and regularflow of split Cooper pairs that are separated by the periodof the drive. These features make the dynamic Cooperpair splitter attractive for a variety of quantum informa-tion processes that require controlled generation of en-tanglement. The Cooper pair splitter may be operatedacross a wide range of driving frequencies. For exam-ple, to generate measurable currents, driving frequenciesin the mega-hertz regime produce pico-ampere currents.On the other hand, slower driving speeds are suitable forsingle-electron detection as in the experiment of Ref. [19].
Perspectives. — We have proposed and analyzed a dy-namic Cooper pair splitter that can generate a noiselessand regular flow of spin-entangled electrons, when oper-ated under optimal conditions. Our proposals appearsfeasible in the light of recent experiments [19, 50], andit may thus pave the way for the controlled and pulsedgeneration of spin-entangled electrons in solid-state ar-chitectures. Furthermore, entanglement witnesses basedon current cross-correlations [46, 47] may be used to ex-perimentally certify the spin-entanglement of mobile elec-trons emitted by a dynamic Cooper pair splitter.
Acknowledgements. — We thank N. Walldorf for his in-volvement at an early stage of the project and A. Ranniand V. F. Maisi for useful discussions. We acknowl-edge support from Aalto Science Institute and Academyof Finland through the Finnish Centre of Excellencein Quantum Technology (project numbers 312057 and312299) and grants number 308515 and 331737. [1] M. Tinkham,
Introduction to Superconductivity (DoverPublications, 2004).[2] G. B. Lesovik, T. Martin, and G. Blatter, Electronicentanglement in the vicinity of a superconductor, Eur.Phys. J. B , 287 (2001).[3] P. Recher, E. V. Sukhorukov, and D. Loss, Andreev tun-neling, Coulomb blockade, and resonant transport of non-local spin-entangled electrons, Phys. Rev. B , 165314(2001).[4] D. Beckmann, H. B. Weber, and H. v. L¨ohneysen, Evi-dence for Crossed Andreev Reflection in Superconductor-Ferromagnet Hybrid Structures, Phys. Rev. Lett. ,197003 (2004).[5] S. Russo, M. Kroug, T. M. Klapwijk, and A. F.Morpurgo, Experimental Observation of Bias-DependentNonlocal Andreev Reflection, Phys. Rev. Lett. ,027002 (2005).[6] L. Hofstetter, S. Csonka, J. Nyg˚ard, andC. Sch¨onenberger, Cooper pair splitter realized ina two-quantum-dot Y-junction, Nature , 960 (2009).[7] L. G. Herrmann, F. Portier, P. Roche, A. L. Yeyati,T. Kontos, and C. Strunk, Carbon Nanotubes as Cooper-Pair Beam Splitters, Phys. Rev. Lett. , 026801(2010).[8] J. Wei and V. Chandrasekhar, Positive noise cross-correlation in hybrid superconducting and normal-metalthree-terminal devices, Nat. Phys. , 494 (2010).[9] L. Hofstetter, S. Csonka, A. Baumgartner, G. F¨ul¨op,S. d’Hollosy, J. Nyg˚ard, and C. Sch¨onenberger, Finite-Bias Cooper Pair Splitting, Phys. Rev. Lett. , 136801(2011).[10] J. Schindele, A. Baumgartner, and C. Sch¨onenberger,Near-Unity Cooper Pair Splitting Efficiency, Phys. Rev.Lett. , 157002 (2012).[11] L. G. Herrmann, P. Burset, W. J. Herrera, F. Portier,P. Roche, C. Strunk, A. Levy Yeyati, and T. Kontos,Spectroscopy of non-local superconducting correlationsin a double quantum dot, arXiv:1205.1972.[12] A. Das, R. Ronen, M. Heiblum, D. Mahalu, A. V. Kre-tinin, and H. Shtrikman, High-efficiency Cooper pairsplitting demonstrated by two-particle conductance reso-nance and positive noise cross-correlation, Nat. Commun. , 1165 (2012).[13] G. F¨ul¨op, S. d’Hollosy, A. Baumgartner, P. Makk, V. A.Guzenko, M. H. Madsen, J. Nyg˚ard, C. Sch¨onenberger,and S. Csonka, Local electrical tuning of the nonlocal signals in a Cooper pair splitter, Phys. Rev. B , 235412(2014).[14] Z. B. Tan, D. Cox, T. Nieminen, P. L¨ahteenm¨aki, D. Gol-ubev, G. B. Lesovik, and P. J. Hakonen, Cooper PairSplitting by Means of Graphene Quantum Dots, Phys.Rev. Lett. , 096602 (2015).[15] G. F¨ul¨op, F. Dom´ınguez, S. d’Hollosy, A. Baumgartner,P. Makk, M. H. Madsen, V. A. Guzenko, J. Nyg˚ard,C. Sch¨onenberger, A. Levy Yeyati, and S. Csonka, Mag-netic field tuning and quantum interference in a Cooperpair splitter, Phys. Rev. Lett. , 227003 (2015).[16] I. V. Borzenets, Y. Shimazaki, G. F. Jones, M. F.Craciun, S. Russo, M. Yamamoto, and S. Tarucha, HighEfficiency CVD Graphene-lead (Pb) Cooper Pair Split-ter, Sci. Rep. , 23051 (2016).[17] L. E. Bruhat, T. Cubaynes, J. J. Viennot, M. C. Darti-ailh, M. M. Desjardins, A. Cottet, and T. Kontos, Cir-cuit QED with a quantum-dot charge qubit dressed byCooper pairs, Phys. Rev. B , 155313 (2018).[18] Z. B. Tan, A. Laitinen, N. S. Kirsanov, A. Galda, V. M.Vinokur, M. Haque, A. Savin, D. S. Golubev, G. B.Lesovik, and P. J. Hakonen, Thermoelectric current ina graphene Cooper pair splitter, Nat. Commun. , 138(2021).[19] A. Ranni, F. Brange, E. T. Mannila, C. Flindt, and V. F.Maisi, Real-time observation of Cooper pair splittingshowing strong non-local correlations, arXiv:2012.10373.[20] G. F`eve, A. Mah´e, J.-M. Berroir, T. Kontos, B. Pla¸cais,D. C. Glattli, A. Cavanna, B. Etienne, and Y. Jin, Anon-demand coherent single-electron source, Science ,1169 (2007).[21] M. D. Blumenthal, B. Kaestner, L. Li, S. Giblin, T. J.B. M. Janssen, M. Pepper, D. Anderson, G. Jones, andD. A. Ritchie, Gigahertz quantized charge pumping, Nat.Phys. , 343 (2007).[22] E. Bocquillon, V. Freulon, J.-M. Berroir, P. Degiovanni,B. Pla¸cais, A. Cavanna, Y. Jin, and G. F`eve, Coherenceand indistinguishability of single electrons emitted by in-dependent sources, Science , 1054 (2013).[23] J. Dubois, T. Jullien, F. Portier, P. Roche, A. Cavanna,Y. Jin, W. Wegscheider, P. Roulleau, and D. C. Glattli,Minimal-excitation states for electron quantum optics us-ing levitons, Nature , 659 (2013).[24] J. D. Fletcher, P. See, H. Howe, M. Pepper, S. P. Giblin,J. P. Griffiths, G. A. C. Jones, I. Farrer, D. A. Ritchie,T. J. B. M. Janssen, and M. Kataoka, Clock-ControlledEmission of Single-Electron Wave Packets in a Solid-State Circuit, Phys. Rev. Lett. , 216807 (2013).[25] L. Fricke, M. Wulf, B. Kaestner, F. Hohls, P. Mirovsky,B. Mackrodt, R. Dolata, T. Weimann, K. Pierz, U. Sieg-ner, and H. W. Schumacher, Self-Referenced Single-Electron Quantized Current Source, Phys. Rev. Lett. , 226803 (2014).[26] T. Jullien, P. Roulleau, B. Roche, A. Cavanna, Y. Jin,and D. C. Glattli, Quantum tomography of an electron,Nature , 603 (2014).[27] N. Ubbelohde, F. Hohls, V. Kashcheyevs, T. Wagner,L. Fricke, B. K¨astner, K. Pierz, H. W. Schumacher, andR. J. Haug, Partitioning of on-demand electron pairs,Nat. Nanotech. , 46 (2015).[28] D. M. T. van Zanten, D. M. Basko, I. M. Khaymovich,J. P. Pekola, H. Courtois, and C. B. Winkelmann, SingleQuantum Level Electron Turnstile, Phys. Rev. Lett. ,166801 (2016). [29] J. P. Pekola, O.-P. Saira, V. F. Maisi, A. Kemppinen,M. M¨ott¨onen, Y. A. Pashkin, and D. V. Averin, Single-electron current sources: Toward a refined definition ofthe ampere, Rev. Mod. Phys. , 1421 (2013).[30] O. Sauret, D. Feinberg, and T. Martin, Quantum masterequations for the superconductor-quantum dot entangler,Phys. Rev. B , 245313 (2004).[31] J. Eldridge, M. G. Pala, M. Governale, and J. K¨onig,Superconducting proximity effect in interacting double-dot systems, Phys. Rev. B , 184507 (2010).[32] B. Hiltscher, M. Governale, J. Splettstoesser, andJ. K¨onig, Adiabatic pumping in a double-dot Cooper-pair beam splitter, Phys. Rev. B , 155403 (2011).[33] N. Walldorf, F. Brange, C. Padurariu, and C. Flindt,Noise and full counting statistics of a Cooper pair splitter,Phys. Rev. B , 205422 (2020).[34] H.-P. Breuer and F. Petruccione, The theory of openquantum systems (Oxford University Press, 2003).[35] B. L. Hazelzet, M. R. Wegewijs, T. H. Stoof, and Y. V.Nazarov, Coherent and incoherent pumping of electronsin double quantum dots, Phys. Rev. B , 165313 (2001).[36] M. B. Plenio and P. L. Knight, The quantum-jump ap-proach to dissipative dynamics in quantum optics, Rev.Mod. Phys. , 101 (1998).[37] Y. Makhlin, G. Sch¨on, and A. Shnirman, Quantum-stateengineering with Josephson-junction devices, Rev. Mod.Phys. , 357 (2001).[38] F. Pistolesi, Full counting statistics of a charge shuttle,Phys. Rev. B , 245409 (2004).[39] M. Albert, C. Flindt, and M. B¨uttiker, Accuracy of thequantum capacitor as a single-electron source, Phys. Rev.B , 041407 (2010).[40] E. Potanina, K. Brandner, and C. Flindt, Optimization ofquantized charge pumping using full counting statistics,Phys. Rev. B , 035437 (2019).[41] C. Flindt, T. Novotn´y, and A.-P. Jauho, Full countingstatistics of nano-electromechanical systems, EPL ,475 (2005).[42] C. Flindt, T. Novotn´y, A. Braggio, M. Sassetti, and A.-P. Jauho, Counting Statistics of Non-Markovian Quan-tum Stochastic Processes, Phys. Rev. Lett. , 150601(2008). [43] C. Flindt, T. Novotn´y, A. Braggio, and A.-P. Jauho,Counting statistics of transport through Coulomb block-ade nanostructures: High-order cumulants and non-Markovian effects, Phys. Rev. B , 155407 (2010).[44] S. Kawabata, Test of Bell’s Inequality using the Spin Fil-ter Effect in Ferromagnetic Semiconductor Microstruc-tures, J. Phys. Soc. Jap. , 1210 (2001).[45] O. Malkoc, C. Bergenfeldt, and P. Samuelsson, Fullcounting statistics of generic spin entangler with quan-tum dot-ferromagnet detectors, EPL , 47013 (2014).[46] P. Busz, D. Tomaszewski, and J. Martinek, Spin correla-tion and entanglement detection in Cooper pair splittersby current measurements using magnetic detectors, Phys.Rev. B , 064520 (2017).[47] F. Brange, O. Malkoc, and P. Samuelsson, MinimalEntanglement Witness from Electrical Current Correla-tions, Phys. Rev. Lett. , 036804 (2017).[48] H. J. Carmichael, S. Singh, R. Vyas, and P. R. Rice,Photoelectron waiting times and atomic state reductionin resonance fluorescence, Phys. Rev. A , 1200 (1989).[49] C. Emary, C. P¨oltl, A. Carmele, J. Kabuss, A. Knorr,and T. Brandes, Bunching and antibunching in electronictransport, Phys. Rev. B , 165417 (2012).[50] F. Brange, A. Schmidt, J. C. Bayer, T. Wagner,C. Flindt, and R. J. Haug, Controlled emission timestatistics of a dynamic single-electron transistor, Sci.Adv. , eabe0793 (2021).[51] T. Brandes, Waiting times and noise in single particletransport, Ann. Physik , 477 (2008).[52] M. Albert, C. Flindt, and M. B¨uttiker, Distributionsof Waiting Times of Dynamic Single-Electron Emitters,Phys. Rev. Lett. , 086805 (2011).[53] E. Potanina and C. Flindt, Electron waiting times of aperiodically driven single-electron turnstile, Phys. Rev.B , 045420 (2017).[54] N. Walldorf, C. Padurariu, A.-P. Jauho, and C. Flindt,Electron Waiting Times of a Cooper Pair Splitter, Phys.Rev. Lett. , 087701 (2018).[55] K. Wrze´sniewski and I. Weymann, Current cross-correlations and waiting time distributions in Andreevtransport through Cooper pair splitters based on a triplequantum dot system, Phys. Rev. B101