Theory of Spin Relaxation in Two-Electron Lateral Coupled Quantum Dots
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Theory of Spin Relaxation in Two-Electron Lateral Coupled Quantum Dots
Martin Raith , Peter Stano , Fabio Baruffa , and Jaroslav Fabian Institute for Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany Institute of Physics, Slovak Academy of Sciences, 845 11 Bratislava, Slovakia German Research School for Simulation Sciences, Forschungszentrum Juelich, D-52425 Germany
A global quantitative picture of the phonon-induced two-electron spin relaxation in GaAs doublequantum dots is presented using highly accurate numerics. Wide regimes of interdot coupling, mag-netic field magnitude and orientation, and detuning are explored in the presence of a nuclear bath.Most important, the giant magnetic anisotropy of the singlet-triplet relaxation can be controlledby detuning switching the principal anisotropy axes: a protected state becomes unprotected upondetuning, and vice versa. It is also established that nuclear spins can dominate spin relaxation forunpolarized triplets even at high magnetic fields, contrary to common belief.
Electron spins in quantum dots [1] are among perspec-tive candidates for a controllable quantum coherent sys-tem in spintronics [2, 3]. Spin qubits in GaAs quantumdots, the current state of the art [4, 5], are coupled totwo main environment baths: nuclear spins, and phonons[6]. The nuclei dominate decoherence, which is on µ stimescales. But only phonons are an efficient energy sinkfor the relaxation of the energy resolved spin states, lead-ing to spin lifetimes as long as seconds [7].The extraordinary low relaxation is boosted by or-ders of magnitude at spectral crossings, unless specialconditions—such geometries we call easy passages—aremet [8, 9]. Spectral crossings seem inevitable in the ma-nipulation based on the Pauli spin blockade [1, 10], thecurrent choice in spin qubit experiments [11]. On theother hand, a fast spin relaxation channel may be de-sired, e.g., in the dynamical nuclear polarization [12–14].The single-electron spin relaxation is well understood[15, 16]: it proceeds through acoustic phonons, in propor-tion to their density of states, which increases with thetransferred energy. The matrix element of the phononelectric field between spin opposite states is nonzero dueto spin-orbit coupling or nuclear spins. At anticrossingsthe matrix element is enhanced by orders of magnitude,even though the anticrossing gap is minute ( ∼ µ eV). Therelaxation rate can be either enhanced or suppressed, de-pending on whether the energy or the matrix elementeffects dominate.The two electron relaxation rates were measured insingle [17–19] and in double [20–22] dots. Theoreticalworks so far mostly focused on single dots [23, 24], orvertical double dots [25, 26], in which the symmetry ofthe confinement potential lowers the numerical demands.A slightly deformed dot was considered in Refs. [27, 28],and a lateral coupled double dot in silicon in Ref. [29].What is key for spin qubit manipulation and most rele-vant for ongoing experiments, is the case of weakly cou-pled and biased coupled dots. In addition, the relativeroles of the spin-orbit and hyperfine interactions in thespin relaxation in GaAs quantum dots has not yet beenestablished.The analysis of the two electron double dot relaxation is challenging because many parameters need to be con-sidered simultaneously: the magnitude and orientation ofthe magnetic field, the orientation of the dot with respectto the crystallographic axes, the strength of the interdotcoupling (parametrized by either tunneling or exchangeenergy) and the bias applied across the double dot (de-tuning). Here we cover all these parameters, includingthe nuclear bath , providing specific relevant predictionsfor experimental setups [30]. Perhaps the most strikingresults are the existence of islands of inhibited spin re-laxation in the magnetic field and detuning maps, andthe switch of the two principal C v axes along which therelaxation shows a minimum or maximum, as detuningis turned on. While singlets and polarized triplets relaxby spin-orbit coupling, the spin-unpolarized triplet relax-ation is dominated by nuclear spins over a wide param-eter range (the spin-orbit induced anisotropy is wipedout), contrary to common belief. The predicted giantspin relaxation anisotropy is a unique and experimentallytestable signature of spin-orbit spin relaxation, which canalso be useful for spin nanodevices, as we argue in thispaper. MODEL
We consider a laterally coupled, top-gated GaAs dou-ble quantum dot patterned in the plane perpendicular toˆ z = [001]. In the two-dimensional and envelope functionapproximation, the Hamiltonian reads H = X i =1 , ( T i + V i + H Z,i + H so ,i + H nuc ,i ) + H C , (1)where i labels electrons. The single-electron terms are T = P / m = ( − i ~ ∇ + e A ) / m, (2) V = (1 / mω min { ( r − d ) , ( r + d ) } + e E · r , (3) H Z = ( g/ µ B σ · B , (4) H so = H br + H d + H d3 , (5)the kinetic energy, the biquadratic confinement poten-tial, the Zeeman term, and the spin-orbit couplings, re-spectively. The position and momentum vectors are two-dimensional, where ˆ x = [100] and ˆ y = [010]. The protoncharge is e and the effective electron mass is m . The con-finement energy, E = ~ ω , and the confinement length, l = ( ~ /mω ) / , define the characteristic scales. Thepotential is minimal at ± d and we call 2 d/l the interdotdistance. The electric field E is applied along the dotmain axis d . Turning on E shifts the potential minimarelative to each other by the detuning energy ǫ = 2 eEd .The magnetic field is B = ( B x , B y , B z ). We use the sym-metric gauge, A = B z ( − y, x ) /
2, and σ = ( σ x , σ y , σ z )are the Pauli matrices. The Land´e factor is g and theBohr magneton is µ B . The Bychkov-Rashba, and thelinear and cubic Dresselhaus Hamiltonian read H br = ( ~ / ml br ) ( σ x P y − σ y P x ) , (6) H d = ( ~ / ml d ) ( − σ x P x + σ y P y ) , (7) H d3 = (cid:0) γ c / ~ (cid:1) (cid:0) σ x P x P y − σ y P y P x (cid:1) + H.c. , (8)parameterized by the spin-orbit lengths l br and l d , and abulk parameter γ c . Nuclei, labeled by n , couple through H nuc = β X n I n · σ δ ( r − R n ) , (9)where β is a constant, and I n is the spin of a nu-cleus at the position R n . The Coulomb interaction is H C = e / πǫ | r − r | , with the dielectric constant ǫ .The Hamiltonian, Eq. (1), and its energy spectrum is dis-cussed in Refs. [31, 32], including our numerical method(configuration interaction) for its diagonalization. Herewe extent it by including nuclear spins, which we treat byaveraging over unpolarized random ensemble. See Sup-plementary [33] for further details.The relaxation is mediated by acoustic phonons H ep = i X Q ,λ s ~ Q ρV c λ V Q ,λ h b † Q ,λ e i Q · R − b Q ,λ e − i Q · R i , (10)with deformation, V df Q ,l = σ e , and piezoelectric potentials, V pz Q ,λ = − eh ( q x q y ˆe λ Q ,z + q z q x ˆe λ Q ,y + q y q z ˆe λ Q ,z ) /Q . Thephonon wave vector is Q , and the electron position vec-tor is R = ( r , z ). The polarizations are given by λ , thepolarization unit vector reads ˆe , and the phonon annihi-lation (creation) operator is denoted by b ( b † ). The massdensity, the volume of the crystal, and the sound veloci-ties are given by ρ , V , and c λ , respectively. The phononpotentials are parameterized by σ e , and h .We define the relaxation rate as the sum of the in-dividual transition rates to all lower-lying states forboth piezoelectric and deformation potentials. Each rate(from | i i to | j i ) is evaluated using Fermi’s Golden Rulein the zero-temperature limit,Γ ij = π ~ ρV X Q ,λ Qc λ | V Q ,λ | | M ij | δ ( ω ij − ω Q ) , (11) interdot distance 2d/l -0.3-0.2-0.100.10.20.3 e n e r gy [ m e V ] detuning ε [meV] ST + T T - S T - T T + S (1,1) S (0,2) a b FIG. 1. Calculated energies of the lowest states for (a) vari-able interdot coupling (at B = 5 T), and (b) detuning (at B = 2 T). Singlet states are given by dashed, triplets by solidlines. The blue strokes mark singlet-triplet anticrossings. In(a), the energy of T is subtracted, and in (b), the quadratictrend in E is subtracted. The green arrows denote pointsof exact compensation and the red oval in (b) shows wherenuclear spins dominate the T relaxation. where M ij = h i | e i Q · R | j i is the matrix element of thestates with energy difference ~ ω ij . Here we are inter-ested in the rates of the singlet ( S ) and the three triplets( T + , T , T − ) at the bottom of the energy spectrum.In numerics we use GaAs parameters: m = 0 . m e ,with m e the free electron mass, g = − . c l = 5290m/s, c t = 2480 m/s, ρ = 5300 kg/ m , σ e = 7 eV, eh =1 . × eV/m, ǫ = 12 . γ c = 27 . , β = 2 µ eVnm , I=3/2. We choose typical lateral dots values, l br =2 . µ m, l d = 0 . µ m, d || [110] and the confinementenergy E = 1 . l = 34 nm. SYMMETRIC DOUBLE DOT
We start with an unbiased double dot. We plot itsspectrum in Fig. 1a) as a function of the interdot cou-pling, which translates into an exponentially sensitive S − T exchange splitting J . Electrical control over J ,necessary e.g. to induce the √ SWAP gate [1], allows fora fast switching between the strong and weak couplingregime, corresponding to the exchange splitting beinglarger and smaller than the Zeeman energy, respectively.During this switching, the ground state changes at an S − T + anticrossing.We cover the freedom of the interdot coupling in Fig. 2.Panel a) shows the relaxation of the first excited state [ S or T + , see Fig. 1a)]. First to note is the strong relaxationsuppression at the S − T + anticrossing as the transferredenergy becomes very small. Remarkably, the anticross-ing does not influence the rate of T , plotted at panel b),at all (the peak close to d = 0 is due to an anticrossingwith a higher excited state). Even though the dominantchannel, T → T + , is strongly suppressed here, its reduc-tion is exactly compensated by the elsewhere negligible T → S channel. The exact compensation arises for the a) 10 a)a)a) [110] γ b) 0 1 2 3 4 5 / l [-] . . . . . E - T [meV] . . . E - E - E - J [meV] b) 0 1 2 3 4 5 / l [-] . . . . . E - T [meV] . . . E - E - E - J [meV] b) 0 1 2 3 4 5 / l [-] . . . . . E - T [meV] . . . E - E - E - J [meV] b) 0 1 2 3 4 5 / l [-] . . . . . E - T [meV] . . . E - E - E - J [meV] [110] γ FIG. 2. Calculated relaxation rates of (a) the first excitedstate ( S or T + , see Fig. 1a)) and (b) the triplet T as a functionof the in-plane magnetic field orientation γ = arccos( B x /B )(angle) and the interdot distance 2 d/l (radius of the polarplot), for a double dot at B = 5 T. The x and y axes cor-respond to crystallographic axes [100] and [010], respectively.The dot orientation d || [110] is marked by a line. The bluehalf circles indicate the S − T + anticrossing, also marked onFig. 1a). The x axis is converted to the tunneling energy T and the exchange J , in addition to 2 d/l . The rate is givenin inverse seconds by the color scale. The system obeys C v symmetry, so point reflection would complete the graphs. relaxation into a quasi-degenerate subspace (we denotesuch cases on Fig. 1 by green arrows) if∆ E ≪ min { E, ~ c λ /l } . (12)Here E is the transition energy and ∆ E is the energywidth of subspace (the anticrossing gap). Equation (12)states that the energy width ∆ E is too small to be re-solved by either phonons with energy E or electron wavefunction scale l [33]. The relaxation then proceeds intothe subspace rather then into its constituent states, sothat any mixing of the states within the subspace is ir-relevant.Further to note on Fig. 2 is the anisotropy of re-laxation, which reflects the anisotropy of the spin-orbitfields. In the weak coupling regime, the relaxation ratesare minimal if the magnetic field orientation is parallel tothe dot main axis, which results in an isle of strongly pro-longed spin lifetimes. Note that this is in contrast to thebiased dot (see below), and to the single-electron case,where the minimal in-plane magnetic field direction, theeasy passage, of a d k [110] double dot is perpendicularto d [9, 34]. The switch can be understood from the effec-tive, spin-orbit induced, magnetic field [9] if written using a) 10 a)a) [110] γ b) 0 2 4 6 8 10 magnetic field [T] b) 0 2 4 6 8 10 magnetic field [T] b) 0 2 4 6 8 10 magnetic field [T] [110] γ FIG. 3. Calculated relaxation rates of (a) the first excitedstate and (b) the triplet T as a function of the in-plane mag-netic field orientation γ (angle) and the magnetic field magni-tude (radius of the polar plot), for a double dot with T = 0.1meV. The layout with respect to the crystallographic axes isthe same as in Fig. 2. The rate is given in inverse seconds bythe color scale. the coordinates along the dot axes x d , y d = ( x ± y ) / √ B so = B × { x d ( l − br − l − d )[110] + y d ( l − br + l − d )[110] } / √ . (13)At the anticrossing, the mixing due to x d is by far domi-nant, so the minimum appears with B along [110]. This x d dominance will be the case for a biased dot, too. Onthe other hand, in a single dot x d and y d induce com-parable mixing, and B so becomes minimal if the largerterm (the one with y d ) is eliminated. Weakly coupledunbiased dot is in this respect similar to a single dot asthe two-electron transitions can be understood as flips ofa particular electron located in a single dot. Since thedirection for the rate minimum switches upon changing d , the system does not show an easy passage, that is alow relaxation rate from weak to strong coupling regime.We plot the magnetic field dependence for a weaklycoupled unbiased double dot in Fig. 3 and observe similarbehavior as in Fig. 2. The relaxation rate is minimal if B || d throughout the shown parametric region. Thisis because the anticrossing and the related directionalswitch happens here at so small magnetic field that it isnot visible at the figure resolution. For completeness, wenote that the T − relaxation behavior is very similar tothe one for T on both Figs. 2 and 3, and we do not showit. BIASED DOUBLE DOT
We now consider a biased double dot. Its spectrum isshown in Fig. 1b) as a function of the detuning. Theground state singlet is in the (1,1) configuration (oneelectron in each dot) for low, and in the (0,2) configu-ration (both electrons in one dot) for large detunings.The crossover, a broad singlet-singlet anticrossing, is akey handle in spin measurement and manipulation [11].The low to large detuning crossover involves S − T ± an-ticrossing, exploited for nuclear-spin pumping [12, 35].We show the detuning and magnetic field influence onthe relaxation in Fig. 4. At the singlet-triplet anticross-ings, we observe that first, the relaxation rate of the firstexcited state dips at the S − T + anticrossing (though thedip is very narrow and hard to see at the figure reso-lution), and second, the T − rate strongly peaks at the S − T − anticrossing. This is a demonstration of the dom-inant effect of the anticrossing on the transition energy,and matrix element, respectively. Third, there are noother manifestations of the S − T ± anticrossings, a factdue to the exact compensation already mentioned before.The anisotropy features of this geometry are striking. Inthe given range of detuning energies, states except T ex-hibit a very distinctive easy passage for a magnetic fieldalong [1¯10], where the relaxation is up to to three ordersof magnitude smaller than with B along [110]. Thoughthe directional switch occurs—rates become minimal fora magnetic field along [110], it is again out of the fig-ure scope (very small and very large detunings). Therates increase at detunings & T is dominated by nuclear spins, thus being isotropic.This is surprising, since the effective (Overhauser) nu-clear magnetic field B nuc is of the order of mT, muchsmaller than the spin-orbit field in Eq. (13), B so ∼ ( l /l so ) B ≈
30 mT at B = 1 T for our parameters. Onetherefore expects the nuclei to lead to much slower re-laxation than the spin-orbit coupling. This was indeedthe case for the unbiased dots and Figs. 2 and 3. Howcan then nuclei dominate here? Looking on Fig. 1b, thishappens when states T and S (1 ,
1) are nearby in energy.Here, the otherwise negligible hyperfine effects take over, because the spin-orbit induced mixing of these two statesis forbidden [27]. Estimating the wave function admix-ture in the lowest order, the nuclei dominate if B so / | E T − E k | . B nuc / | E T − E S | , (14)with k being the closest state to which T is coupled bythe spin-orbit interaction. The above condition general-izes in an obvious way for other states than T and there a) 10 a)a)a)a) [110] γ b) 10 b) 10 b)b)b)b)b)b) [110] γ c) 1.5 1.6 1.7 1.8 1.9 2 detuning [meV] c) 1.5 1.6 1.7 1.8 1.9 2 detuning [meV] c) 1.5 1.6 1.7 1.8 1.9 2 detuning [meV] c) 1.5 1.6 1.7 1.8 1.9 2 detuning [meV] c) 1.5 1.6 1.7 1.8 1.9 2 detuning [meV] [110] γ FIG. 4. Calculated relaxation rates of (a) the first excitedstate, (b) T , and (c) T − as a function of the in-plane magneticfield orientation γ (angle) and detuning energy (radius of thepolar plot), for a double dot with 2 d/l = 4 .
35 ( T = 10 µ eV),chosen along Ref. [11], and B = 2 T. The layout with respectto the crystallographic axes is the same as in Fig. 2. Therate is given in inverse seconds by the color scale. The bluelines indicate the singlet-triplet anticrossings, in line with themarks in Fig. 1b). The dashed red lines in panel b) confinethe area where hyperfine coupling dominates. are additional cases of nuclear dominance in our system.However, they happen on parameter regions too small tobe visible on the resolution of Fig. 4, so we discuss themonly in the Supplementary material [33]. CONCLUSIONS
Our predictions are experimentally observable. Untilnow the spin-orbit origin, and especially its induced di-rectional anisotropy of the spin relaxation in weakly cou-pled two-electron dots has not yet been experimentallyestablished. With employing vector magnets it shouldnow be possible to overcome earlier experimental chal-lenges and change the magnetic field orientation whilekeeping the sample fixed and detect the anisotropy [36].The spin-orbit/nuclear induced relaxation can be maskedby cotunneling and smeared by a finite temperature. Theformer is reduced in the charge sensing readout setups[37], in which the coupling to the leads can be madesmall. The latter effect is small for experimentally rele-vant sub Kelvin temperatures, such that the directionalanisotropies are well preserved.Our results demonstrate control over the spin-orbitinduced anticrossing gaps (easy passages appear if thegaps are closed) by sample and magnetic field geome-try. It offers electrical tunability of spin relaxation, bychanging the double dot orientation (in the Supplemen-tary material [33], we suggest a spin current measure-ment device exploiting easy passage). In addition, suchcontrol may be especially useful when dealing with hy-perfine spins. Indeed, in the polarization scheme con-sidered in Ref. [14], the nuclear spin polarization is pro-portional to non-hyperfine assisted spin relaxation (seeEq. (7) therein) and so would benefit from a setup withmaximized spin-orbit induced relaxation rates (out of theeasy passage). On the other hand, the adiabatic pumpingscheme demonstrated in Ref. [35], relies on the S - T + an-ticrossing being solely due to the nuclear spins (and notthe spin-orbit coupling), suggesting improved efficiencyin an easy passage configuration. We propose a similarnon-adiabatic nuclear pumping scheme based on the easypassage in the Supplementary material [33]. All these ex-amples illustrate the potential benefits which intentionalcontrol of spin relaxation, based on our results, may offer. ACKNOWLEDGMENTS
This work was supported by DFG under grant SPP1285 and SFB 689. P.S. acknowledges support by meta-QUTE ITMS NFP 26240120022, CE SAS QUTE, EUProject Q-essence, APVV-0646-10 and SCIEX.
SUPPLEMENTAL INFORMATION
In this supplementary material, we provide furtherdetails illustrating the main text, and derivations ofsome of its results. In Sec. A we comment on our nu-merical method and derive the discretized form of theelectron-nuclear Hamiltonian, Eq. (9). In Sec. B we de-rive Eq. (12), the condition for the exact compensationand illustrate the exact compensation showing channelresolved relaxation rates. In Sec. C we compare the nu-clear vs. spin-orbit induced relaxation rates and discussthe additional cases to the one mentioned in the maintext, where the nuclear spins dominate the relaxation.Finally, in Sec. D we suggest schemes for dynamic nuclearpolarization and detection of spin polarization, which arebased on easy passages.
Numerical method
We use the exact diagonalization of Eq. (1) in the con-figuration interaction method. In this work, the two-electron basis consists of 1156 Slater determinants, gen- erated by 34 single electron orbital states. The discretiza-tion grid is typically 135 × − . The diagonalization procedure wasdescribed in detail in Ref. [32]. Here we extend it addingnuclear spins, for which we now derive the discretizedform of the Hamiltonian in Eq. (9), H discnuc .Consider a basic element of the spatial grid, a rectan-gular box with lateral dimensions h x and h y . Such vol-ume elements are labeled by the index k = 1 , ..., M , with M their amount. In the two dimensional approximationone assumes that the electron wave function along the z direction is fixed to ψ ( z ). Let it be, for concreteness, theground state of a hard-wall confinement of width w = 11nm. As follows from Eq. (9), the discretized Hamiltonianis diagonal in the spatial index. It has matrix elements( H discnuc ) kk ′ = δ kk ′ β X n ∈ k I n · σ h k, ψ | δ ( r − R n ) | k, ψ i , (15)where the sum is over nuclei inside the volume element k . To proceed further, we discretize the delta function intwo dimensions as 1 /h x h y , introduce the nuclei volumedensity as 1 /v , and get( H discnuc ) kk ′ = δ kk ′ ( β/v ) X n ∈ k v I n · σ | ψ ( z n ) | /h x h y . (16)We now replace the sum over (typically many) nuclearspins by an effective spin I and get the discretized formof the Hamiltonian as( H discnuc ) kk ′ = δ kk ′ ( β/v ) I · σ . (17)By the central limit theorem, the effective spins are com-pletely described by their average and dispersion, whichfollow from the corresponding characteristics of the nu-clear spin ensemble. For random unpolarized nuclearspins, which we consider, it holds h I n i = 0 , h I n · I m i = δ nm I ( I + 1) , (18)so that the effective spins have zero average and the fol-lowing dispersion h I k · I k ′ i = δ kk ′ I ( I + 1) /N. (19)Here N is the number of nuclei in the grid volume el-ement, N = h x h y h z /v , where the effective extensionalong z is defined by the wave function profile h − z = Z d z | ψ ( z ) | . (20)For the hard-wall potential one gets h z = 2 w/ I drawn from a random Gaussian ensembledescribed by Eq. (A5), for which we diagonalize the two-electron Hamiltonian. Having the two-electron spectrumallows us to calculate the matrix elements M ij and en-ergy differences ω ij which enter the relaxation rates Γ ij in Eq. (11). Exact compensation of relaxation rate channels
In general, the relaxation rate channels significantlychange at spectral anticrossings because of the strongmixing of states. We consider the total relaxation rateby summing over the individual relaxation channels of alllower lying states. Therefore, a change in one relaxationchannel may be compensated by another channel, suchthat the total relaxation rate is smooth (no peak or dip)across the anticrossing. This generally happens if a staterelaxes into a quasi-degenerate subspace of anticrossingstates. We exemplify such a situation on Fig. 1b) by agreen arrow and consider relaxation of T (state i ) at thisdetuning. We write the total relaxation of T asΓ i = X j π ~ ρV X Q ,λ Qc λ | V Q ,λ | | M ij | δ ( ω ij − ω Q ) , (21)where the sum includes the two quasi-degenerate states j = T + and j = S . The exact compensation arises ifone can approximate the argument of the delta functionby a common energy difference ω ij ≈ ω ij ≈ ω . Indeed,the relaxation can be then written asΓ i = π ~ ρV X Q ,λ Qc λ | V Q ,λ | h i | M P M † | i i δ ( ω − ω Q ) , (22)where P j = P j | j ih j | is the projector on the quasi-degenerate subspace, which is not influenced by mixingof the basis states j . The condition for the approxima-tion Eq. (B2) to be valid is that both under-integral fac-tors in Eq. (B1), the phonon density of states as well aselectron overlap integral, are not changed much by theslight shift of the transition energy. The phonon densityof states scales as a certain power (albeit different forpiezoelectric and deformation potential) in the phononwave vector, which translates into the condition ω j j /c λ ≪ ω ij /c λ . (23)On the other hand, the natural scale for the electron wavefunction is the confinement length l , so that the overlap M will not change much if ω j j /c λ ≪ /l . (24)Denoting ω j j = ∆ E/ ~ we get Eq. (12) of the maintext. For the interdot coupling denoted on Fig.1a andthe dominant piezoelectric phonons we have E = 125 µ eV, ~ c t /l = 48 µ eV, while ∆ E is just 7 µ eV, so thatthe exact compensation condition is well satisfied.To illustrate the exact compensation, we plot in Fig. 5the individual relaxation channels as a function of inter-dot distance. The parameters are chosen the same as inFig. 2 of the main text. We find the exact compensationat the S − T + anticrossing for the T and the T − relax-ation. In the case of the unpolarized triplet, the dip of r e l a x a ti on r a t e [ / s ] T → T + T_ → T T_ → T T → T + T_ → T + T_ → T + T → ST → S T + ↔ ST + ↔ ST_ → ST_ → S xxyyBB xxyyBB xxyyBB FIG. 5. Calculated channel resolved relaxation rates vs. in-terdot distance in units of l for both parallel (top) and per-pendicular to d (bottom) in-plane magnetic field orientation( B = 5 T, zero detuning). The relaxation channels of T and T − are in blue and black color, respectively. The relaxationrate of the first excited state is red. the T → T + channel is compensated by a peak of the T → S channel. For T − , the dip and peak occurs in the T − → S and T − → T + channels, respectively. Note thatif the in-plane magnetic field is perpendicular to the dotmain axis d (lower panel), the relaxation channels for T − and T do not vary at all, as the S − T + anticrossing gapvanishes, ∆ E = 0, and the exact compensation is trivial. Hyperfine versus spin-orbit induced relaxation
Comparing the value of the nuclear and spin-orbiteffective fields, we estimate the relaxation due to theformer is typically three orders of magnitude smaller,( B nuc /B so ) ∼ − , if the external field is of the orderof Tesla. However, in a weakly coupled detuned doubledot the nuclear spins can dominate over the spin-orbit in-duced relaxation in some cases, when Eq. (14) is satisfied.We plot in Fig. 6 the spin relaxation rates enabled byspin-orbit and hyperfine coupling, respectively. Panel a)gives the relaxation rate of the first excited state. The hy-perfine coupling becomes relevant only close to the S − T + anticrossing along the easy passage. Here, the wide dipis narrowed (red versus the blue curve). However, therate remains reasonably low, such that the easy passagesurvives. Adding the nuclear dominated area to Fig. 4a) of the main text would barely be visible. Panel b)shows the rate of T . We find that the hyperfine-inducedrelaxation is dominant for any in-plane magnetic field detuning [meV] detuning [meV] detuning [meV] r a t e [ / s ] interdot distance 2d/l S/T + T T_ T a bc d FIG. 6. Calculated spin-orbit induced relaxation rates foran in-plane magnetic field orientation parallel (black curves)and perpendicular (blue curves) to the dot main axis d . Thered curves show the hyperfine-induced spin relaxation. (a)-(c) Weakly coupled double dot ( T = 10 µ eV) as a functionof detuning for B = 2 T. The panels display the relaxationrates for the first excited state, the unpolarized triplet, and T − respectively. (d) Unbiased double dot as a function ofinterdot distance (in units of l ) for B = 5 T. The relaxationrate of T is shown. orientation if the unpolarized triplet is close in energyto the first excited singlet, as shown in Fig. 4 b) of themain text. Panel c) displays the relaxation of T − . Atthe S − T − anticrossing, the spin-orbit induced relax-ation strongly peaks unless the in-plane magnetic fieldorientation is perpendicular to the dot main axis. At theanticrossing, also the hyperfine-induced rate is enhanced.Displacing the magnetic field from the easy passage, thespin-orbit rate quickly gains on magnitude, therefore thenuclear-dominated area on Fig. 4 c) would cover onlya single point at its current resolution. In panel d) weshow the relaxation rate for an unbiased dot. We choose T as an example, the state which is most prone to haverelaxation dominated by nuclear spins in the biased dot.Here, the relaxation due to the spin-orbit coupling is sev-eral orders of magnitude larger than due to the nuclei,for any orientation of the external field. We observe asimilar difference in rates for other states in this setup aswell. Easy passage exploitation examplesDynamical nuclear spin polarization
We sketch two schemes of dynamical nuclear spinpumping in Fig. 7. The first is the one originally pro-posed by Reilly et. al. in Ref. [35]. Here, the double dot
FIG. 7. Sketches of two schemes of dynamical nuclear spinpumping. The blue arrows indicate the path which the systemstate follows during one pump cycle. is initialized in the S (0 , state. Then the system is adi-abatically brought through the anticrossing (step 1), bywhich a nuclear spin is flipped, assuming the anticross-ing is due to the (transverse component of) the nucleareffective field (and not due to the spin-orbit coupling).Placing the system into the easy passage, which was notdone in the experiment, thus offers improved scheme effi-ciency. The cycle is finished by resetting the system intothe S (0 , , achieved by a fast transition (step 2) and asubsequent relaxation (step 3).We propose here a non-adiabatic version of the scheme,which does not require the increasingly slower gapcrossovers, by ending the step 1 inside the anticrossing.This necessarily requires to monitor the anticrossing po-sition, which however is possible. Again, the scheme ismost efficient if the spin-orbit contribution to the an-ticrossing gap is minimized, what happens in the easypassage configuration. Spin polarization detection
Here we propose a device which allows to detect thespin polarization of a lead using a weakly coupled dou-ble dot. The dot is connected to source and drain leadssuch that the current passes only through the left dot.The system is biased such that only (1,1) and (0,2) oc-cupations are allowed if the right dot is brought belowin energy and the current is allowed to flow. The sys-tem is periodically brought above both source and drainleads so that the right dot is emptied. After such a reset,when the right dot is lowered in energy by a gate, theelectron which is traversing the device may tunnel intothe right dot and becomes trapped (we assume its spin ispreserved). Now, if an electron with the same spin orien-tation enters the left dot, a T + , (1 , triplet state is formed,with a long lifetime such that the electron in the left dottunnels out and the current flows. If, on the other hand,a spin opposite electron enters the left dot, the systemquickly collapses into the S (0 , state and the current isblocked until the reset (see the sketch in Fig. 8). As theresult, the higher the spin polarization of the electrons inthe source lead, the higher the current on average. The FIG. 8. Spin polarization detection scheme. (a) The doubledot is in a T + , (1 , state and the current is enabled. (b) Forthe S (0 , state the current is blocked. scheme requiresΓ T → S & Γ left → lead ≫ Γ T + → S , (25)where Γ are the rates for transitions corresponding to theindexes. The conditions in Eq. (25) are only achievablein the easy passage configuration while Γ T → S must bedominated by the hyperfine-induced relaxation. [1] D. Loss and D. P. DiVincenzo, Phys. Rev. A , 120(1998)[2] I. ˇZuti´c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. , 323 (2004)[3] J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, andI. ˇZuti´c, Acta Phys. Slov. , 565 (2007), arXiv:0711.1461[4] R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha,and L. M. K. Vandersypen, Rev. Mod. Phys. , 1217(2007)[5] R. Brunner, Y.-S. Shin, T. Obata, M. Pioro-Ladri`ere,T. Kubo, K. Yoshida, T. Taniyama, Y. Tokura, andS. Tarucha, Phys. Rev. Lett. , 146801 (2011)[6] A. V. Khaetskii and Y. V. Nazarov, Phys. Rev. B ,125316 (2001)[7] S. Amasha, K. MacLean, I. P. Radu, D. M. Zumb¨uhl,M. A. Kastner, M. P. Hanson, and A. C. Gossard, Phys.Rev. Lett. , 046803 (2008)[8] V. N. Golovach, A. Khaetskii, and D. Loss, Phys. Rev.Let. , 016601 (2004)[9] P. Stano and J. Fabian, Phys. Rev. Lett. , 186602(2006)[10] X. Hu and S. Das Sarma, Phys. Rev. A , 062301 (2000)[11] J. M. Taylor, J. R. Petta, A. C. Johnson, A. Yacoby,C. M. Marcus, and M. D. Lukin, Phys. Rev. B , 035315(2007)[12] A. Pfund, I. Shorubalko, K. Ensslin, and R. Leturcq,Phys. Rev. Lett. , 036801 (2007)[13] M. S. Rudner and L. S. Levitov, Phys. Rev. Lett. ,036602 (2007) [14] M. S. Rudner and L. S. Levitov, Phys. Rev. Lett. ,246602 (2007)[15] J. M. Elzerman, R. Hanson, L. H. Willems van Beveren,B. Witkamp, L. M. K. Vandersypen, and L. P. Kouwen-hoven, Nature , 431 (2004)[16] P. Stano and J. Fabian, Phys. Rev. B , 045320 (2006)[17] T. Fujisawa, D. G. Austing, Y. Tokura, Y. Hirayama,and S. Tarucha, Nature , 278 (2002)[18] S. Sasaki, T. Fujisawa, T. Hayashi, and Y. Hirayama,Phys. Rev. Lett. , 056803 (2005)[19] T. Meunier, I. T. Vink, L. H. W. van Beveren, K.-J.Tielrooij, R. Hanson, F. H. L. Koppens, H. P. Tranitz,W. Wegscheider, L. P. Kouwenhoven, and L. M. K. Van-dersypen, Phys. Rev. Lett. , 126601 (2007)[20] J. R. Petta, A. C. Johnson, A. Yacoby, C. M. Marcus,M. P. Hanson, and A. C. Gossard, Phys. Rev. B ,161301 (2005)[21] F. H. L. Koppens, J. A. Folk1, J. M. Elzerman, R. Han-son, L. H. W. van Beveren, I. T. Vink, H. P. Tranitz,W. Wegscheider, L. P. Kouwenhoven, and L. M. K. Van-dersypen, Science , 1346 (2005)[22] A. C. Johnson, J. R. Petta, J. M. Taylor, A. Yacobi,M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C.Gossard, Nature , 925 (2005)[23] J. I. Climente, A. Bertoni, G. Goldoni, M. Rontani, andE. Molinari, Phys. Rev. B , 081303(R) (2007)[24] V. N. Golovach, A. Khaetskii, and D. Loss, Phys. Rev. B , 045328 (2008)[25] D. Chaney and P. A. Maksym, Phys. Rev. B , 035323(2007)[26] K. Shen and M. W. Wu, Phys. Rev. B , 235313 (2007)[27] M. Florescu and P. Hawrylak, Phys. Rev. B , 045304(2006)[28] O. Olendski and T. V. Shahbazyan, Phys. Rev. B ,041306(R) (2007)[29] L. Wang and M. W. Wu, J. Appl. Phys. , 043716(2011)[30] We present the results for the state-of-the art GaAs quan-tum dots. We have also calculated the rates for silicon(unpublished), where the rates are orders of magnitudesmaller but qualitatively the results are similar. Differ-ences may arise for isotopically purified Silicon dots,where nuclei effects are reduced.[31] F. Baruffa, P. Stano, and J. Fabian, Phys. Rev. Lett. , 126401 (2010)[32] F. Baruffa, P. Stano, and J. Fabian, Phys. Rev. B ,045311 (2010)[33] See Supplemental Material at http://...[34] M. Raith, P. Stano, and J. Fabian, Phys. Rev. B ,195318 (2011)[35] D. J. Reilly, J. M. Taylor, J. R. Petta, C. M. Marcus,M. P. Hanson, and A. C. Gossard, Science , 817(2008)[36] S. Amasha (private communication).[37] C. Barthel, D. J. Reilly, C. M. Marcus, M. P. Hanson,and A. C. Gossard, Phys. Rev. Lett.103