Inducing spin-correlations and entanglement in a double quantum dot through non-equilibrium transport
aa r X i v : . [ c ond - m a t . s t r- e l ] D ec Inducing Spin-Correlations and Entanglement in a Double Quantum Dot through NonequilibriumTransport
C. A. B¨usser and F. Heidrich-Meisner Department of Physics and Arnold Sommerfeld Center for Theoretical Physics,Ludwig-Maximilians-University Munich, 80333 Munich, Germany ∗ Institute for Theoretical Physics II, Friedrich-Alexander University Erlangen-Nuremberg, 91058 Erlangen, Germany
For a double quantum dot system in a parallel geometry, we demonstrate that by combining the effects ofa flux and driving an electrical current through the structure, the spin correlations between electrons localizedin the dots can be controlled at will. In particular, a current can induce spin correlations even if the spins areuncorrelated in the initial equilibrium state. Therefore, we are able to engineer an entangled state in this double-dot structure. We take many-body correlations fully into account by simulating the real-time dynamics using thetime-dependent density matrix renormalization group method. Using a canonical transformation, we provide anintuitive explanation for our results, related to Ruderman-Kittel-Kasuya-Yoshida physics driven by the bias.
PACS numbers: 73.23.Hk, 72.15.Qm, 73.63.Kv
Introduction:
Considerable progress in nanotechnology inthe last decades has made possible the fabrication of new ar-tificial structures [1, 2] such as quantum dots (QDs), quan-tum rings, or molecular conductors. The physics of quantumdots in a parallel geometry is intriguing, since it allows one tostudy interference effects between electrons traveling throughdifferent paths, most notably realized in the Aharanov-Bohmeffect. Such structures have been studied in several experi-ments [3, 4]. Besides the interest in practical applications innanoelectronics or in fundamental many-body physics such asthe Kondo effect [4–6], double quantum dots (DQD) also playa vital role in the context of quantum information processing[7–9]. Generating, controlling, and detecting entangled statesin condensed matter systems is one of the challenges for futurequantum computation applications [10]. Various proposalsfor entangling spatially separated electrons have been put for-ward, such as, for instance, by splitting Cooper pairs [11, 12]or by manipulating spins in quantum dots [13]. In a DQD, anentangled state can be realized by putting the electrons into asinglet state [7, 8, 14]. Means of detecting entangled states ofelectrons were discussed in, e.g., Refs. [11, 15].In this work, we demonstrate that an entangled statebetween electrons localized in a DQD embedded in anAharonov-Bohm interferometer can be induced and con-trolled by sending an electrical current through the structure.In the presence of a flux, the initial state can even be fullyuncorrelated yet the nonequilibrium dynamics results in non-zero spin correlations in the steady state. The sign and thestrength of such steady-state spin correlations depend on volt-age, interactions, and the flux. The generation of entangle-ment through nonequilibrium dynamics in quantum dots, withdifferent set-ups, has been discussed in Refs. [16]. An ad-ditional motivation for our work stems from the current in-terest, both from theory [17, 18] and from experiment (see,e.g., [19]), in the nonequilibrium dynamics of nano-structureswith strong electronic correlations. We emphasize that wetreat both interactions and nonequilibrium dynamics in a well-controlled manner using the time-dependent density matrixrenormalization group (DMRG) method [20]. As we will see, -0.5-0.2500.250.50.751 J ( t ) time -0.2-0.100.1 S ( t ) N=41 - φ=π
N=41 - φ=0 C ( t ) C - N=41 - φ=π (a)(b) U=0.5U/
Γ=2 φ −V/2 V/2 FIG. 1. (Color online) (a) Current J ( t ) for φ = 0 and π . (b)Spin correlation S ( t ) and concurrence C ( t ) . All for N = 41 , U = 0 . , V = 0 . . Inset in (a): Sketch of the DQD structure. the effect of inducing spin correlations is the largest at volt-ages ∼ Θ( W/ ( W is the bandwidth of the reservoirs) whereKondo correlations cease to matter [2, 5]. The model:
We model the quantum dots as Anderson im-purities resulting in the Hamiltonian [depicted in the inset ofFig. 1(a)], H = H l + H hy1 + H hy2 + H int , (1) H l = X α = L,R N − X i =1; σ h − t ( c † αiσ c αi +1 σ + h.c. ) i + X α = L,R N X i =1; σ µ α n αiσ , (2) H hy1 = − t ′ [ d † σ c L σ + c † R σ d σ + h.c. ] , (3) H hy2 = − t ′ [ d † σ c L σ + e iφ c † R σ d σ + h.c. ] , (4) H int = X j =1 , σ [ U n jσ n j ¯ σ + V g n jσ ] . (5)The system size is N + 2 where N is the number of sitesin the left or right lead. The two dots are at the center ofthe system labeled by j = 1 , . The Hamiltonian consists offour parts: First, the non-interacting leads H l with a constanthopping matrix element t = 1 used as the unit of energy( ¯ h = 1 , e = 1 ). Second, the terms H hy1 and H hy2 give riseto the hybridization between the localized levels of the dotsand the leads. We consider fully symmetric tunnel couplings,i.e., | t ′ | = | t ′ | = t ′ (see the supplementary material [21] for adiscussion of asymmetric couplings). We define the tunnelingstrength by Γ = 2 πt ′ ρ leads ( E F ) = 2 t ′ , where ρ leads ( E F ) is the local density of states (LDOS) of the leads at the Fermienergy E F . In the hopping matrix element between the sec-ond dot and the right lead we incorporate an arbitrary phase φ . Finally, there is the interacting region H int with the twoquantum dots, which are both subject to the same Coulombrepulsion U and a gate potential V g = − U/ such that bothdots are kept at half filling. The operator c † αlσ ( c αlσ ) creates(annihilates) an electron at site l in the α = L, R lead withspin σ while d † jσ ( d jσ ) acts on dot j ; n αlσ = c † αlσ c αlσ asusual. In Eq. (2), µ L and µ R mimic the chemical potentials ofthe leads.The ground state and the linear conductance of DQDsEq. (1) were extensively studied in Ref. [22]. A closely re-lated DQD model with a finite flux φ and with spin-polarizedelectrons was discussed in Ref. [23].The phase included in Eq. (4) may have a different mean-ing depending on the specific physical realization. The mostobvious one is to associate φ with a magnetic flux that piercesthe ring structure containing the two dots and the first site fromeach lead as shown in Fig. 1(b). As usual, one can use a gaugetransformation such that the flux appears in only one of thefour hopping matrix elements. Another situation describedby Eq. (4) is a single quantum dot with two levels where bysymmetry the levels can couple with a phase difference to theleads.We use DMRG [20] to obtain the steady state in the pres-ence of a finite bias voltage by time-evolving the wave-function | Ψ( t ) i and then measuring its properties such as thecurrent and spin correlations as a function of time t . Thismethod has been successfully used to study nonequilibriumtransport through nano-structures with electronic correlations[17, 24, 25]. We evaluate the spin correlations from [26]S ( t ) = h Ψ( t ) | ~S · ~S | Ψ( t ) i . (6)The current between two sites in the leads is defined as J l,m ( t ) = it X σ h Ψ( t ) | c † lσ c mσ − c † mσ c lσ | Ψ( t ) i . (7)In the figures, we display the current J = ( J L ,L + J R ,R ) / averaged over the first link in the left and right lead.Our simulations start from the system in equilibrium witha finite Γ = 0 and a charge per spin of h n jσ i = 0 . on bothdots. At time t = 0 , we turn on a bias voltage V = µ L − µ R that drives the system out of equilibrium. We work at V/2 (a.2)
V/2 (b.2)(b.1) −channel s −channel a −channel a −channel s (a.1) φ = 0 (a) (b) φ = π FIG. 2. (Color online)
Illustration of the canonical transformationEq. (8). (a) φ = 0 . (b) φ = π . (a.1), (b.1): V = 0 ; (a.2), (b.2): V = 0 . The application of the bias leads to a ladder structure wherethe bias acts like a transverse hopping matrix element between thesymmetric ( s -channel) and antisymmetric states ( a -channel) definedin Eq. (8). large values of Γ = 0 . such that the transient dynamics toreach the steady state is short [25]. The two quantum dots aretreated as a super-site permitting the use of a Trotter-Suzukibreakup of exp ( − iHt ) [27]. The time step is δt ∼ . and weenforce a fixed discarded weight [27] of − or less, keepinga maximum of 2000 DMRG states. All runs are performed atan overall half filling of dots and leads. Results:
In Fig. 1, we elucidate the time dependence ofthe current and spin correlations, comparing the behavior of φ = 0 to φ = π . Similar to a single quantum dot [25], thecurrent undergoes transient dynamics, and then takes a quasi-stationary value (i.e., a plateau in time), which we shall referto as the steady-state regime. Note that on finite systems, thereis a system-size dependent revival time, resulting in a decay ofthe steady state current and a sign change (realized for t > ∼ .For a discussion of transient time scales as well as an analysisof time dependent data for currents, see Ref. [25].For the spin correlations shown in Fig. 1(b), we first observethat in the initial state, S > for φ = 0 whereas the corre-lation vanishes for φ = π . The application of the bias voltagedoes virtually not affect the value of S for φ = 0 , which re-mains positive. The more interesting behavior is realized for φ = π . As a function of time, S decreases and approachesa roughly constant value. The transient time is comparableto the one for the current and is of order / Γ . Moreover, thetransients are suppressed by increasing the bias, similar to asingle quantum dot [25]. This finite and large spin correlationbetween the spins localized in the dots that emerges in thesteady state and that is induced by driving a current throughthe structure is the main aspect of our work. It implies thatnonequilibrium dynamics can be used to prepare a DQD in acorrelated and thus entangled state.To link the spin correlations to entanglement we use theconcurrence C [28, 29]. For instance, the concurrence ap-proaches C = 1 if the spin correlation is − / and if thereare no charge fluctuations on the dots [21]. In Fig. 1(b) weinclude the concurrence versus time calculated for φ = π . Weobserve that for t = 0 the concurrence is zero showing thatthe dots are not entangled. Applying the bias, and after reach-ing the steady state for the spin correlations, the concurrencetakes a value C ∼ . corresponding to a finite entangle-ment between the dots.The qualitative behavior of the spin correlations can be un-derstood by using a canonical transformation of the states ofthe leads, which is given by (see, e.g., [23, 30]): c γlσ = ( c Rlσ ± c Llσ ) / √ , (8)where γ = s, a are the symmetric and antisymmetric com-binations, respectively. The result of this transformation issketched in Fig. 2, where the leads shown there now repre-sent the new states obtained from Eq. (8). In the absence ofa bias voltage, there is no direct coupling between these newstates, as depicted in Figs. 2(a.1) and (b.1). Most importantly,the dots are coupled to only the symmetric states for φ = 0 ,whereas for φ = π , dot j = 1 is coupled to the symmetricstates and dot j = 2 to the antisymmetric ones. For φ = 0 , theRuderman-Kittel-Kasuya-Yoshida (RKKY) interaction givesrise to a ferromagnetic correlation between the dots since eachpath that connects them involves an odd number of sites andsince the leads are at half-filling [22]. For φ = π , the dots arepart of two decoupled subsystems and therefore, S vanishes.Upon applying a bias, one effectively obtains a ladder ge-ometry where the voltage acts as a transverse coupling be-tween the symmetric and antisymmetric states of Eq. (8) asshown in Figs. 2(a.2) and (b.2). For φ = 0 , the coupling V only marginally affects the correlations. By contrast, for φ = π and V = 0 , the dots are now connected through pathswith an even number of sites in the effective leads and there-fore, in the ground state of such a geometry, one expects afinite negative spin correlation. Our numerical results shownin Fig. 1(b) unveil that the same behavior occurs in nonequi-librium as well. While here we focus on fully symmetric tun-nel couplings, the main results can be recovered in the case ofasymmetric couplings [21], and therefore, fine-tuning of pa-rameters is not necessary to observe a change of S inducedby a bias V .After qualitatively explaining the emergence of finite spin-correlations in the current-carrying stationary state, we nextstudy the dependence of the steady-state properties on thebias potential. We denote the steady-state values by h S i and h J i , obtained from averaging over time-dependent data in thesteady-state regime (compare Ref. [25]). Figure 3(a) shows h J i /V versus V for phases φ = 0 and π . For φ = 0 , h J i /V approaches a constant value at low bias [22]. For φ = π ,the linear conductance vanishes due to the Aharanov-Bohmeffect [23]. A finite voltage causes a finite current to flow inboth cases, but h J i /V for φ = 0 is always larger than in the φ = π case.In Fig. 3(b), we display the steady-state spin correlations h S i versus V . First, let us emphasize that data for thesteady-state values obtained from systems of different lengthsare included, showing that all our main results are quantita-tively robust against finite-size effects. For φ = 0 , a constantvalue of h S i > is found. A slight decrease appears for < J > / V N=25 -- φ=π
N=35 -- φ=π
N=41 -- φ=π
N=41 -- φ=0 V -0.3-0.2-0.100.1 < S > U/ Γ -0.18-0.16-0.14-0.12 < S > N=35 -- φ=π (a)(b)
U=0.5U/
Γ=2
Γ=0.25
V=0.5
FIG. 3. (Color online) (a) h J i /V vs. V for φ = 0 and φ = π in units of twice the conductance quantum G . (b) Steady-state spincorrelations h S i vs. bias V . The figure shows data for differentsystem sizes N = 25 , , for φ = π . In (a) and (b), U/ Γ = 2 .Inset in (b): Steady-state spin correlations vs. U/ Γ for V = 0 . . V > ∼ , which we trace back to the variation of the LDOSof the leads seen by the dots. Since in our simulations wework with tight-binding bands with a finite band-width andband curvature, this LDOS decreases with V . For the caseof φ = π , the value of the steady-state correlations can betuned by the bias voltage and in fact, h S i increases with U . Therefore, to obtain a strong correlation a large voltageis needed putting the system out of the Kondo regime. Asthe figure clearly shows, we can get to h S i ≈ − . for U/ Γ = 2 . We therefore realize a mixed state with singlet cor-relations dominating over triplet correlations. If there were nocharge fluctuations then this value of S would correspondto a Werner state [31, 32] with 50% of the weight in the sin-glet. In our case, however, the current needed to obtain theentangled state typically induces charge fluctuations at large V . Therefore, h S i ≤ − / implies an even larger relativecontribution of the singlet over the triplet than in a situationwithout any charge fluctuations. The fact that a finite voltageunavoidably induces charge fluctuations is the reason why thesteady-state spin correlations do not reach their largest possi-ble negative value of − / .The steady-state values further depend on U/ Γ . To elu-cidate this, we plot h S i versus U in the inset of Fig. 3(b)for a fixed value of V = 0 . , by increasing U/ Γ to 4, i.e.,in a regime where charge fluctuations are still relevant evenin equilibrium. As expected, the larger U , the more stronglycharge fluctuations are suppressed, leading to larger steady-state spin correlations. Therefore, either a large U/ Γ at a fixedvoltage or applying a large voltage order of V ∼ t inducesthe largest steady-state correlations. Fortunately, many exper-iments with DQDs realize U/ Γ > ∼ [4, 5], thus relaxing therequirement on voltage. An important role of U is to define alocal spin as in many other quantum information applicationof QDs [8].So far we have investigated the dependence of correlationson V , U and φ in nonequilibrium, comparing the cases of φ = 0 to φ = π . An additional degree of tunability can beadded if the phase can take arbitrary values (see Fig. S1 inthe supplementary material [21]). As expected from the dis-cussion of Fig. 2, h S i is positive for small φ at V = 0 andthen decreases to zero as φ = π is approached. This transi-tion to the uncorrelated case of φ = π is continuous. At afinite voltage, it is possible to go from positive steady-statecorrelations to negative ones by changing φ . For the param-eters of Fig. 3(a), the steady-state correlations change sign at φ c ≈ . π (see Fig. S1). This value depends both on U and V . To summarize, the steady-state correlations can be tunedboth in sign and magnitude by changing V , φ , and U . Wehave further verified that the steady-state correlations are in-dependent of the intial conditions [21].Based on the qualitative picture developed so far, we con-clude that the steady-state correlations are a result of mixingthe symmetric and antisymmetric states of lead electrons innonequilibrium. At finite U , this may be viewed as an RKKYeffect in nonequilibrium. A discussion on how to estimate theeffective indirect coupling J eff ( V ) induced by the bias can befound in the supplementary material [21]. As is well knownfrom the physics of the RKKY effect in equilibrium the spincorrelation induced by indirect exchange is destroyed for tem-peratures larger than Γ [33]. For the nonequilibrium versionof RKKY discussed here, we expect that temperatures shouldbe smaller than the effective strength J eff ( V ) shown in [21]for thermal fluctuations not to affect the induced correlations.Finally, we study the behavior of S under quenches ofparameters of the Hamiltonian Eq. (1). We proceed as before,i.e., a finite bias voltage V > is turned on at t = 0 , and inaddition we instantaneously change some of the tunnel cou-plings at a time t q ≥ .We fnnd that if we disconnect the quantum dots from theleads at time t q > by setting t ′ = t ′ = 0 after the steadystate has been established, as expected, the spins remain in acorrelated state after isolating them from the reservoirs (seeFig. S4(b) in [21]).In a second example, after reaching the steady state, we iso-late one of the dots while the current continues to flow throughthe other. This results in the loss of the spin correlations af-ter a short transient time (see Fig. S4(c) in [21]). Therefore,control over the tunneling matrix elements allows one to putthe system back into its original uncorrelated state. Both thegeneration of entanglement and the removal happen on shorttimes scales, similar to the proposals discussed in [16]. Summary:
In this work, we demonstrated that spin correla-tions between spatially separated electrons localized in a par-allel DQD embedded in the rings of an Aharonov-Bohm in-terferometer can be induced and modified by driving a currentthrough the structure. The steady-state correlations depend onvoltage, the flux, and Coulomb interactions. Control over theindividual tunneling couplings would allow one to isolate theentangled spins from the environment or to remove the entan- glement again. The mechanism behind this time-dependentformation of correlations can be thought of as an RKKY ef-fect in nonequilibrium. Our results may be relevant for appli-cations of DQD structures in quantum information processing.
Acknowledgments -
We thank S. Andergassen, E. Dagotto,L.G.G.V. Dias da Silva, A. E. Feiguin, I. Hamad, G. B. Mar-tins, G. Roux, and L. Vidmar for helpful discussions. We areindebted to G. Burkard, F. Marquardt, and V. Meden for a crit-ical reading of the manuscript and valuable comments. Thiswork was supported by the
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DEPENDENCE OF STEADY-STATE SPIN CORRELATIONSON THE PHASE
Figure 4 shows the dependence of the spin correlations onthe phase φ , both in equilibrium ( V = 0 ), and in the steadystate ( V = 0 . ). At a finite voltage, it is possible to go frompositive spin correlations to negative ones by changing φ . Forthis set of parameters, the steady-state correlations changesign at φ c = 0 . . See main article for further discussion. CONCURRENCE
In this Section we use the concurrence to quantify the en-tanglement (see Refs. [26] and [27] in the main paper). Firstwe define the single fermion operator at the dots 1,2 as N si = n i ↑ + n i ↓ − n i ↑ n i ↓ . (9)This operator projects onto the subspace with exactly onefermion on the dot- i . Using N si the concurrence (see Ref.[27]) can be written as C ( t ) = max (cid:26) , − − S ( t ) h Ψ( t ) | N s N s | Ψ( t ) i (cid:27) . (10)Note that the concurrence takes its maximum value 1 when S ( t ) / h Ψ( t ) | N s N s | Ψ( t ) i → − / . One example to getthis is h S i = − / and h Ψ( t ) | N s N s | Ψ( t ) i = 1 .In Figure 5 we present the results for the average of the con-currence in the steady state as a function of the bias V and thephase φ . In panel (a) we see that for V > . the concur-rence starts to be non-zero showing that the singlet starts to berelevant. Increasing the bias the concurrence grows monoton-ically at the same time at which the effective RKKY mech-anism becomes stronger. Note that while h S i saturates at V ∼ , h C i increases since h N s N s i decreases. φ / π -0.2-0.100.1 < S > V=0V=0.5U=0.5U/
Γ=2
FIG. 4. Spin correlations h S i as a function of phase φ for V = 0 and in the steady state with V = 0 . for the parameters of Fig. 3. V < C > φ /π < C > (a)(b) U = 0.5U/
Γ = 2 φ = π
U = 0.5U/
Γ = 2
V = 0.5
FIG. 5. Concurrence h C i as a function of (a) bias V and (b) phase φ in the steady state with V = 0 . for the parameters of Fig. 3. In panel (b) we present the concurrence vs φ . For φ = 0 thesystem is in the state where the spins of the dots are parallel.As before there is a critical value of the phase, close to φ/π ∼ . , where the concurrence starts to be non-zero. Increasing φ from that value the concurrence grows reaching the valueexpected from panel (a) for V = 0 . . EFFECTIVE RKKY COUPLING
As discussed in the main article an effective spin coupling,similar to the RKKY effect, emerges when the bias is applied.To estimate the strength J eff of the effective coupling betweenthe localized spins in the dots in the steady state, we proceedas follows. First, in the set-up shown in Fig. 2(b.1) of themain article, we connect the impurities by an additional term J eff ~S · ~S and we then calculate the equilibrium correlations S = S ( J eff ) in this reference system. In the next step, weuse the numerically determined function h S i = h S i ( V ) and by equating it to S ( J eff ) , we obtain J eff = J eff ( V ) ,keeping all other parameters fixed in both the time-dependentsimulation and in the reference system. The results are pre-sented in Fig. 6 and its inset. QUENCHES
In this section we show how robust is the steady-statespin correlations against quenching certain parameters of theHamiltonian Eq. (1). As explained in the main text we pro-ceed as before. The finite bias voltage
V > is turned on at t = 0 . In addition we perform quenches of either the phase φ or some of the tunnel couplings at a time t q ≥ .The first example, presented in Fig. 7(a), consists of chang-ing the phase from to π . We show results for t q = 0 and V J e ff N=41 -- φ=π U/ Γ J e ff N=35 -- φ=π
U=0.5U/
Γ=2
Γ=0.25
V=0.5
FIG. 6. Effective J eff for DQD system as a function of V (mainpanel) and U (inset) calculated for φ = π . Same parameters as inFig. 3(b) of the main article. and observe that in both cases, the spin correlations evolvefrom their initial constant and positive value to a negativevalue that only depends on parameters in the steady state (i.e., V and φ ), but as this comparison shows, is virtually indepen-dent of the specific transient dynamics.In the second case, presented in Fig. 7(b), we simply dis-connect the quantum dots from the leads at time t q > afterthe steady state has been established. The spins remain in acorrelated state after isolating them from the reservoirs.Third, after reaching the steady state, we isolate one of thedots while the current continues to flow through the other one.As a consequence the spin correlations is destroyed after ashort transient time [see Fig. 7(c)]. ASYMMETRIC TUNNEL COUPLINGS
In this section we show that the key results of our workdescribed in the main text for a DQD with symmetric tunnelcouplings carrying over to the asymmetric case. We start fromthe most general form of the Hamiltonian from Eqs. (3) and(4) of the main text. The hybridization between the localizedlevels of the dots and the leads can be written as H hy1 = − t ′ L d † σ c L σ − t ′ R d † σ c R σ + H . c ., (11) H hy2 = − t ′ L d † σ c L σ − t ′ R e iφ d † σ c R σ + H . c . , (12)where tunnel matrix elements t αj , are considered that can ingeneral be different for each dot ( j = 1 , ) and lead ( α = L, R ). The tunneling strength is then defined by Γ j = t ′ Lj + t ′ Rj , j = 1 , . For each dot we introduce an angle θ j via θ j =arctan( t ′ Lj /t ′ Rj ) , and therefore, the terms from Eqs. (11) and(12) can be re-expressed as H hy j = − p Γ j d † jσ (sin θ j c L σ + cos θ j c R σ + H . c . ) . (13)We next generalize the canonical transformation of the leadoperators that was discussed in the main text in the followingway: c µiσ = (sin θ c Liσ + cos θ c Riσ ) , (14) c νiσ = (cos θ c Liσ − sin θ c Riσ ) . (15) S ( t ) No quench, φ=0
No quench, φ=πφ -quench at t q =0 φ -quench at t q =10 time -0.2-0.15-0.1-0.050 S ( t ) t’ =t’ =0 time t’ =0 (a)(b) (c)t q t q t q t q U=0.5U/
Γ=2 φ=π φ=π
FIG. 7. Behavior of the spin correlations in various quantumquenches. (a) Quench of the phase φ from φ = 0 to φ = π at atime t q = 0 , (dashed and dot-dashed curves). (b) Quench of thehopping matrix elements t ′ and t ′ to zero at t q = 20 . (c) Quenchof just t ′ to zero at time t q = 20 , effectively decoupling dot 1 whilethe current continues to flow through dot 2. For comparison, the datafrom Fig. 1(b) without any additional quenches are included (lineswith open/solid circles are for φ = π and φ = 0 , respectively). In allpanels, the time t q at which the quench is performed is indicated byarrows. In this figure, U/ Γ = 2 , V = 0 . , and N = 41 . In the special case of θ = π/ , this reduces to Eq. (8) fromthe main text with µ = s and ν = a . Note that the oper-ator c µiσ defined in Eq. (14) appears directly in H hy1 fromEq. (13). Therefore, the Quantum Dot (QD) j = 1 is just con-nected to the µ channel. The resulting configuration, i.e., QD1 coupled to the µ -channel and QD j = 2 coupled to eitherthe µ -, ν -, or both channels is depicted in Fig. 8The Hamiltonian of the leads H l with chemical potential µ R = − µ L = V / is, under this transformation, given by H l = H c + H V , (16) H c = − t X γ = µ,ν N − X i =1; σ (cid:16) c † γiσ c γi +1 σ + H . c . (cid:17) (17) H V = N X i =1; σ h t p ( c † νiσ c µiσ + H . c . ) + V ν n νiσ + V µ n µiσ ] , (18)with t p = − V sin θ cos θ , (19) V µ = − V ν = V (cid:0) cos θ − sin θ (cid:1) . (20)Similar to the case of fully symmetric couplings shown inFig. 2 of the main text, the bias potential connects the chan-nels µ and ν through an effective coupling t p . This couplingtakes its maximum value in the case of symmetric couplingsfor QD j = 1 , i.e., t ′ L = t ′ R and therefore, θ = π/ .Next, the second QD is connected to the new channels µ Set θ /π θ /π t ′ l t ′ r t ′ l t ′ r φ/π | t ′ µ | / Γ | t ′ ν | / Γ t p /V V µ /V h S i A 0.25 0.25 0.353 0.353 0.353 0.353 0.0 1.000 0.000 0.500 0.000 + B 0.09 0.89 0.139 0.480 0.169 -0.470 1.0 0.996 0.004 0.267 0.422 + C 0.48 0.47 0.031 0.499 0.497 -0.047 0.2 0.997 0.003 0.067 -0.496 + D 0.25 0.25 0.353 0.353 0.353 0.353 1.0 0.000 1.000 0.500 0.000 − E 0.14 0.64 0.213 0.452 -0.213 0.452 0.0 0.000 1.000 0.385 0.318 − F 0.07 0.43 0.110 0.488 0.488 0.110 0.5 0.090 0.910 0.212 0.452 − G 0.12 0.63 0.184 0.465 0.459 -0.198 0.2 0.049 0.951 0.342 0.364 − TABLE I. Relationship between the bare parameters of the system [see Eqs. (11) and (12)] and the tunnel matrix elements of the effectivechannels µ and ν [compare Eqs. (13) and (21)]. In this table we show a few examples for which the discussion of the main article applies evenfor an asymmetric coupling. We choose Γ = Γ = 0 . for simplicity. In the last column we list the expected sign of the spin correlations. and ν through: H hy2 = X σ (cid:16) t ′ µ d † σ c µ σ + t ′ ν d † σ c ν σ + H . c . (cid:17) . (21)with t ′ µ = p Γ (cid:0) cos θ cos θ + sin θ sin θ e iφ (cid:1) , (22) t ′ ν = p Γ (cid:0) − cos θ sin θ + sin θ cos θ e iφ (cid:1) . (23)It is straightforward to verify that | t ′ µ | + | t ′ ν | = Γ .Figure 9 shows, for φ = 0 , π/ and π , the square modu-lus of t ′ µ and t ′ ν as a function of θ and θ for U = 0 . , Γ = Γ = 0 . , N = 35 . For φ = 0 and φ = π we observediagonal stripes where t ′ µ or t ′ ν vanish resulting in the situ-ation depicted in Figs. 2 (a) and (b) from the main text. For φ = 0 it is obvious that, if θ = θ , the canonical transforma-tion will decouple both QDs from the ν -channel at the sametime as both dots have the same asymmetry.In order to reproduce the features of Fig. 2 of the main textfor the case of asymmetric couplings, we require that either t ′ µ or t ′ ν be close to zero. To illustrate this we have selectedeight parameter sets for the tunneling matrix elements and theflux φ that reproduce such situations, which we list in Table I.Each set of parameters contained in this table is labeled by aletter A,B, . . . , G.Set A is the completely symmetric situation with φ = 0 thatwe discussed in the main text. In this case, t ′ µ / √ Γ = 1 and t ′ ν = 0 . Therefore, QD j = 1 and j = 2 are connected to thesame effective lead and the spin correlations are positive in the t’ t’ −V/2 V/2 φ −channel−channel V µνµ V t p µ2 ν ν2 (b)(a) FIG. 8. Sketch of the representation of the DQD structure and itscoupling (a) the original leads and (b) the effective leads defined inEqs. (14) and (15). We refer to the effective leads as the µ - and ν -channel, respectively. steady state. A similar behavior is realized for sets B and C.Set B corresponds to a situation in which φ = π while set forC, we choose φ = 0 . π . Note that in set B the phase changesthe sign of t ′ R and the situation is thus similar to set A. In setC, all four couplings, t ′ L , t ′ R , t ′ L , and t ′ R , are different fromeach other and for the selected value of the phase φ = 0 . π ,no signs are affected.The examples labeled D to G are cases in which t ′ µ issmall, and therefore, where, according to the picture put for-ward in the discussion of Fig. 2(b) of the main text, we expectnegative spin correlations in the steady state. Set D is the com-pletely symmetric case with θ = θ = π/ and φ = π . Thiscase was analyzed in the main article and is shown here forcomparison. For set E, although φ = 0 , there is a differenceof π/ between θ and θ , resulting in a behavior similar to setD, despite the left-right asymmetry present in set E. Note that t ′ L is negative as a consequence, and therefore, the steady-state spin correlations are also negative, in accordance withthe discussion of the main article. The set F realizes a casewith φ = π/ and a small t ′ ν ∼ . . In this example, left-right symmetry is broken but there are still just two differentvalues for the four tunnel matrix elements t ′ αj . Finally, in setG, φ = 0 . π and all four tunnel matrix elements are different.The very small value of t ′ ν implies negative steady-state spincorrelation as before.To verify our predictions for the sign of steady-state spincorrelations that are based on the numbers listed in Table I,we have calculated the spin correlation as a function of time.The results are shown in Fig. 10. Sets A, B and C, as theyall have a small value of t ′ µ , lead to a spin correlation that ispositive in the initial state ( t = 0 ) whose sign and absolutevalue are barely affected by the bias at all. Sets D, E, F andG all have a small value for t ′ ν . As a consequence, the dotsare predominantly connected to one of the two channels µ or ν [compare Fig. 2(b.1) of the main text] and spin correlationsvanish in equilibrium, i.e., at for V = 0 . Upon applying a bias,negative spin correlations emerge, as expected. Note the smallvalue of the steady-state spin correlation in set F, consistentwith the observation that in this case, the value of the couplingbetween the effective leads t p is the smallest.As a conclusion, even for asymmetric tunnel couplings be-tween the DQD and the leads the application of a voltage can |t’ µ2 | / Γ for φ =0
0 0.25 0.5 0.75 1 θ / π θ / π |t’ µ2 | / Γ for φ = π /2
0 0.25 0.5 0.75 1 θ / π θ / π |t’ µ2 | / Γ for φ = π
0 0.25 0.5 0.75 1 θ / π θ / π |t’ ν2 | / Γ for φ =0
0 0.25 0.5 0.75 1 θ / π θ / π |t’ ν2 | / Γ for φ = π /2
0 0.25 0.5 0.75 1 θ / π θ / π |t’ ν2 | / Γ for φ = π
0 0.25 0.5 0.75 1 θ / π θ / π FIG. 9. Square modulus of the couplings between QD j = 2 and the effective channels µ and ν defined in Eq. (14) and (15). All curvesare for Γ = 0 . , U = 0 . , V g = − U/ and N = 35 . To reproduce the features of Figs. 2(a) and (b) of the main text, we need to require | t ′ ν | = 0 or | t ′ µ | = 0 , respectively. Note that t ′ µ and t ′ ν do not depend on Γ . time -0.2-0.100.1 S ( t ) ABC0 5 10 15 20 25 30-0.2-0.100.1 DEFG
U = 0.5U/ Γ = U/ Γ = 2 V = 0.5
FIG. 10. Spin correlations as a function of time for the parametersets presented in Table I. Note that for sets with small values of t ′ ν ,such as A, B and C, the spin correlations do practically not changewhen the bias is applied. By contrast, whenever t ′ µ is small, as isthe case in sets D, E, F and G, the spin correlations in the steady stateare negative. Sets A and D are the cases discussed in the main text,compare Fig. 1(b). induce and change spin correlations, which is the main resultof our work. The main requirement to obtain a large effect ofthe bias V on the nonequilibrium spin correlations is a smallvalue of either t µ2