Full Counting Statistics of Generic Spin Entangler with Quantum Dot-Ferromagnet detectors
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Full Counting Statistics of Generic Spin Entangler with QuantumDot-Ferromagnet detectors.
O. Malkoc, C. Bergenfeldt, P. Samuelsson
Department of Physics, Lund University, Box 118, S-221 00 Lund, Sweden
PACS – Electronic transport in quantum dots
PACS – Entanglement and quantum nonlocality
PACS – Superconducting mesoscopic systems
Abstract –Entanglement between spatially separated electrons in nanoscale transport is a fun-damental property, yet to be demonstrated experimentally. Here we propose and analyse the-oretically the transport statistics of a generic spin entangler coupled to a hybrid quantum dot-ferromagnet detector system. We show that the full distribution of charges arriving at the fer-romagnetic terminals provide complete information on the spin state of the particles emitted bythe entangler. This provides means for spin entanglement detection via electrical current correla-tions, with optimal measurement strategies depending on the a priori knowledge of ferromagnetpolarization and spin-flip rates in the detector dots. The scheme is exemplified by applying it toAndreev and triple dot entanglers.
Introduction -
Entanglement between spatially sep-arated quantum systems constitutes an indispensableresource for quantum information processing [1]. Innanoscale electronic systems, a promising arena for quan-tum information and computation, the ultimate carriersof quantum information are individual electrons. Con-trolled creation, spatial separation and detection of entan-gled electrons thus constitute key elements in nanoscalequantum information processing. During the last one anda half decade, a large number of schemes for transport gen-eration and detection of electronic entanglement have beenproposed [2,3]. However, a clear-cut experimental demon-stration is still lacking. The main reason is the paramountdifficulty to, in a single nanosystem, generate, coherentlycontrol and unambigously detect the entanglement.An early key proposal for spin entanglement generationis the quantum dot based Andreev entangler [4]; a super-conductor coupled to two quantum dots, further coupledto normal leads. In the transport state Cooper pairs, elec-tron spin singlets, tunnel out from the superconductor, viathe dots, into the leads. At ideal operation, each Cooperpair is coherently split without altering the spin proper-ties. This gives a source of pairs of spatially separated,maximally spin entangled electrons in the leads. Subse-quent theoretical works extended on or further analyseddifferent properties of the entangler [4–8].Recently, important steps were taken towards an ex- E N T A N G L E R D E T E C T O R γγ γ A B Γ Γ P A+ P A- P B+ P B-A BAB A B ^ ^ ^
Fig. 1: Schematic of the combined entangler-detector system.The generic entangler emits single (dashed line) or split pairsof (solid line) particles, with spin dependent rates described bymatrices ˆ γ A , ˆ γ B and ˆ γ AB respectively, into the detector quan-tum dots A and B. The dot A (B), with a single, spin degener-ate level at energy ε A ( ε B ), is further tunnel coupled with thesame rate Γ A (Γ B ) to two ferromagnetic leads with polarisa-tions (cid:126)p A + = − (cid:126)p A − ( (cid:126)p B + = − (cid:126)p B − ). perimental demonstration of an Andreev entangler. In aseries of experiments [9–14], splitting of Cooper pairs intotwo quantum dots, formed in semiconductor nanowires orcarbon nanotubes, was reported. Efficient splitting of thep-1 a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec . Malkoc, C. Bergenfeldt, P. Samuelssonpairs was clearly demonstrated by current [13] and crosscorrelation measurements [12]. The experiments spurredfurther theoretical investigations on various aspects ofCooper pair splitters [15–21].Importantly, in none of the experiments reported [9–14]were the spin properties of the emitted pairs directly in-vestigated. To verify that the Cooper pair splitters alsowork as Andreev entanglers, emitting spin-singlets, non-local spin sensitive detection is necessary. To this aim,albeit challenging, a natural extension of the experimentswould be to couple the dots to ferromagnetic (FM) leads,see fig. 1. By performing a set of current cross correlationmeasurements [22–25] with non-collinear FM-polarization[22,26] the entanglement can be tested by a Bell inequalityor even quantified by spin state tomography [27, 28]. Tofacilitate such an experiment under realistic conditions,in the presence of spurious tunneling processes, spin-flipscattering in the dots and limited magnitude of the po-larization, several questions need to be carefully adressed.Most importantly i) how are the spin properties of thepair emitted by the entangler manifested in the cross cor-relators of the currents at the FM-leads and, if possible,ii) how can system parameters and detector settings beoptimized to allow for an unambiguous detection of theentanglement of the emitted state?In this work we provide answers to these questions byconsidering the full statistics of charge transfer betweenthe entangler and the FM-leads. To make the scheme ap-plicable beyond Cooper pair splitters we consider a genericentangler-detector setup, shown in fig. 1. The entangleremits arbitrary single and two-particle spin-states into thedots. The two dots together with the FM-leads constitutethe detector. To avoid cross-talk between the two detectordots as well as back-action of the detector on the entan-gler we consider a weak entangler-detector coupling. Im-portantly, working within a spin dependent quantum mas-ter equation formalism [29–32] we can treat both chargingeffects and spin-flip scattering in the dots, extending onearlier works [15,26,33] on statistics of entanglers coupled,via a single non-interacting dot, to FM-electrodes.The charge transfer statistics allows us to identify theindividual particle tunneling events [37] as well as theirspin properties. Based on this statistics we show howthe current cross correlations provide direct informationon the spin properties of the emitted, entangled two-particle state. In line with earlier work [22–25] we findthat spurious single particle tunneling does not affect thecorrelations. Moreover, depending on how well the FM-polarizations and the spin-flip rate are characterized, wepropose measurement strategies to optimize the entan-glement detection. To demonstrate the versatility of ourscheme we apply it both to the Andreev entangler [4] anda triple quantum dot entangler [34]. Entangler-Detector system - The combined entangler-detector system is shown in fig. 1. The detector subsystemconsists of two quantum dots, A and B, with each dot α = A, B coupled to two FM-leads α + , α − via tunnel barriers with rates Γ α + = Γ α − = Γ α /
2. Each dot has a single spindegenerate level, at energy ε A and ε B respectively. Doubleoccupancy of dot A or B is prevented by strong on-siteCoulomb interaction. The FM-leads have polarisations (cid:126)p α + = − (cid:126)p α − = ≡ p(cid:126)n α with identical magnitude p and unitvectors (cid:126)n A and (cid:126)n B non-collinear.The generic entangler is acting as a source of bothsingle electrons and split pairs of spin-correlated elec-trons, see fig. 1. The pair emission process is char-acterized by a 4 × γ AB , with elements γ σσ (cid:48) ,ττ (cid:48) AB where σ, σ (cid:48) , τ, τ (cid:48) = ↑ , ↓ . This describes emissionof pairs with a spin density matrix ˆ γ AB / tr[ˆ γ AB ] at arate tr[ˆ γ AB ]. As a key example, emission of spin sin-glets | Ψ S (cid:105) = ( |↑ A ↓ B (cid:105) − |↓ A ↑ B (cid:105) ) / √ γ givesˆ γ AB = γ | Ψ S (cid:105)(cid:104) Ψ S | . The emission of single particles is cor-respondingly described by 2 × γ α with ma-trix elements γ σσ (cid:48) α . Throughout the paper we consider γ σσ (cid:48) ,ττ (cid:48) AB , γ σσ (cid:48) A , γ σσ (cid:48) B (cid:28) Γ A , Γ B so that back-tunnelling fromthe dots to the entangler can be neglected. Moreover, weassume γ σσ (cid:48) α ≤ γ σσ (cid:48) ,ττ (cid:48) AB , achievable for relevant entanglers[4, 34]. In addition, we account for spin flip scattering inthe dots with a rate η , taken to be the same for A and B.The FM-leads are all kept at the same potential. More-over, a large bias is applied between the entangler and theFM-leads, in order to have all detector-entangler energylevels well inside the bias window. The temperature of theleads is much smaller than the bias as well as the distancefrom the detector-entangler energy levels to the edges ofthe bias window. This allows us to neglect back-tunnelingfrom the FM-leads into the dots, known to complicate theentanglement detection [35, 36]. Full transport statistics -
As we describe in detail be-low, in this high-bias regime the transport properties canbe described exactly within a quantum master equationapproach to the reduced spin density matrix of the dots.The full distribution of charge transferred to the FM-leads(during a long measurement time) is conveniently charac-terised [37] by a cumulant generating function F χ where χ = { χ A + , χ A − , χ B + , χ B − } denotes the set of lead count-ing fields. To leading order in the rate matrices we find F χ = (cid:88) α,m Tr (cid:104) ˆ Q ζαm ˆ γ α (cid:105) (cid:0) e iχ αm − (cid:1) (1)+ (cid:88) n,m Tr (cid:104) ( ˆ Q ζAn ⊗ ˆ Q ζBm )ˆ γ AB (cid:105) (cid:16) e i ( χ An + χ Bm ) − (cid:17) . where ⊗ denotes the direct product, ˆ Q ζαm = (1 / ζ α (cid:126)n αm · (cid:126)σ ] is a 2 × (cid:126)σ = (ˆ σ x , ˆ σ y , ˆ σ z )a vector of Pauli matrices and 0 ≤ ζ α ≤ ζ α = p α (1 − η α ) is a productof the FM-lead polarization p α and 1 − η α , where η α = η/ (Γ α + η ) is the dimensionless spin-flip rate in dot α ,ranging from 0 for negligible spin-flip scattering to 1 forcomplete spin randomization.Eq. (1) is the key technical result of our paper. It al-lows for a compelling and physically clear picture of thep-2ull Counting Statistics of Generic Spin Entangler with Quantum Dot-Ferromagnet detectors.transport statistics through the entangler-detector systemand provides means to identify the spin properties of theemitted pairs. The generating function F χ in Eq. (1) de-scribes a set of independent Poisson transfer processes ofsingle and pairs of particles: • Each term ∝ − e i ( χ Am + χ Bn ) describes a pair of par-ticles arriving, one particle to lead Am and one to Bn ,with a transfer rate Tr[( ˆ Q ζAm ⊗ ˆ Q ζBn )ˆ γ AB ]. The transferrate depends on the spin properties of the emitted pair,via the rate matrix ˆ γ AB . In particular, for an emittedsinglet ˆ γ AB = γ | Ψ S (cid:105)(cid:104) Ψ S | we have the two-particle rate( γ/ − ζ A ζ B (cid:126)n Am · (cid:126)n Bn ]. This rate is dependent onthe relative orientation of the polarizations via (cid:126)n Am · (cid:126)n Bn ,clearly demonstrating the non-local character of the spin-correlations. • Each term ∝ − e iχ αm describes a single particle arrivingat terminal αm , with a transfer rate Tr[ ˆ Q ζαm ˆ γ α ]. Similarto the two-particle term, the transfer rate depends on thespin properties of the emitted particle via ˆ γ α . Detector efficiency vs entanglement suppression -
As isclear from Eq. (1), both finite spin-flip scattering η α > p α < ζ α <
1. Importantly, the transfer rates in F χ can be rewritten as follows, providing a different picture:Making use of the formal quantum operation approach[1] we can write the detector matrix as ˆ Q ζαm = E α ( ˆ Q αm )where E α (ˆ q ) = ζ α ˆ q + (1 − ζ α )Tr[ˆ q ]ˆ1 / × q and ˆ Q αm = (1 / (cid:126)n αm · (cid:126)σ ]the ideal detector matrix, for efficiency ζ α = 1. Notingthat we can write E α (ˆ q ) = [1 / ζ α )ˆ q + [1 / − ζ α )(ˆ σ x ˆ q ˆ σ x + ˆ σ y ˆ q ˆ σ y + ˆ σ z ˆ q ˆ σ z ), we can write the single par-ticle transfer rate as Tr[ E α ( ˆ Q αm )ˆ γ α ] = Tr[ ˆ Q αm E α (ˆ γ α )],describing perfect detection of a depolarized rate matrixˆ γ α . This is readily extended to the two-particle transferrate, which can be writtenTr[ E ( ˆ Q Am ⊗ ˆ Q Bn )ˆ γ AB ] = Tr[( ˆ Q Am ⊗ ˆ Q Bn ) E (ˆ γ AB )] (2)where E = E A ⊗ E B describes two independent, local de-polarization operations. For clarity, the depolarized two-particle rate matrix can be written explicitly E (ˆ γ AB ) = ζ A ζ B ˆ γ AB + ζ A (1 − ζ B )2 Tr B [ˆ γ AB ] ⊗ ˆ1+ ζ B (1 − ζ A )2 ˆ1 ⊗ Tr A [ˆ γ AB ] + (1 − ζ A )(1 − ζ B )4 ˆ1 ⊗ ˆ1 (3)where Tr α [ .. ] denotes a partial trace over the spin de-grees of freedom in dot α . Taking again the exam-ple of (maximally entangled) spin singlets emitted witha rate γ , i.e. ˆ γ AB = γ | Ψ S (cid:105)(cid:104) Ψ S | , the depolarized rate E (ˆ γ AB ) = γ [ ζ A ζ B | Ψ S (cid:105)(cid:104) Ψ S | + [1 / − ζ A ζ B )ˆ1 ⊗ ˆ1] de-scribes emission of Werner states [39], entangled only for ζ A ζ B > /
3. This clearly illustrates the following: the twoparticle transfer rate is the same for maximally entangledstates detected with reduced efficiency as for partially en-tangled states detected with unit efficiency. As we nowdiscuss, this insight greatly helps to develop measurementstrategies for an unambiguous entanglement detection.
Cross correlations and entanglement detection -
From F χ the different low frequency cumulants are obtained bysuccessive derivatives with respect to the counting fields.For the average electrical current at terminal Am (andsimilarly for B n ) we have I Am = − ie∂ χ Am F χ | χ =0 giving I Am = e Tr (cid:104) ˆ Q ζAm (ˆ γ A + Tr B [ˆ γ AB ]) (cid:105) . (4)The average current provides information about the singleparticle processes through A via ˆ γ A as well as the local,reduced single particle properties of the emitted pairs, viaTr B [ˆ γ AB ]. Consequently, I Am and I Bn are local quantitiesand can not provide full information on the emitted two-particle state, in particular not on the entanglement.Turning instead to the non-local cross correlations be-tween currents at reservoirs αm and βn , obtained as S αm,βn = − e ∂ χ αm ∂ χ βn F χ | χ =0 , we have S Am,Bn = e Tr (cid:104) ( ˆ Q ζAm ⊗ ˆ Q ζBn )ˆ γ AB (cid:105) (5)From Eqs. (5) and (1) it is clear that S Am,Bn is di-rectly proportional to the corresponding two-particle emis-sion rate. In particular, S Am,Bn provides direct informa-tion about the spin properties of the individual pairs, viaˆ γ AB . Moreover, S Am,Bn does not contain any informationabout the spurious single-particle emission or correlationbetween emitted pairs (contributes only to next order inˆ γ α / Γ β , ˆ γ AB / Γ β ). This illustrates in a compelling way thata long time measurement, with a large number of emittedpairs collected in the leads, effectively [22–25] constitutesan average over a large number of identically prepared pairspin states.Importantly, the form of the cross correlator in Eq. (5)allows in principle for entanglement detection via e.g. [27]a complete tomographic reconstruction of ˆ γ AB or a testof a Bell inequality [22–25]. In both cases, one needs toperform a set of measurements with different polarizationsettings (cid:126)n A and (cid:126)n B . However, the interpretation of themeasurement result, in particular the answer to the ques-tion “is the emitted state entangled?”, depends both onthe method of detection as well as an accurate knowledgeof the detector efficiencies. This is clearly illustrated byconsidering separately two cases: • When the detector efficiencies ζ α are accurately known,i.e. both the FM-polarizations p α and the spin-flip rates η α can be faithfully determined, a quantum spin tomogra-phy is in principle viable [28] for arbitrary ζ α . In contrast,a Bell inequality test can only be performed for a lim-ited range of efficiencies. Interestingly, as was discussedby Eberhardt already two decades ago [38], an a prioriknowledge about ζ α allows one to optimize polarizationsettings (cid:126)n α , increasing the efficiency range for which aBell inequality violation is possible. • When the efficiencies ζ α are not known, a quantum spintomography can give an incorrect two-particle state. Inparticular, the reconstructed state can have an entangle-ment larger than the emitted state, opening up for anp-3. Malkoc, C. Bergenfeldt, P. Samuelssonincorrect conclusion that entanglement has been detected.This “false detection” scenario can be illustrated by con-sidering emission of Werner states κ | Ψ S (cid:105)(cid:104) Ψ S | + [1 / − κ )ˆ1 ⊗ ˆ1. An underestimation of the detector joint efficiency ζ A ζ B by a factor 1 /κ will then lead to tomographically re-constructed singlet state | Ψ S (cid:105)(cid:104) Ψ S | , maximally entangled.In contrast, a Bell test with unknown detector efficien-cies can not lead to “false detection” of entanglement [40].However, for unknown or ill-characterized efficiencies it isdifficult to identify detector settings for an optimal viola-tion, making a Bell test experimentally more demanding. Quantum master equation -
We now turn to the deriva-tion of the transport statistics, in terms of the reduceddensity operator of the state in the dots, ρ = ρ ( t ). In thehigh-bias limit under consideration, the dynamics of ρ canbe described exactly by a Liouville equation on Lindbladform dρdt = L H ( ρ ) + L ( ρ ) + L ( ρ ) + L η ( ρ ) + L χF M ( ρ ) . (6)Here the term L H ( ρ ) = − i ¯ h [ H d , ρ ] describes the free evo-lution of the dot state, with H d = (cid:80) ασ ε α d † ασ d ασ and d † ασ ( d ασ ) creating (annihilating) electrons in dot α , withspin σ . The terms L ( ρ ) and L ( ρ ) describe the injection,from the entangler to the dots, of single and two-particlestates respectively and are given by L ( ρ ) = (cid:88) ασσ (cid:48) γ σσ (cid:48) α (cid:20) d † ασ ρd ασ (cid:48) − { d ασ (cid:48) d † ασ , ρ } (cid:21) (7)and, L ( ρ ) = (cid:88) τστ (cid:48) σ (cid:48) γ σσ (cid:48) ,ττ (cid:48) AB (cid:104) d † Aσ d † Bτ ρd Bτ (cid:48) d Aσ (cid:48) (8) − { d Bτ (cid:48) d Aσ (cid:48) d † Aσ d † Bτ , ρ } (cid:21) . To preserve the trace and ensure positivity of the den-sity matrix the emission rate matrices must be Hermitianˆ γ α = ˆ γ † α , ˆ γ AB = ˆ γ † AB . Entangler examples with detailedderivations of the one and two-particle rate matrices aregiven below. Spin flip scattering in the dots, with a rate η , is accounted for by the term L η ( ρ ) = η (cid:88) ασ (cid:20) d ασ ρd † ασ − { d † ασ d ασ , ρ } (cid:21) (9)The last term L χF M ( ρ ) accounts for the coupling to theFM-reservoirs. In order to describe the full charge trans-fer statistics we have included counting fields χ αm , with m = ± , in the terms describing tunnelling out to the FM-reservoirs. This gives L χF M ( ρ ) = (cid:88) αmσ (cid:48) σ Γ α (cid:20) d ασ ρd † ασ (cid:48) Q σ (cid:48) σαm e iχ αm − { d † ασ d ασ , ρ } (cid:21) (10) where Q σσ (cid:48) αm = ( ˆ Q αm ) σσ (cid:48) .Working in the local spin-Fock basis {| (cid:105) , | σ A (cid:105) , | τ B (cid:105) , | σ A τ B (cid:105)} , with σ, τ = ↑ , ↓ , Eq. (6) canbe written as a linear matrix equation ∂ t (cid:126)ρ χ = ˆ M χ (cid:126)ρ χ .Here ˆ M χ is a χ -dependent transition rate matrix and the( χ -dependent) vector (cid:126)ρ χ = [ ρ , (cid:126)ρ A , (cid:126)ρ B , (cid:126)ρ AB ], where ρ isthe matrix element for both dots empty and (cid:126)ρ α ( (cid:126)ρ AB ) avector with the elements for only dot α (both dot A andB) occupied, including one (two) particle spin coherences.Following ref. [29] the generating function F χ can thenbe obtained from the eigenvalue problemˆ M χ (cid:126)ρ χ = F χ (cid:126)ρ χ , (11)To leading order in γ σσ (cid:48) α / Γ α , γ σσ (cid:48) ,ττ (cid:48) AB / Γ α , it is possible tosolve Eq. (11) analytically, giving Eq. (1) above. Transfer rate matrices -
We now turn to a discussionof the single and two-particle transfer rate matrices ˆ γ α and ˆ γ AB . As pointed out above, we consider an entanglersubsystem which is weakly coupled to the dots A and B.Together with the high bias limit, this implies that singleand pairs of particles which have tunneled out of the en-tangler will only tunnel out to the FM-leads, and neverback to the entangler. In addition, we make the assump-tion that the many-body state of the entangler has a welldefined energy E e . We can then evalute the rate matriceswithin a T-matrix formulation of time-dependent many-body perturbation theory [41]. Discussing explicitly thekey quantity, the two-particle rate matrix ˆ γ AB , we havethe spin dependent golden rule resultˆ γ AB = ˆ T Γ A + Γ B ( ε A + ε B − E e ) + (Γ A + Γ B ) / . (12)The spin dependence is contained in the matrix ˆ T , whichhas elements( ˆ T ) σσ (cid:48) ,ττ (cid:48) = Tr { ρ e H (2) T | σ A τ B (cid:105)(cid:104) τ (cid:48) B σ (cid:48) A | H (2) T } , (13)where ρ e is the density matrix of the isolated entan-gler, H (2) T the effective two-particle entangler-dot tunnel-ing Hamiltonian and the trace is running over the degreesof freedom of both the entangler and the dots. The sec-ond factor in Eq. (12) is (cid:82) dE A dE B ν A ( E A ) ν B ( E B ) δ ( E A + E B − E i ) where ν α ( E α ) = (Γ α /π )[( E α − (cid:15) α ) + Γ α / − is the density of states of dot α = A, B , broadened by thecoupling to the FM-leads. The single particle rate matri-ces ˆ γ α can be calculated in a similar (and simpler) man-ner. Note that in evaluating Eq. (12), the polarizationand tunnel rate conditions (cid:126)p α + = − (cid:126)p α − and Γ α + = Γ α − allow us to treat the two FM-reservoirs coupled to dot α as one effective, normal reservoir. For the same reason,spin-flip scattering in the dots have no effect on the resultin Eq. (12).The expression in Eq. (12) opens up for a treatment of awide range of two-particle spin entanglers coupled to quan-tum dot detectors, including entanglers with a possiblyp-4ull Counting Statistics of Generic Spin Entangler with Quantum Dot-Ferromagnet detectors.mixed spin state or spin-dependent entangler-dot tunnel-ing. The only information required to evaluate the transferrate matrices is the effective one and two-particle tunnel-ing rates and the energy and spin-properties of the isolatedentangler state. To demonstrate the viability of our ap-proach we analyse two proposed quantum dot based spinentanglers. Andreev entangler -
We first consider an Andreev entan-gler, of large interest due to the recent Cooper pair splitterexperiments [9–14], discussed above. For completeness weshow a schematic of the Andreev entangler-dot detectorsystem in fig. 2, including relevant energies and tunnel-ing rates. In line with our earlier assumptions we here E A B P Γ A Γ B P E e = 0 ε A ε B A B
Δ-Δ a) b) J Fig. 2: a) Schematic of Andreev entangler-detector system.Split Cooper pairs are emitted from the superconductor (SC)into dots a and B with a rate J . For other tunnel rates we referto fig. 1. b) Energy level diagram of the combined entangler-detector system, with superconducting gap ∆ and ground state E e = 0 as well as dot level energies (cid:15) A , (cid:15) B shown. consider the case where the dominating two-particle pro-cess is emission of split Cooper pairs into the two dots.The processes where the two particles tunnel to the samedot are suppressed due to large on-site Coulomb interac-tion. Moreover, considering dot energies (cid:15) A , (cid:15) B well insidethe superconducting gap ∆ the single particle rates aresmaller than or of the order of the two-particle, pair tun-neling rates [4], i.e. γ σσ (cid:48) α ≤ γ σσ (cid:48) ,ττ (cid:48) AB .We recall that the state of the isolated entangler is thesuperconducting ground state, with an energy E e heretaken to be zero. Moreover, the effective two-particle tun-neling Hamiltonian can be conveniently be written [42] H (2) T = J (cid:104) b ( d † A ↑ d † B ↓ − d † A ↓ d † B ↑ ) + h.c. (cid:105) . Here J is thetunneling element, depending on the properties of the su-perconductor and the coupling to the dots, and b thedestruction operator of a Cooper pair in the superconduc-tor with the properties (cid:104) b (cid:105) = (cid:104) b † (cid:105) = (cid:104) b † b (cid:105) = 1, wherethe average is taken with respect to the superconductingground state. We then directly obtain ˆ T = J | Ψ S (cid:105)(cid:104) Ψ S | with | Ψ S (cid:105) the spin singlet state and, writing out explicitly,the two-particle rate matrix in Eq. (12) asˆ γ AB = 2 J (Γ A + Γ B )( ε A + ε B ) + (Γ A + Γ B ) / | Ψ S (cid:105)(cid:104) Ψ S | . (14)Along the same lines one can obtain ˆ γ α . With ˆ γ AB andˆ γ α we can then via Eq. (1) evaluate the full, spin depen-dent transport statistics for the Andreev entangler. We stress that from the expressions for the current, Eq. (4),and cross correlations, Eq. (5), we reproduce known re-sults [4, 6] in the parameter limits corresponding to ourassumptions. Triple dot entangler -
As a second example we considertriple-dot entangler proposed in ref. [34]. Here the entan-gler consists of a quantum dot with a single, spin degener-ate level at energy (cid:15) d and an on-site interaction strength U , coupled to a normal lead. The entangler dot is furthercoupled to the detector dots via tunnel barriers with rates t A = t B = t . A schematic of the entangler-detector sys-tem is shown in fig. 3. As stated above, double occupancyof the detector dots A and B is prohibited by strong on-site interactions. To have a two particle rate larger thanor of the order of the single particle rates the level en-ergies (cid:15) d , (cid:15) A and (cid:15) B are tuned to meet the two-particleresonance condition 2 (cid:15) d + U ≈ (cid:15) A + (cid:15) B . Moreover, singleparticle resonances, (cid:15) α ≈ (cid:15) d , (cid:15) d + U , are avoided.Under these assumptions, together with the high biascondition, the state of the entangler dot is simply thedouble occupied level d † e ↑ d † e ↓ | (cid:105) , with d † eσ creating an elec-tron with spin σ in the entangler dot. The effectivetwo-particle tunneling Hamiltonian is further given by H (2) T = [2 t /U ]( d e ↑ d e ↓ [ d † A ↑ d † B ↓ − d † A ↓ d † B ↑ ] + h.c. ), wherein the prefactor t comes from the two entangler-detectordot tunnel events and U is the ”energy cost” for the virtualstate created by the first particle, tunneling off resonance.The two-particle rate matrix in Eq. (12) then becomesˆ γ AB = 8 t U Γ A + Γ B δ(cid:15) + (Γ A + Γ B ) / | Ψ S (cid:105)(cid:104) Ψ S | . (15)where δ(cid:15) = 2 (cid:15) d + U − ( (cid:15) A + (cid:15) B ), the energy away fromtwo-particle resonance. Along the same lines one can ob- a) A B Γ A Γ B Γ S b) E t t A B P B P A ε A ε B ε d ε d +U Fig. 3: a) Schematic of triple dot entangler-detector system.The entangler dot is coupled to the detector dots via tunnelbarriers with rates t A , t B and to a normal lead with rate Γ S .For other tunnel rates we refer to fig. 1. b) Energy level di-agram of the combined entangler-detector system, with (cid:15) A , (cid:15) B the detector dot level energies, (cid:15) d the entangler dot level energyand U the entangler dot on-site Coulomb interaction strength.At 2 (cid:15) d + U ≈ (cid:15) A + (cid:15) B two particles tunnel resonantly from theentangler dot to the detector dots A and B, with an effectiverate ∝ t /U . tain ˆ γ α . From the expressions for the current, Eq. (4),we reproduce known results [34] in the parameter limitscorresponding to our assumptions.p-5. Malkoc, C. Bergenfeldt, P. Samuelsson Conclusions -
We have investigated the full count-ing statistics of spin dependent single and two-particletransfer in a generic entangler coupled to quantum dot-ferromagnet detectors. From the full statistics we haveidentified individual charge transfer events. Moreover,we have demonstrated that the current cross correlatorscan be used to determine the two-particle spin state evenin the presence of spin-flip scattering and limited fer-romagnet polarization. For the future, it would be in-teresting to investigate in more detail how the obtainedresults are modified when relaxing assumptions such asequal dot-ferromagnet couplings and ferromagnet polar-izations as well as single occupancy of detector dots, inorder to strengthen the connection to experimentally re-alizable systems. We note that during the preparationof the present manuscript we became aware of the recentwork [43] where related aspects of entanglement detectionin Cooper pair splitters were discussed.
Acknowledgements -
We acknowledge support from theSwedish VR.
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