Nuclear spin pumping and electron spin susceptibilities
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r Nuclear spin pumping and electron spin susceptibilities
J. Danon and Yu. V. Nazarov Dahlem Center for Complex Quantum Systems,Freie Universit¨at Berlin, Arnimallee 14, 14195 Berlin, Germany Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, The Netherlands
In this work we present a new formalism to evaluate the nuclear spin dynamics driven by hyperfineinteraction with non-equilibrium electron spins. To describe the dynamics up to second order in thehyperfine coupling, it suffices to evaluate the susceptibility and fluctuations of the electron spin. Ourapproach does not rely on a separation of electronic energy scales or the specific choice of electronicbasis states, thereby overcoming practical problems which may arise in certain limits when using amore traditional formalism based on rate equations.
INTRODUCTION
In recent years, considerable theoretical and experi-mental work is aimed at the controlled manipulation ofelectron spins in nanoscale solid state devices [1]. Thisresearch is motivated by actual applications, such as indigital information storage and read-out [2, 3], but alsoby the possibility of using the spin of electrons as com-putational units (qubits) in a quantum computer [4].One of the mechanisms influencing the electron spindynamics in these nano-devices is the hyperfine inter-action between the electron spin and the nuclear spinsof the device’s constituent material. For spin qubitshosted in semiconductor quantum dots [5], hyperfine in-teraction has been identified as the main source of deco-herence, causing the spin coherence time to be in thens range [6–8]. Much recent experimental and theo-retical work is aimed at suppressing this hyperfine in-duced decoherence [9, 10]. In other semiconductor nano-structures [11, 12] — possibly also metallic structures [13]— hyperfine interaction can even dominate the electronictransport properties and spin dynamics. Understand-ing the role of hyperfine interaction in spintronic devicestherefore is crucial in the development of the field.Traditionally, the interplay between electronic and nu-clear spins is treated in the convenient framework of rateequations [14, 15]. The rates of hyperfine transitions flip-ping nuclear spins ‘up’ and ‘down’ are separately cal-culated using Fermi’s golden rule, and the balancing ofthese rate yields the net nuclear spin pumping in thesystem. Although this approach works very well in manycases [16, 17], it can become cumbersome when the elec-tron spin dynamics are more complicated. Strong dy-namical nuclear spin effects have been observed in GaAsdouble quantum dots in the spin blockade regime un-der conditions of electron spin resonance [18]. Nuclearspin flips in this setup are due to second order processes,and some of the transitions have a vanishingly small en-ergy difference between initial and virtual state, whichcannot be dealt with in a standard Fermi golden ruleapproach [19]. The situation is even worse for systemswith strong spin-orbit coupling. In InAs nanowire double quantum dots in the spin blockade regime, signatures ofstrong dynamic polarization have been observed too [20].In this case, the presence of too many comparable energyscales makes it impossible to choose a unique set of ba-sis states to describe the electron dynamics in [21]. Itis clear that a basis-independent description of the cou-pled electron-nuclear spin dynamics, not involving anyseparate transition rates, is highly desirable.In this work we present an alternative method to eval-uate the spin dynamics of electrons and nuclei coupledto each other by hyperfine interaction. We show that, inorder to describe the nuclear spin dynamics up to secondorder in the hyperfine coupling, it suffices to evaluate thefluctuations and susceptibility of the electron spins in thesystem, which can be done using linear response theory.Our approach does not rely on the calculation of sepa-rate transition rates with Fermi’s golden rule, nor on thechoice of electronic basis states to work in. As a result,our formalism may be applicable in cases where Fermi’sgolden rule cannot be used, and thus provides a usefulalternative.Since a Fermi’s golden rule (FGR) calculation oftenhas to be adapted to specific circumstances [14–17], it isdifficult to pick a single implementation of it and makea good comparison to our approach. Therefore we willillustrate our ideas with two example systems: (i) We willshow how in a simplest toy system our formalism and asmart implementation of FGR produce identical results.(ii) We will investigate a more complicated system, whichcannot be dealt with in an FGR approach, and show howour formalism straightforwardly produces an equation forthe hyperfine driven nuclear spin dynamics.
MAIN RESULT
Let us start with presenting the main result of ourwork: an equation for the hyperfine driven dynamics ofthe expectation value of a nuclear spin coupled to (many)electronic spins. Under conditions which we will specifybelow, this equation takes the closed form d h ˆ K a i dt = ε abc S b h ˆ K c i − Q ab h ˆ K b i + 13 P a , (1)where { a, b, c } ∈ { x, y, z } , and we use the conventionthat over repeated indices still has to be summed. Theoperator ˆ K is the nuclear spin operator for the nucleusunder consideration, and ε abc is the permutation tensor.The vectors S and P , and the matrix Q are defined as S a = Av h ˆ S a ( r n , t ) i Q ab = ( Av ) ( δ ab R cc − R ba ) P a = ( Av ) K ( K + 1) ε abc χ bc , (2) A being the material-specific hyperfine coupling energy,1 /v the density of nuclei, K the total nuclear spin, r n theposition of the nucleus, and ˆ S the electron spin densityoperator, ˆ S ( r , t ) = ˆ ψ † α ( r , t ) σ αβ ˆ ψ β ( r , t ). Note that wehave set for transparency ¯ h = 1, or, in other words, weexpress all energies in terms of corresponding frequencies.We further assumed for simplicity only one type of spin-carrying nuclei present, having a constant density. Thesymbols χ and R represent correlation functions of thelocal electronic spin density, R ab = Z t h [ ˆ S a ( r n , t ) , ˆ S b ( r n , t ′ )] + i c dt ′ χ ab = − i Z t h [ ˆ S a ( r n , t ) , ˆ S b ( r n , t ′ )] − i c dt ′ , (3)where the square brackets denote the (anti)commutatorof two operators, i.e., [ ˆ A, ˆ B ] ± = ˆ A ˆ B ± ˆ B ˆ A , and the sub-script c means that we have to use the ‘connected’ partof the expectation value, i.e., remove the contribution ofthe averages h AB i c = h AB i − h A ih B i .Note that, since hyperfine interaction works two-way,the electronic dynamics, and thereby S , Q and P , can inturn also be affected by the state of the nuclear spins inthe system. This creates a feedback mechanism in Eq.(1), which makes the equation non-linear. However, in allpractical implementations, one can make use of the largedifference in time scales of the electronic and nuclear spindynamics. This allows to treat the ensemble of nuclearspins effectively as a static ‘classical’ magnetic field whenevaluating the electronic correlators.The first term in Eq. (1) is first order in the hyperfinecoupling A and gives rise to a precession of the nuclearspin around the direction of the time-averaged local elec-tron spin polarization. The precession frequency dependson the magnitude of this polarization | v h ˆ S ( r n , t ) i| , aswell as on the strength A of the hyperfine coupling.The other two terms are second order in A , and de-scribe the effect on the nucleus of fluctuations of electronspin. The correlation functions R ab in the matrix Q areordered as the ‘classical’ noise of electron spin ˆ S ( r n , t ). It persists even if the quantum operator of spin is replacedwith a classical fluctuating field affecting the nuclear spin.Since such fluctuations cause random rotation of this spinand therefore isotropization of its density matrix, the ef-fect of R ab is relaxation of the nuclear spin. The cor-relators χ ab in the vector P have the same structure asa Kubo formula and give the stationary susceptibility,that is, the response of ˆ S a in the point r n to a con-stant magnetic field in the b -direction, concentrated inthe same point. One can interpret the χ ab as the ‘quan-tum’ noise of electron spin: the part of the fluctuations inˆ S ( r n , t ) which solely exists due to the non-commutativityof the spin operators. It is worth to note that pumpingof the nuclear spin is produced by the asymmetric partof the tensor χ ab only. The Onsager relations state thatsusceptibilities are symmetric for an equilibrium system, χ ab = χ ba . Therefore, pumping can only take place if theelectron part of the system is driven out of equilibriumby some external agent.In order to find an explicit expression for d h ˆ K i /dt asgiven by Eq. (1), one needs to find the elements of S , Q and P . The first step is to find the steady-state electronspin density h ˆ S ( r n ) i , which immediately yields the vec-tor S . The elements of Q and P correspond to the localfluctuations and susceptibility of this electron spin den-sity, and can be evaluated using linear response theory,as we will illustrate below. Let us emphasize here thatnone of the steps in such a calculation involves the eval-uation of any separate transition rate between differentelectronic-nuclear levels or depends on the specific choiceof electronic basis states. DERIVATION
We will now show how to derive Eq. (1) from a secondorder perturbation theory in the hyperfine interaction.We start from an effective equation of motion for the re-duced density matrix ˆ ρ k of the nuclear spins which readsup to second order in the perturbation ˆ H hf ( t ) (see theAppendix for a derivation) d ˆ ρ k dt = Tr e n − i [ ˆ H hf ( t ) , ˆ ρ e ⊗ ˆ ρ k ] − − Z t [ ˆ H hf ( t ) , [ ˆ H hf ( t ′ ) , ˆ ρ e ⊗ ˆ ρ k ] − ] − dt ′ o , (4)where ˆ ρ e is the electronic part of the (quasi-)stationarydensity matrix described by the unperturbed Hamilto-nian ˆ H . This Hamiltonian includes all processes anddegrees of freedom which are relevant for the electronspin dynamics, such as (e.g. in a transport setup) all leadsand tunnel couplings to these leads. The trace Tr e { . . . } denotes a trace over the electronic degrees of freedom,where the horizontal bar indicates that from the result-ing correlation functions only the fully connected contri-bution should be taken into account (see Appendix).The perturbation ˆ H hf ( t ) accounts for the hyperfine in-teraction which couples the nuclear spins to the localelectron spin density. Assuming that the electrons of in-terest have s -type orbitals, we useˆ H hf ( t ) = Av X n ˆ S a ( r n , t ) ˆ K an ( t ) , (5)where the index a determines, as above, the spin compo-nent of the operators, a ∈ { x, y, z } . The operators arerepresented in the interaction picture (see the Appendix),which means that their time-dependence is governed byˆ H , i.e., ˆ S ( r n , t ) = e i ˆ H t ˆ S ( r n ) e − i ˆ H t . We assume that the nuclear spins evolve on a time scale which is muchlonger than that of the electron spin dynamics, so thatfor all times of interest we can use ˆ K n ( t ) = ˆ K n . We seethat, if we focus on the dynamics of one specific nuclearspin (as we will do below), our perturbative approach re-quires that Av is much smaller than the energy scalesin the unperturbed Hamiltonian ˆ H .We then substitute the perturbation (5) into the equa-tion of motion (4), make use of the fact that a trace isinvariant under cyclic permutation of the operators in itsargument, and then finally arrive at d ˆ ρ k dt = − iAv X n h ˆ S a ( r n , t ) i [ ˆ K an , ˆ ρ k ] − − ( Av ) X n,m (cid:26) Z t h ˆ S a ( r n , t ) ˆ S b ( r m , t ′ ) i c dt ′ [ ˆ K an , ˆ K bm ˆ ρ k ] − − Z t h ˆ S b ( r m , t ′ ) ˆ S a ( r n , t ) i c dt ′ [ ˆ K an , ˆ ρ k ˆ K bm ] − (cid:27) , (6)where now brackets are used to denote the trace over theelectronic part of the density matrix, h . . . i ≡ Tr e { . . . ˆ ρ e } in general, and h . . . i c ≡ Tr e { . . . ˆ ρ e } for the fully con-nected terms. From (6) we can derive equations of motionfor the expectation values of the nuclear spin operators. A single nucleus, K = 1 / Let us first focus on the transparent case of one singlenucleus with nuclear spin K = 1 /
2, interacting with alocal electron spin density. Up to first order, i.e., usingonly the first term in (6), we find d h ˆ K a i dt (1) = Tr (cid:26) ˆ K a d ˆ ρ k dt (cid:27) = − iS b Tr (cid:8) [ ˆ K a , ˆ K b ] − ˆ ρ k (cid:9) = ε abc S b h ˆ K c i , (7)describing a precession of the nuclear spin around the lo-cal electron spin density at the position r n of the nucleus, with the vector S defined as S ≡ Av h ˆ S ( r n , t ) i . Obvi-ously, the traces appearing in (7) and the correspondingaverages h . . . i involve tracing and averaging over the nu-clear spin part of the system only. Since no confusion ispossible, we do not label these traces and averages sepa-rately with a k .The exchange of angular momentum between electronsand the nucleus is to leading order described by the sec-ond term in (6), i.e., second order in ˆ H hf . We writethe correlation functions of the electron spin in terms ofthe quantities R ab and χ ab introduced in (3), and makeagain use of the cyclic invariance of the trace. For theconvenient case of K = 1 /
2, we can simplify products ofnuclear spin operators using ˆ K a ˆ K b = i ε abc ˆ K c + δ ab .Along these lines we simplify the second order term, d h ˆ K a i dt (2) = − ( Av ) n ( R bc + iχ bc )Tr (cid:8) ˆ K a ˆ K b ˆ K c ˆ ρ k − ˆ K a ˆ K c ˆ ρ k ˆ K b (cid:9) − ( R bc − iχ bc )Tr (cid:8) ˆ K a ˆ K b ˆ ρ k ˆ K c − ˆ K a ˆ ρ k ˆ K c ˆ K b (cid:9)o = − ( Av ) n R bc (cid:10) [[ ˆ K a , ˆ K b ] − , ˆ K c ] − (cid:11) + iχ bc (cid:10) [[ ˆ K a , ˆ K b ] − , ˆ K c ] + (cid:11)o = ( Av ) n R ba h ˆ K b i − R bb h ˆ K a i + ε abc χ bc o , (8)which is, combined with Eq. (7), identical to the expression given in Eq. (1). Many nuclei,
K > / When following the same derivation for the case of many nuclei and spin higher than K = 1 /
2, it is generally notpossible to derive a closed set of equations for d h ˆ K n i /dt . The first order equations do not change and are still givenby Eq. (7). The second order equation however does change, and becomes d h ˆ K an i dt (2) = ( Av ) n R ba h ˆ K bn i − R bb h ˆ K an i + ε abc χ bdnn h [ ˆ K dn , ˆ K cn ] + i + 2 X m = n ε abc χ bdnm h ˆ K dm ˆ K cn i o , (9)where the correlator χ has acquired two more indices, χ abnm = − i Z t h [ ˆ S a ( r n , t ) , ˆ S b ( r m , t ′ )] − i c dt ′ . (10)This correlation function now describes the non-local sus-ceptibility of the electron spin density, i.e., the linear re-sponse in the spin coordinate ˆ S a at the position r n dueto a perturbation along ˆ S b at another position r m .We notice two practical problems with Eq. (9). Firstof all, for K > / K an ˆ K bn = i ε abc ˆ K cn + δ ab to simplify products of nuclear spin op-erators, and secondly, also correlations h ˆ K dm ˆ K cn i between different nuclei play a role. This makes it impossible toderive from Eq. (9) a closed set of equations for d h ˆ K n i /dt .However, typically the number of nuclear spins in theensemble is large ( N ∼ –10 ). When focusing onthe dynamics of one single nuclear spin, as we do inEq. (9), the presence of the large number of other nu-clear spins can, to first approximation, be consideredas a static ‘classical’ field. In other words, we can as-sume the fluctuations in the nuclear field h ˆ K dm ˆ K cn i c tobe much smaller than the average values of the field h ˆ K dm ih ˆ K cn i . This assumption then allows us to replace h ˆ K dm ˆ K cn i → h ˆ K dm ih ˆ K cn i in the last term of Eq. (9). Weemphasize here that in a standard FGR calculation, thesame assumption is implicitly used: Transition rates be-tween different spin states of a single nucleus are evalu-ated, ignoring any possible corrections due to correlationsbetween this spin and the other nuclear spins.With this simplification, the last term of Eq. (9)now reads ( Av ) P m ε abc χ bdnm h ˆ K dm ih ˆ K cn i and describesan effective electron mediated internuclear spin-spin cou-pling. This effect can be interpreted as a correction tothe first-order equation (7): The effective field aroundwhich the n -th nuclear spin precesses is adjusted S a → S a +( Av ) P m χ abnm h ˆ K bm i , i.e., it acquires a contributiondue to the effective field produced by all other nuclearspins, mediated by the electron spin [22].The third term in (9) contains products of spin op-erators of the same nucleus, and must thus be treatedon different footing. In most realistic experimental cir-cumstances, the temperature is much larger than thenuclear Zeeman energy. This means that the (ther-mal) equilibrium density matrix of the nuclear spins can be approximated to be isotropic, i.e., ˆ ρ k ∝ . Aslong as we focus on small deviations from this equilib-rium, i.e., small polarizations h ˆ K a i , the nuclear spin den-sity matrix is always close to isotropic. In this case,quadratic terms like h ˆ K an ˆ K an i c will dominate the contri-bution from h [ ˆ K dn , ˆ K cn ] + i c in (9), since they are of or-der unity, h ( ˆ K an ) i c ≈ K ( K + 1). We thus can con-centrate on these quadratic contributions, and approxi-mate h ˆ K cn ˆ K dn i ≈ δ cd K ( K + 1). We see that then finallyEq. (9) reduces to the simple result of Eq. (8).Let us repeat here that this reduction is based on twoassumptions: (i) we neglect the correlations between thenuclear spin under consideration and all other nuclearspins (as is also done in an FGR calculation), and (ii) weonly allow for small deviations of ˆ ρ k from equilibrium,so that h ˆ K cn ˆ K dn i ≈ δ cd K ( K + 1). In fact, one does notnecessarily have to use these assumptions and make thissimplification. Equation (6) gives the time-evolution upto second order in ˆ H hf of the full nuclear spin density ma-trix in terms of the electron spin fluctuations and suscep-tibilities. One could solve this Equation (numerically) forany initial condition ˆ ρ k and calculate the time-dependentexpectation value of any desired nuclear spin operator. IMPLEMENTATION
With the formalism presented in Eqs (1)–(3), we areable to express the hyperfine driven nuclear spin dynam-ics in terms of the susceptibility and fluctuations of theelectron spin density. Let us now explain how one couldcalculate the necessary electronic correlation functionsusing linear response theory. For a given setup, we writedown the full set of Bloch equations describing the evolu-tion of the electronic density matrix under ˆ H , and solvefor the steady state solution ˆ ρ (0) ≡ ˆ ρ e . The elements S a are then simply found as the equilibrium expectationvalues S a = Av h ˆ S a ( r n ) i ˆ ρ (0) = Tr { ˆ S a ( r n )ˆ ρ (0) } . Next,we add to the set of Bloch equations the first-order effectof a perturbation ˆ H ′ = Λ · ˆ S ( r n ), i.e., we add the terms i [ˆ ρ (0) , ˆ H ′ ] − , and solve for the new steady state densitymatrix ˆ ρ (1) . The spin susceptibility χ ab needed in Eq. (1)is then found as the part of h ˆ S a ( r n ) i ˆ ρ (1) = Tr { ˆ S a ( r n )ˆ ρ (1) } which is linear in Λ b . The fluctuations R ab are found in asimilar way, the terms to add to the Bloch equations thenread [ˆ ρ (0) , ˆ H ′ ] + − { ˆ H ′ ˆ ρ (0) } ˆ ρ (0) , where the last term re-moves the connected part of the electron spin correlators.We will illustrate the procedure below.From this short outline, it is already clear that thisapproach has several properties which are different froma traditional FGR calculation: (i) For an FGR calcula-tion, there should be an obvious basis to calculate alloccupation probabilities in. Here we can choose any ba-sis which is most convenient for evaluating the steadystate electronic density matrix. (ii) When calculatingseparate FGR transition rates, it should a priori be clearwhich quantization axis to choose for the nuclear spins.In our formalism, as soon as ˆ ρ (1) is found for the twotypes of perturbations, all elements of χ and R can beread off immediately, yielding the dynamics for the fullvector h ˆ K n i , not just the polarization along a single axis.(iii) For FGR to work, all levels involved in the hyperfinetransitions (including possible virtual states in higher or-der processes) should be well separated in energy, i.e., thesplittings should be larger than all decay rates and otherpossible incoherent processes present. In our method allincoherent processes are accounted for in the set of Blochequations, we do not have to compare them with theother energy scales. (iv) In an FGR calculation, whenfocusing on the nuclear spin dynamics of a small subsys-tem being part of a larger system (e.g. a quantum dotcoupled to multiple reservoirs), one usually includes onlythe energy splittings inside the subsystem in the actualperturbation theory. All other processes relevant for theelectron spin dynamics (such as the coupling to the reser-voirs) have to be accounted for ‘afterward’ in an effectivedensity of states, which is not always possible. In ourapproach, we treat the coherent and incoherent parts ofthe dynamics in the subsystem on equal footing. One ad-vantage is that the perturbation theory in our approachdoes not break down when the energy splittings in thesubsystem vanish, as long as the energies correspondingto the incoherent dynamics are large enough.Let us mention here also a drawback of our method.Our main result (1) describes only the dynamics of aver-age nuclear fields h ˆ K i . Temporal fluctuations and otherstochastic properties of the fields are not accessible viathis approach. In an FGR calculation, one can separatethe rates for flipping nuclear spins up and down, andderive from this separation a Fokker-Planck equation de-scribing the stochastics of the nuclear fields [19]. EXAMPLE APPLICATIONS
We will now illustrate the above with a calculation ofthe nuclear spin pumping in two example systems. First,we will consider nuclear spin pumping in a trivial toymodel concerning a single quantum dot coupled to twoleads. We will show how a smart implementation of FGR can produce an identical result in this case. Then wewill investigate a more complicated setup, involving adouble quantum dot, which cannot be dealt with in anFGR approach. We will show however how our formalismcan be applied without any problem, yielding a generalequation for the nuclear spin pumping in this setup.
Single dot: Fermi’s golden rule with modifications
We consider a single quantum dot connected to twoleads, as illustrated in Fig. 1. Due to the strong Coulombrepulsion, only one excess electron can occupy the dot.If we apply an external magnetic field B ext = B ext ˆ z , theenergy levels for spin up and down in the dot are split bythe electronic Zeeman energy. We take the tunnel rateinto the dot Γ in to be very fast and equal for both spindirections. The outward tunnel rates Γ ↑ , ↓ are differentfor the two spin directions (to provide a physical picturewith this assumption: one could assume the right leadto be ferromagnetic). Since the electron occupying thedot will thus have on average a non-zero spin polariza-tion along the z -direction, hyperfine interaction with thenuclear spins is expected to produce a non-zero nuclearspin polarization along this axis.In typical experiments, the temperature ( ∼
100 mK ∼ µ eV) is much smaller than the orbital level spacingin the dot ( ∼ s -type character. As long as also the energyscale of the hyperfine interaction is much smaller than theorbital level spacing, we can project ˆ H hf to the orbitalground state, and writeˆ H hf ( t ) = X n A n ˆ S a ( t ) ˆ K an , (11)where the hyperfine coupling coefficients are defined as A n = Av | ψ ( r n ) | , i.e., in terms of the envelope wavefunction ψ ( r ) of the orbital ground state. The operatorˆ S is now the electron spin operator, instead of spin den-sity. A usual further simplification to make is to assumethat | ψ ( r n ) | is approximately constant in the dot. Thisimplies that | ψ ( r n ) | ≈ ( N v ) − , where N ∼ –10 isthe number of nuclear spins in the dot. In this case, allcoupling coefficients simply reduce to A n = A/N .Let us now calculate the nuclear spin pumping ratealong the z -direction, d h ˆ K zn i /dt , for this system. Since allnuclear spins are coupled to ˆ S with the same coefficient, h ˆ K zn i for the n -th nucleus equals the ensemble averagednuclear spin h ˆ K z i . We only take into account a finitepolarization in the z -direction, so we use the equation d h ˆ K z i dt = 13 P z − Q zz h ˆ K z i , (12)with the functions P z = A n K ( K + 1)( χ xy − χ yx ) and nuclei FIG. 1. Energy diagram for a trivial single quantum dot setupin which nuclear spin pumping is to be expected. The dot iscoupled to two leads with different chemical potentials. Onesingle-electron level lies within the bias window, which is splitinto two sublevels by the Zeeman energy (we assumed a nega-tive g -factor). The tunnel rate from the left lead into the dot,Γ in is large and equal for both spin directions. The outwardtunnel rates Γ ↑ , ↓ are different for spin up and down. Q zz = A n ( R xx + R yy ). Note that χ and R contain nowcorrelators of the electron spin, not electron spin density.A finite nuclear polarization in the z -direction behaveson the time scale of the electronic dynamics as a staticcontribution to the magnetic field experienced by theelectrons. We incorporate this contribution into an ef-fective field B eff = B ext − A h ˆ K z i (we assume a negative g -factor and a positive hyperfine coupling constant, as isthe case in GaAs). The coherent time-evolution of theelectronic 2 × ∂ t ˆ ρ = i [ B eff ˆ S z , ˆ ρ ] − , where we express B eff in terms of a frequency. To this we add the incoherenttunnel rates Γ ↑ and Γ ↓ , which, to a good approximation,include the presence of the leads and the left and righttunnel barriers into the equations of motion for the elec-tron on the dot. Together, this yields the set of equations ∂ t ρ ↑ = − Γ ↑ ρ ↑ + (Γ ↓ ρ ↓ + Γ ↑ ρ ↑ ) ∂ t ρ ↓ = − Γ ↓ ρ ↓ + (Γ ↓ ρ ↓ + Γ ↑ ρ ↑ ) ∂ t ρ ↑↓ = ( iB eff − Γ) ρ ↑↓ ∂ t ρ ↓↑ = ( − iB eff − Γ) ρ ↓↑ , (13)where Γ ≡ Γ ↑ +Γ ↓ . These equations can be solved for thestationary situation ∂ t ˆ ρ (0) = 0, yielding ρ (0) ↑ ( ↓ ) = Γ ↓ ( ↑ ) / Γand ρ (0) ↑↓ = ρ (0) ↓↑ = 0. We only need the response functions χ xy and χ yx , so we add the terms i [ˆ ρ (0) , Λ x ˆ S x + Λ y ˆ S y ] − to Eq. (13) and solve for the new stationary solution ˆ ρ (1) .The correlator χ xy is then simply given by the term inTr { ˆ S x ˆ ρ (1) } linear in Λ y , and in a similar way we find χ yx .The results are combined to χ xy − χ yx = Γ ↓ − Γ ↑ B + Γ . (14)Now we evaluate R xx and R yy . To this end, we add theterms [ˆ ρ (0) , Λ x ˆ S x +Λ y ˆ S y ] + − { (Λ x ˆ S x +Λ y ˆ S y )ˆ ρ (0) } ˆ ρ (0) to Eq. (13), and then look for the linear responses in ˆ S x and ˆ S y , yielding R xx + R yy = Γ2 B + Γ . (15)Combining all together using Eq. (12), we thus find d h ˆ K z i dt = A n Γ ↓ ( − h ˆ K z i ) − Γ ↑ ( + h ˆ K z i )4( B ext − A h ˆ K z i ) + Γ , (16)where for simplicity we assumed K = 1 / |↑⇓ n i and |↓⇑ n i , where thedouble arrow indicates the nuclear spin along the z -axis.Neglecting the nuclear Zeeman energy, the two states areseparated by the energy ¯ hB eff , and both are broadenedby their respective electronic decay rate Γ ↑ ( ↓ ) . Hyper-fine induced transitions from |↑⇓ n i to |↓⇑ n i contribute positively to d h ˆ K zn i /dt , and their rate is given byΓ + = 2 π | h↓⇑ n | A n ˆ S − ˆ K + n | ↑⇓ n i | p i D , (17)the square of the relevant matrix element, multiplied bythe chance p i of finding the system initially in |↑⇓ n i andthe effective ‘density of states’ for this transition D .For the occupation probability p i , we take p i = ( −h ˆ K z i ) × ρ (0) ↑ . In this simple case, it is possible to incorpo-rate all incoherent dynamics (the two decay rates) into asmartly chosen D : We take the sum of the level broad-enings of the two electronic levels |↑i and |↓i . Assuminga Lorentzian shape for these broadenings, we thus use D = 1 π Γ B + Γ , (18)so that we can writeΓ + = A n Γ ↓ ( − h ˆ K z i )4 B + Γ . (19)We derive similarly an equation for Γ − and then combinethe two to write for the net nuclear spin pumping rate d h ˆ K z i dt = A n Γ ↓ ( − h ˆ K z i ) − Γ ↑ ( + h ˆ K z i )4( B ext − A h ˆ K z i ) + Γ . (20)We see that this result coincides with Eq. (16). The factthat the FGR approach in this case also works for smallenergy splittings, i.e., in the regime B eff < ∼ Γ, is due tothe fact that all incoherent effects in this model can beincorporated relatively simply into D . Double dot: Beyond Fermi’s golden rule
Let us now illustrate what can happen when the setupbecomes more complicated. We consider a double quan-tum dot connected to two leads with different chemical
FIG. 2. Energy diagram of a more complicated setup. Twoquantum dots are coupled to each other and to two leads withdifferent chemical potentials. The two dots have different ef-fective g -factors, so that the application of an external mag-netic field yields different Zeeman energies in the dots. Thedouble dot is tuned to the (0 , → (1 , → (0 , → (0 , in . potentials, as depicted in Fig. 2. The dots are tuned suchthat an electronic transport cycle involves the chargestates (0 , → (1 , → (0 , → (0 , n, m ) de-notes a state with n ( m ) excess electrons on the left(right)dot. The electronic levels in the two dots are both splitby a Zeeman energy in a spin up and spin down level. Weassume that the splitting in the left dot is larger than inthe right dot (e.g. due to a size-related difference in ef-fective g -factors), and that the (1 ,
0) spin down level hasa too high energy to play a role. The (1 ,
0) and (0 , d and coupledto each other with a tunnel coupling t . We again assumethe outward tunneling rates to be much slower than the inward, Γ ≪ Γ in , so that the system effectively will neveroccupy the (0 ,
0) charge state.We will focus on nuclear spin pumping in the right dot.A non-zero nuclear spin pumping rate is expected alongthe direction of the magnetic field since the average localelectron spin polarization along this direction is finite.Denoting the level in the left dot by | L i and the twolevels in the right dot by |↑i and |↓i , the Hamiltonian forthe three-level system readsˆ H = d | L i h L | + B eff |↓i h↓| + t | L i h↓| + t |↓i h L | , (21)where the feedback of the nuclear spins on the electronspin dynamics again is incorporated into the effectivemagnetic field B eff = B ext − A h ˆ K z i . Using this Hamilto-nian and adding the effect of the incoherent decay rateΓ, we can write down the equations of motion for ˆ ρ , andsolve for ˆ ρ (0) , yielding ρ (0) L = 4 d + 4 t + Γ d + 8 t + Γ , ρ (0) ↑ = 4 t d + 8 t + Γ ,ρ (0) L ↑ = ( ρ (0) ↑ L ) ∗ = 4 dt + 2 it Γ4 d + 8 t + Γ , (22)and zero for the other five elements. We then proceedas we did above, and add consecutively the perturba-tions i [ˆ ρ (0) , ˆ H ′ ] − and [ˆ ρ (0) , ˆ H ′ ] + − { ˆ H ′ ˆ ρ (0) } ˆ ρ (0) to theequations of motion. We solve in both cases for the newsteady-state solution ˆ ρ (1) and extract the linear responsesneeded. Combining all together, we find d h ˆ K z i dt = 4 A n t Γ( B + 2 t + Γ )( − h ˆ K z i )(4 d + 8 t + Γ )[4( B eff d − B + t ) + (5 B − B eff d + 4 d + 4 t )Γ + Γ ] , (23)where B eff = B ext − A h ˆ K z i . This equation gives the nuclear spin pumping in the right dot along the z -axis, withoutany restriction imposed on the relative magnitude of the parameters B eff , d , t , and Γ.Let us now try to evaluate the same pumping using an FGR approach. The most transparent electronic basis to useis {| L i , |↑i , |↓i} since in this basis we have well-defined incoherent in- and outward tunneling rates. We see howeverthat the stationary density matrix in this basis (22) has off-diagonal elements. In order to proceed, we have to makeseveral assumptions. Let us consider the limit of Γ ≫ t . In this case the off-diagonal elements of ˆ ρ (0) can be neglected,and we can approximate ˆ ρ (0) ≈ | L i h L | . We further assume that we have three well-separated levels, i.e., B eff , d ≫ Γ.In this limit the nuclear spin pumping is resulting from a second order transition: (i) tunneling from the initial state | L ⇑ n i to the virtual state |↓⇑ n i , and then (ii) an electron-nuclear spin flip-flop from |↓⇑ n i to the final state |↑⇓ n i .We thus use a standard second order Fermi golden rule to write for the pumping rate d h ˆ K z i dt = 2 π | h↓⇑ n | A n ˆ S − ˆ K + n | ↑⇓ n i h↑⇓ n | ˆ H T | L ⇓ n i | ( E ↓⇑ n − E L ⇑ n ) p L ⇑ n D , (24)where ˆ H T = t | L i h↓| + t |↓i h L | is the tunneling part ofthe Hamiltonian ˆ H . The occupation probability of theinitial state is p L ⇑ n = − h ˆ K z i . The ‘density’ D for this transition is mainly set by the level broadening of thefinal state, so D = π Γ / [4( B eff − d ) + Γ ]. This yields d h ˆ K z i dt = A n t Γ( − h ˆ K z i ) d [4( B eff − d ) + Γ ] , (25)which coincides with (23) in the limit B eff , d ≫ Γ ≫ t .(Note that B eff − d ≫ Γ is not a necessary condition.)There are other limits in which considering the rightrates to calculate, combined with a good approximationfor ˆ ρ (0) and a smart choice for D , can result in the rightpumping rate. In general however, it is not trivial to un-derstand how to incorporate all incoherent dynamics into D . Even if one would simply diagonalize the stationarydensity matrix ˆ ρ (0) and try to evaluate all hyperfine tran-sitions in the new basis, one would have to transform theincoherent processes in correct effective densities of state.In the approach using the electron spin susceptibility andfluctuations, this is all done on the fly and its validity isnot restricted to certain limits of the parameter space.Let us emphasize here that the fact that a correctlyadapted FGR approach did work in our single dot ex-ample but not in the double dot example, is only due tothe complexity of the coherent and incoherent electronspin dynamics in the two cases. The success of the FGRapproach is not inherently connected to the number ofquantum dots in a setup: Also for more complicated sin-gle dot systems, the FGR approach might break down.The two examples given above both consider localizedelectron spins. Our approach however, works in princi-ple equally well for systems with delocalized electrons.In this case, as explained above, the susceptibility andfluctuations needed are correlators of electron spin den-sity instead of simply electron spin. This however doesnot change the complexity of the formalism: One onlyhas to find the right set of (Bloch) equations to describethe electron spin density. CONCLUSIONS
To conclude, we presented a new approach to evaluatethe nuclear spin dynamics driven by hyperfine interactionwith non-equilibrium electron spins. To describe the dy-namics up to second order in the hyperfine coupling, itsuffices to evaluate the susceptibility and fluctuations ofthe electron spin. This approach does not rely on a sep-aration of electronic energy scales or the specific choiceof electronic basis states, thereby overcoming practicalproblems which may arise under certain circumstanceswhen using a more traditional formalism based on rateequations.
ACKNOWLEDGMENTS
J. D. would like to thank P. W. Brouwer for helpfuldiscussions. This work was supported by the Alexander von Humboldt Foundation and the ‘Stichting voor Fun-damenteel Onderzoek der Materie’.
APPENDIX: DERIVATION OF EQ. (4)
Here we will show how to derive the time evolutionequation (4) using second order perturbation theory inthe interaction picture. The total Hamiltonian of the sys-tem ˆ H is split into an unperturbed part ˆ H and a smallperturbation ˆ H ′ . In the interaction picture, all opera-tors acquire a time-dependence governed by ˆ H , so thatfor any operator ˆ A we have ˆ A I ( t ) = e i ˆ H t ˆ Ae − i ˆ H t (weassumed for simplicity a time-independent ˆ H ). The re-maining part of the total Hamiltonian, the perturbationˆ H ′ , governs the time-dependence of the wave function, i∂ t | ψ I i = ˆ H ′ I ( t ) | ψ I i , where ˆ H ′ I ( t ) is the Hamiltonian ofthe perturbation in the interaction picture. The time-evolution operator for | ψ I i thus readsˆ U I ( t, t ′ ) = T (cid:20) exp (cid:26) − i Z tt ′ ˆ H ′ I ( τ ) dτ (cid:27)(cid:21) , (26)where T is the time-ordering operator. The relation be-tween ˆ U I and the ‘regular’ time-evolution operator in theSchr¨odinger picture ˆ U is straightforward to derive andreads ˆ U ( t, t ′ ) = e − i ˆ H t ˆ U I ( t, t ′ ) e i ˆ H t ′ .Let us now consider the evolution of the density ma-trix of a system described by the Hamiltonian ˆ H + ˆ H ′ .We assume that at some time in the past t → −∞ thesystem was still unperturbed and could be described by astationary density matrix ˆ ρ which is determined solelyby ˆ H . The perturbation is then switched on adiabat-ically, formally done by replacing ˆ H ′ → e ηt ˆ H ′ , where η is an infinitesimally small positive real number. Thedensity matrix at any later time t can then be writtenas ˆ ρ ( t ) = ˆ U ( t, t )ˆ ρ ˆ U ( t , t ), where ˆ U describes the time-evolution due to ˆ H + e ηt ˆ H ′ . In terms of evolution in theinteraction picture, we see that we can writeˆ ρ ( t ) = e − i ˆ H t ˆ U I ( t, t )ˆ ρ ˆ U I ( t , t ′ ) e i ˆ H t , (27)where we used that e i ˆ H t ˆ ρ e − i ˆ H t = ˆ ρ since ˆ ρ is perdefinition stationary under ˆ H .Introducing ˆ ρ I ( t ) = e i ˆ H t ˆ ρ ( t ) e − i ˆ H t , the ‘interaction’density matrix, we note that it obeys the simple relationˆ ρ I ( t ) = ˆ U I ( t, t )ˆ ρ ˆ U I ( t , t ′ ) , (28)and that for the expectation value of any observable h A ( t ) i = Tr { ˆ A ˆ ρ ( t ) } = Tr { ˆ A I ( t )ˆ ρ I ( t ) } . (29)From Eqs (28) and (26) it follows immediately that wecan write as time-evolution equation for ˆ ρ I ( t ) d ˆ ρ I dt = − i [ ˆ H ′ I ( t ) , ˆ ρ I ( t )] − . (30)If one knows the stationary density matrix ˆ ρ , one caniteratively use Eq. (30) to approximate d ˆ ρ I dt = − i [ ˆ H ′ I ( t ) , ˆ ρ ] − − Z t [ ˆ H ′ I ( t ) , [ ˆ H ′ I ( t ′ ) , ˆ ρ ] − ] − dt ′ + . . . (31)In our case, we would like to obtain a time-evolutionequation for a reduced density matrix, i.e., separate thesystem into two subsystems, S (1) and S (2) , and trace outthe degrees of freedom of one of the subsystems. Weassume that initially, at t , the density matrix can beseparated, ˆ ρ = ˆ ρ (1)0 ⊗ ˆ ρ (2)0 , where the superscripts labelthe subsystems. We then trace in Eq. (31) over the de-grees of freedom of S (2) , which yields up to second orderin ˆ H ′ for the reduced density matrix d ˆ ρ (1) I dt = − i [Tr { ˆ H ′ I ( t )ˆ ρ (2)0 } , ˆ ρ (1)0 ] − − Z t Tr (cid:8) [ ˆ H ′ I ( t ) , [ ˆ H ′ I ( t ′ ) , ˆ ρ (1)0 ⊗ ˆ ρ (2)0 ] − ] − (cid:9) dt ′ . (32)In this expression, tracing out the degrees of freedom of S (2) in general introduces correlation functions of thosecoordinates of S (2) to which ˆ H ′ is coupled. 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