Nuclear spin polarization in a single quantum dot pumped by two laser beams
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Nuclear spin polarization in a single quantum dot pumped by two laser beams
Xiao-Feng Shi ∗ Department of Physics, University of California San Diego, La Jolla, California 92093-0319, USA (Dated: March 2, 2018)We theoretically investigate dynamic nuclear spin polarization in a self-assembled quantum dotpumped optically by two laser beams. With the assumption that a noncollinear interaction betweenthe hole spin and nuclear spins leads to nuclear spin polarization, we find that both weak andstrong nuclear spin polarizations can arise, depending on the intensities and central frequencies ofthe lasers. For weak nuclear spin polarization, we use a perturbation method to show that thedistribution of the nuclear spin Overhauser field may become significantly narrower. Using MonteCarlo simulations to study a single quantum dot, we find that strong nuclear spin polarization canalso be generated via appropriate optical pumping.
PACS numbers: 73.21.La, 78.67.Hc, 72.25.Fe
I. INTRODUCTION
Dynamic nuclear spin polarization by optical pump-ing on single quantum dots (QDs) is extensively studiedin recent years, due to its importance for improvingtechniques in quantum computing, and for understandingthe nuclear spin environment of the electron (hole) spinqubit. For instance, an efficient suppression of nuclearspin fluctuation by optical pumping allows one to pro-long the electron spin coherence time in a QD.
Also,significant nuclear spin polarization may be achieved even by optical excitation of spin-forbidden transitions. In order to understand the dynamic nuclear spin polar-ization induced by continuous optical pumping on singleQDs, a microscopic theory was recently introducedto explain relevant experiments.
However, this the-ory is restricted to single laser pumping and only treatsweak nuclear spin polarization, while experiments involv-ing two laser beams or strong nuclear spin polariza-tion remain to be explained microscopically.In this paper, we study dynamic nuclear spin po-larization in a single QD charged with one electronand pumped by two narrow-linewidth continuous wavelasers (see Fig. 1). We find that both weak and strongnuclear spin polarizations can be generated. Specifically,when the laser with central frequency ω and moder-ate Rabi frequency Ω is off-resonant (resonant) withthe transition between the exciton and the electron spineigenstate | x − ( x +) i , the nuclear spins tend to haveweak polarization for Ω ≪ Ω , but strong polariza-tion for Ω ≫ Ω . In the former case, we derive aFokker-Planck equation for the evolution of the prob-ability density of nuclear spin polarization. We use theFokker-Planck equation to show that the nuclear spinfluctuation can be reduced, thus enhancing the electronspin coherence. Also, we perform numerical study ofseveral interesting phenomena observed in laser spec-troscopy experiments. For the case of strong nuclearspin polarization, the perturbation method for derivingthe Fokker-Planck equation breaks down. We then cre-ate a toy model comprising a small QD in order to obtainunbiased results using the Green’s function Monte Carlo simulation.
Our Monte Carlo simulation shows thatat least as high as 50% of the nuclear spin polarizationdegree can be generated by pumping optically on the sin-gle QD in our toy model. We note that a 50% degree ofnuclear spin polarization is close to the large nuclear spinpolarization degree observed in experiments by opticallypumping single QDs.
The paper is organized as follows. Section II briefly in-troduces the system Hamiltonian. Section III solves theequation of motion for the nuclear spin population. InSec. IV, we study the case corresponding to the experi-ments in Ref. 13, where only weak nuclear spin polariza-tion was observed. In Sec. V, we first identify the factorsresponsible for large nuclear spin polarization. Then wenumerically study a small QD of artificial size to showthat large nuclear spin polarization can indeed be gener-ated by optical pumping on single QDs. We conclude inSec. VI.
II. MODEL
Consider a self-assembled InAs QD charged with oneelectron, where the confined electron interacts with N indium nuclear spins and N arsenic nuclear spins. Defin-ing z as the growth direction of the QD, we apply anin-plane static magnetic field B in the x direction andlabel the electron spin eigenstates as | x ±i , as shown inFig. 1. Two linearly polarized coherent laser beams, onewith the polarization of the electric vector along x di-rection, and the other along the y direction, selectivelycouple the two electron spin states to a common trionlevel, denoted as | T −i . The trion consists one heavyhole and two electrons, with the two electrons in thesinglet configuration. The Hamiltonian of the system isˆ H = ˆ H eh + ˆ H n + ˆ H HI , (1)where ˆ H eh is the Hamiltonian of the electron-hole system,ˆ H n the nuclear spin Zeeman term, and ˆ H HI the hyperfineinteraction between the electron/hole spin and the nu-clear spins in the QD, ˆ H HI = X j a e ,j (cid:16) ˆ S x e ˆ I xj + ˆ S y e ˆ I yj + ˆ S z e ˆ I zj (cid:17) + X j a h ,j | β | (cid:26) ˆ S z h ˆ I zj + 2 | β |√ h ˆ S x h ( ˆ I xj cos δ + ˆ I yj sin δ ) + ˆ S y h ( ˆ I yj cos δ − ˆ I xj sin δ ) io , (2)where a e(h) ,j is the hyperfine interaction strength be-tween the electron (hole) spin S e(h) and the nuclear spin I j , the superscript x, y or z denotes component of thespin moment in the corresponding direction, and β = | β | e iδ is the heavy-light hole mixing coefficient. Be-sides the dipole-dipole coupling in the hole spin-nuclearspin interaction, there are extra hyperfine couplings withstrength proportional to | β | in Eq. (2) that arise fromthe mixing between the heavy and light hole bands. Theband-mixing is caused by an in-plane strain of the QDwhen one grows QDs in the Stranski-Krastanow growthmode during molecular beam epitaxy. Observed val-ues of | β | for typical In(Ga)As QDs range from 0 .
02 to0 . The angle δ is determined by the straindetail of the QD and can in principle take any valuebetween 0 and 2 π . For simplicity, we have ignored the in-trinsic interactions (including dipole-dipole interactions)between nuclear spins in ˆ H n . The latter will be consid-ered later when studying relevant experiments, involvingnuclear spin depolarization. For study involving small nuclear spin polariza-tions, we consider a QD containing N = 9500 InAsmolecules (estimated from Ref. 13). Without loss of gen-erality, we assume g e(h) = 0 .
49 ( − . | x −i and | T −i may be slightly off-resonant with the transi-tion between the electronic and excitonic states, whilethe laser connecting | x + i and | T −i is resonant unlessotherwise specified. For this case with a strong externalmagnetic field of | B | = 2 .
64 T, we can prove that thelevel | T + i decouples from the rest of the system (Ap-pendix A), resulting in a three-level electron-hole systemwhich we call a Λ system (ΛS). III. NUCLEAR SPIN DYNAMICS
Following Refs. 13, 26, and 47, we assume that theeffective noncollinear hyperfine interactions between theelectron/hole spin and nuclear spins can lead to dynamicnuclear spin polarization in a single QD. For simplicity,we only consider the hole spin-nuclear spin noncollinearinteraction of the form ˆ S x h ˆ I yj here, which appeared inEq. (2). We neglect the influence of the transverse part of FIG. 1. (Color online) Schematic of an electron-hole system.The x -polarized laser connecting | x −i and | T −i has centralfrequency ω and Rabi frequency Ω , and the y -polarized laserconnecting | x + i and | T −i has central frequency ω and Rabifrequency Ω . the electron (hole) spin-nuclear spin hyperfine interactionon the nuclear spin dynamics as discussed in Appendix A.Using the theory developed in Ref. 31, we derive anequation for the time evolution of the nuclear spin popu-lation ˆ P ( t ), i.e., the diagonal part of the reduced densitymatrix of the nuclear spins, ddt ˆ P ( t ) ≈ − X j nh ˆ I − j , ˆ I + j ˆ W α j , + ˆ P ( t ) i + h ˆ I + j , ˆ I − j ˆ W α j , − ˆ P ( t ) io , (3)where α j denotes In (As) when there is an indium (ar-senic) atom at site j , andˆ W α j , ± = (cid:12)(cid:12)(cid:12)(cid:12) | β | a h ,j sin δ √ | β | ) (cid:12)(cid:12)(cid:12)(cid:12) Z e ± ig j µ N Bt ′ × Tr[ ˆ S x h,I ( t ′ ) ˆ S x h ˆ ρ ( ss )Λ (0)] dt ′ , (4)ˆ S x h,I ( t ) = ˆ U † ( t ) ˆ S x h ˆ U ( t ) , ˆ U ( t ) = T e − i R t [ ˆ H eh,r ( t ′ )+ ˆ H n ] dt ′ , where µ N is the nuclear magneton, ˆ ρ ( ss )Λ is the reduceddensity matrix for the steady state of the ΛS, Tr de-notes trace over the Λ-system degrees of freedom, T isthe time ordering operator, ˆ H eh,r ( t ′ ) is obtained by per-forming a rotating frame transformation on ˆ H eh (see Ap-pendix A), and the integration is from − t to t . However,since the scale of t is much larger than the time scale ofthe ΛS, we can perform the integration from −∞ to ∞ and the final integration can be calculated through thequantum regression theorem. There is a feedback loop between the nuclear spinsand the ΛS. First, the rate ˆ W α j , ± in Eq. (3) for flip-ping the nuclear spins depends on the steady state ofthe ΛS. Second, the steady state of the ΛS depends onthe nuclear spin state, since the eigenenergies of the twoelectron spin eigenstates | x ±i in the ΛS are shifted byan effective magnetic field contributed by ˆ S x e P a e ,j ˆ I xj inˆ H HI . Here we neglect the nuclear spin Overhauser fieldcontributed from the hole spin-nuclear spin hyperfine in-teraction since | a h ,j | is much smaller than | a e ,j | .The effective magnetic field, i.e., the nuclear spin Over-hauser field, can be written as [see Eq. (A7)] h = [ A In Is In + A As Js As ] , (5)where I = 9 / , J = 3 / A In(As) is the hyperfine constantof the indium (arsenic) nuclear spins, and s In , s As arethe expectation values ofˆ s In = 1 N I X j ∈ In ˆ I xj , ˆ s As = 1 N J X j ∈ As ˆ J xj . (6)Because of the nuclear spin Overhauser field h , the nom-inal detunings∆ , = E T − E x − ω , ∆ , = E T − E x − ω (7)are replaced by the actual detunings∆ = ∆ , + h , ∆ = ∆ , − h E x , E x and E T are the respective eigenenergies (whenthere is no net nuclear spin Overhauser field) of | x i , | x −i and | T −i , and ω is the central frequency of the laserbeam that couples | x ∓i and | T −i . For different adiabaticnuclear spin Overhauser fields on the ΛS, the Zeeman en-ergy of the electron spin is effectively shifted by differentamounts, giving different flipping rates in Eq. (3). When h keeps on changing, the flipping rates in Eq. (3) keepon changing. This feedback loop between the ΛS andthe nuclear spins in the QD will not cease whenever thedistribution of h is changing. IV. SMALL NUCLEAR SPIN POLARIZATIONA. Approximation when | s In(As) | ≪ In practice, it is not easy to solve the distribution of h from Eq. (3) since the number of nuclear spins is large.Nonetheless, when the polarizations of the two speciesof nuclear spins are much smaller than 1, we can obtainsolution for the evolution of the joint probability densityof the nuclear spin polarizations p s ( s In , s As ) = Tr h ˆ P ( t ) δ ˆ s In ,s In δ ˆ s As ,s As i , (9)where δ ˆ s In(As) ,s In(As) is the delta function. In particular,we derive a Fokker-Planck equation, ∂∂t p s ≈ X α =In,As ∂∂s α (cid:20) ∂∂s α D α p s ( s In , s As ) − v α p s ( s In , s As ) (cid:21) , (10)where v α and D α , given by Eq. (B11), are known as thedrift and diffusion coefficients, respectively. From the −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50.20.40.60.811.21.41.61.822.2 h (GHz) F li pp i ng r a t e ( / s ) Γ In,p Γ As,p
FIG. 2. (Color online) Flipping rate Γ
In(As),p as a func-tion of the nuclear spin Overhauser field h in a QD with N = 9500 InAs molecules. Parameters: B = − .
64 T, g e(h) = 0 . − . , | β | = 0 . , δ = π/ , a e ,j = − a h ,j = A j /N , and γ s = 0 . , ∆ , = ∆ , = 0 , Γ = 0 . , Ω =0 . , Ω = 1 .
35 (unit: GHz). Here the gyromagnetic ratiois g j µ N = 0 . . / T, and the hyperfine constant A j = 13 . .
1) GHz (taken from Ref. 52) when there is anindium (arsenic) nuclear spin at j . relation between s In , s As and h in Eq. (5), we can obtainthe distribution of h from p s ( s In , s As ).In order to study the absorption coefficient of the probelaser in laser spectroscopy experiments on single QDs, we consider the dynamics of the mean of the nuclear spinOverhauser field. To simplify the calculation, we neglectthe finite width of the distribution of the polarization s In and s As . Then the mean of nuclear spin polarization s In(As) obeys (Appendix B) ddt s
In(As) ≈ v In(As) − γ dep s In(As) , (11)where v In = − Γ In , p (cid:20) s In − I + 1)3 s In , (cid:21) , Γ In , p = D ˆ W In , + + ˆ W In , − E ,s In , = * ˆ W In , + − ˆ W In , − ˆ W In , + + ˆ W In , − + , (12)and similarly for v As . Here, h· · · i denotes the respec-tive expectation value. In Eq. (11) we have added a nu-clear spin depolarization channel with rate γ dep ≥
0. Thereason for adding this term is that, in the nuclear spinHamiltonian ˆ H n , we have ignored the direct dipole-dipoleinteractions which play an important role in depolarizingthe nuclear spins. A similar depolarization term wasalso used in the numerical fitting of the experimental re-sults in Ref. 13. By choosing the electron spin dephasingrate γ s = 0 .
06 GHz (this value is taken from the sup-plementary material of Ref. 13), the rate of spontaneousdecay from | T −i to either | x + i or | x −i as Γ = 0 . = 0 .
24 (1 .
35) GHz for theprobe (pump) laser beam, we calculate the flipping rate −1 −0.5 0 0.5 1 1.502468 (a) ∆ = −0.3 GHz −1 −0.5 0 0.5 10246 p h ( ss ) (b) ∆ = 0 −2 −1.5 −1 −0.5 0 0.5 102468 (c) ∆ = 0.675 GHz −4 −3 −2 −1 0 1 2 3 40246 h (GHz)(d) ∆ = 2 GHz(solid); −2 GHz(dash−dot) Optically pumpedthermal
FIG. 3. (Color online) Solid (dashed) curves show the steadydistributions of the nuclear spin Overhauser field h built upwith (without) optical pumping. ∆ , = − . , , .
675 and ± A In = 14 . Γ In(As) , p as a function of h for ∆ , = ∆ , = 0, andobtain the results shown in Fig. 2. B. Numerical results
There are two important observations made in Ref. 13.The first is an enhanced electron spin coherence time,which was observed indirectly via a deeper dark statedip of the probe laser absorption. The second observa-tion concerns a series of phenomena caused by dynamicnuclear spin polarization, which were observed via thechange of the probe laser absorption in reference to theexpected absorption when there is no dynamic nuclearspin polarization. We will study these two types of phe-nomena.For the first observation where optical pumping re-duces the nuclear spin fluctuation, we will study thesteady distribution of the nuclear spin Overhauser field.For the second observation, we will simulate the timeevolution of the absorption coefficient of the probe laseras the laser frequency is tuned. The input for the sim-ulation is the laser detuning, which partially determinesthe steady state of the electron-hole system. The outputis the expectation value of the absorption coefficient forthe probe laser. Since the absorption of the probe laser isproportional to the imaginary part of the density matrix element h x − | ˆ ρ ( ss )Λ | T −i , we define χ i = Im |h x − | ˆ ρ ( ss )Λ | T −i| , (13)as the absorption coefficient of the probe laser. The keypoint is to calculate the expectation value of the nu-clear spin Overhauser field, since the actual detuningsfor the optical transitions take into account the nuclearspin Overhauser field, which shifts the electron spin Zee-man energy. As for the input parameters, γ s is in prin-ciple determined by the fluctuation of h . However, asmentioned in Sec. IV A, we neglect the finite width of p s ( s In , s As ) when studying the second type of observa-tions. This means that we have lost the information onthe evolution of the nuclear spin fluctuation. As a re-sult, we apply a constant electron spin decoherence rate γ s taken from the estimation in Ref. 13 for each set ofchosen parameters.
1. Narrowed distribution of the nuclear spin field
Following Ref. 31, we assume that the system containsonly indium nuclear spins, since the momentum of an in-dium nuclear spin is three times that of an arsenic nuclearspin, and A In > A As . In this case, we rescale the indiumnuclear spin hyperfine constant as shown in Appendix C.Equation (10) becomes ∂∂t p s = ∂∂s In (cid:20) ∂∂s In D In p s − v In p s (cid:21) , (14)where the difussion coefficient is given by D In = Γ In,p N I (cid:20) I + 1)3 − s In s In,0 (cid:21) . (15)As in Refs. 31, 53, and 56, the solution for the steadydistribution of nuclear spin polarization is p (ss)s ( s In ) = N D In ( s In ) exp (cid:20)Z s In − v In ( s ′ In ) D In ( s ′ In ) ds ′ In (cid:21) , (16)where N is the normalization factor. From p (ss)s ( s In )we can evaluate the steady distribution of the nuclearspin Overhauser field p (ss)h ( h ). Using the same param-eters (except ∆ , ) in Fig. 2, we plot p (ss)h ( h ) for dif-ferent nominal detunings ∆ , = − . , , .
675 and ± s In is ∼ ∓ .
067 for ∆ , = ± | s In | ≪ n -times narrower p (ss)h ( h ) indicates n -times −3 −2 −1 0 1 2 3−0.500.51 h ( GH z ) (a) from leftfrom right−3 −2 −1 0 1 2 300.050.10.15 ∆ (GHz) χ i (b) FIG. 4. (Color online) (a) Nuclear spin Overhauser field,and (b) χ i as a function of detuning ∆ , when we changethe laser detuning by 0 .
04 GHz at 4 s interval. Data shownwith solid (dash-dot) curves correspond to changing ∆ , fromleft (right) to right (left). Here we use γ dep = 0 . /s . Otherparameters except ∆ , are the same as in Fig. 3. smaller an electron spin decoherence rate. Specifically,∆ , = 0 .
675 GHz in Fig. 3(c) is equal to Ω /
2, cor-responding to the detuning of the second pump in thetwo-pump experiment shown in Fig. 3c of Ref. 13, wherea significant enhancement of the electron spin coherenceis observed. Besides a narrowed distribution of the nu-clear spin Overhauser field, we also note that there aretwo separated peaks in p (ss)h ( h ) when ∆ , = 0, indicatingbistability of the nuclear spin polarization. Finally,we note that p (ss)h ( h ) for | ∆ , | and −| ∆ , | are symmet-rical to each other about the line h = 0, as shown inFig. 3(d) for illustration.
2. Laser spectroscopy experiments
Here, we study systems with two different sets of pa-rameters and show that, as they were captured, threephenomena observed experimentally in Ref. 13. The firstcase is for showing that the absorption curve will shiftwhen the probe laser frequency is changed step by step.In the second case, we will show that (i) the line-shapesof χ i for forward and backward scanning are not symmet-ric to each other when the pump laser is off-resonance,and (ii) there can be an abrupt switching of the probelaser absorption when we stop changing the laser fre-quency at a rising edge of the absorption curve. We notethat these three phenomena agree with the respectiveexperiments (see Fig. 1, Fig. 2a, and Fig. 2c of Ref. 13, −3 −2 −1 0 1 2 3−1.5−1−0.500.51 h ( GH z ) (a) from leftfrom right−3 −2 −1 0 1 2 300.020.040.060.080.10.120.14 ∆ (GHz) χ i (b) FIG. 5. (Color online) Same as Fig. 4. Parameters arethe same as in Fig. 4 except γ s = 0 . , ∆ , = − . , Ω =0 . γ dep = 0 . /s . respectively).For the first case, the parameters ∆ , , Γ , γ s , Ω , Ω are set as 0 , . , . , . , .
35 GHz, respectively. Weset γ dep = 0 . /s here. We change the central frequency ω of the probe laser by 40 MHz at 4 s interval, from highto low and from low to high, corresponding to chang-ing ∆ , from negative to positive and from positive tonegative, respectively. By using Eq. (11), the calculatedOverhauser field h and the absorption coefficient χ i ofthe probe laser is shown in Fig. 4(a) and (b), respectively.The data shown with a solid (dash-dot) curve correspondto changing ∆ , from left to right (from right to left),respectively. One can see that the absorption curve shiftstowards the direction of changing ∆ , . This result agreeswith the experimental observation in Ref. 13.For the second case, we choose a detuned pump laserwith ∆ , = − . x -axis, while Fig. 2a of Ref. 13 uses laser frequency),and choose the Rabi frequency of the pump laser Ω as0 . γ s = 0 .
22 GHz, as estimated fromRef. 13 (see Fig. 1 of its supplementary information).Also, we set γ dep = 0 . /s , which is much smaller thanthe depolarization rate used in Fig. 4. Interestingly, wenote that in the supplementary material of Ref. 13, asmaller (comparing with the one used for the resonantpump laser) nuclear spin depolarization rate for a de-tuned pump laser is also used for numerical simulation.We have chosen the parameters different from those inFig. 4 in order to reproduce the experimental results inRef. 13. Indeed, when we calculate the nuclear spin Over- ∆ =2.3 GHz0 100 200 300 400 500−1.4−1.2−1 0 100 200 300 400 5000.020.030.040.050.060.07 χ i stop at ∆ =2.335 GHz0 200 400 600 800−1.4−1.2−1−0.8 h ( GH z ) ∆ =2.34 GHz0 100 200 300 400−1.1−1−0.9−0.8−0.7 t(s) ∆ =2.35 GHz 0 500 1000 0.0250.03 (a)(d)(b)(c) swithcingswithcing FIG. 6. (Color online) Left and right panels of (a): h and χ i asa function of t after stopping changing ∆ , from 3 . .
04 GHz as in Fig. 5] to ∆ , = 2 . , at 2 . , .
34 and2 .
35 GHz, respectively. The inset of the right panel of (d)shows the longer time behavior of χ i . One can see that theabsorption coefficient has a switching behavior in (b) and (c). hauser field and the absorption coefficient of the probelaser shown in Fig. 5, we can see that the line-shapes of χ i for forward and backward scanning are not symmetricto each other.In the second case, we also study a switching behaviorof the probe laser absorption. As indicated by the arrowin Fig. 5(b), we now change ∆ , from right to left step bystep until reaching ∆ , = 2 . , , webegin to record the laser absorption as a function of time.we present h and χ i as a function of time in the left andright panel of Fig. 6(a), respectively, where we find noabrupt switching of χ i . We then repeat the same calcu-lation as in Fig. 6(a) when the point of ∆ , at which westop changing ∆ , is set as 2 . , .
34 or 2 .
35 GHz. Therespective results are shown in Fig. 6(b), (c) and (d). Wecan see that there is an obvious switching behavior of χ i at around t = 310 s (or t = 610 s) after we stop scan-ning the probe laser in Fig. 6(b) [or (c)]. The switchingbehavior here qualitatively agrees with the experimentalobservation in Fig. 2c Ref. 13. Interestingly, our simula-tion reveals that there would be no such behavior when −1 −0.5 0 0.5 1−2−1012 (a) −1 −0.5 0 0.5 1−2−1012 (b) −1 −0.5 0 0.5 1−2−1012 (c) −1 −0.5 0 0.5 1−2−1012 (d)s In v I n / Γ I n , p −0.01 0 0.01 0.02 0.03−0.2−0.100.10.2−0.02 −0.01 0 0.01 0.02−0.200.2 −0.06 −0.04 −0.02 0−0.500.51−0.08 −0.06 −0.04 −0.02 0−0.500.5 FIG. 7. (Color online) v In / Γ In,p as a function of s In . ∆ , = − . , , .
675 and 2 GHz in (a), (b), (c) and (d), respectively.Other parameters are the same as in Fig. 3. In each subfig-ure, the right panel zooms in the left panel, showing the cross-ings [which gives the position of a local maximum of p (ss)s ( s In )]between the curve v In / Γ In,p and the horizontal dashed line.One can see that there are two such crossings in each panel,denoted by the circles. we stop shifting the probe laser frequency too far fromthe rising edge, say, at ∆ , = 2 .
35 GHz, as shown inFig. 6(d).The hysteresis in Figs. 4 and 5 and the switching effectsin Fig. 6 result from the nonlinear feedback between theelectron-hole system and the nuclear spin bath, where thefeedback is controlled by the hole spin-nuclear spin non-collinear hyperfine interaction. In Appendix A, we showthat the spin flip-flops due to the hyperfine interactionmay be neglected for strong magnetic field. However,the feedback is much less effective in polarizing the nu-clear spins if the external magnetic field is lower than,for instance, 0 . s In(As) , in Eq. (B9) that is responsible for the nonlinear feedbackwill vanish as ∼ B , thus causing the hysteresis andswitching effects through the noncollinear interaction inour model to vanish. In such a case, the dynamics of thenuclear spins will be controlled primarily by the hyperfineflip-flops. It has also been noted that similar hystere-sis of nuclear spin polarization in an optically pumpedQD could be explained by exploiting the spin flip-flopsthrough the electron spin-nuclear spin hyperfine interac-tion [see, e.g., Refs. 4, 8, 14, and 57]. −101 ∆ = 0(b) −101 ∆ = 0.675 GHz(c) −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1−0.500.511.5 ∆ = 2 GHz(d) s In v I n / Γ I n , p −101 ∆ = −0.3 GHz(a) FIG. 8. (Color online) Same as Fig. 7, except that here Ω =1 .
35 GHz and Ω = 0 .
24 GHz. −0.500.51 ∆ = −0.3 GHz(a) −0.500.5 ∆ = 0(b) −1−0.500.5 ∆ = 0.675 GHz(c) −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1.5−1−0.500.5 ∆ = 2 GHz(d) s In v I n / Γ I n , p FIG. 9. (Color online) Same as Fig. 7, except that here Ω =Ω = 0 .
24 GHz.
V. LARGE NUCLEAR SPIN POLARIZATION
Now, we turn to the cases in which the nuclear spindynamics controlled by Eq. (3) can induce significantnuclear spin polarization, where we cannot apply theFokker-Planck equation from the preceding section sinceits derivation requires | s In(As) | ≪
1. In order to simplifythe discussion, we follow Sec. IV B 1 and assume a system −0.500.51 s ∆ = −0.3 GHz(a) −1−0.500.51 s ∆ = 0(b) −1−0.500.5 s ∆ = 0.675 GHz(c) −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1.5−1−0.500.5 ∆ = 2 GHz(d) v I n / Γ I n , p s In FIG. 10. (Color online) Same as Fig. 7, except that hereΩ = Ω = 1 .
35 GHz. with only indium nuclear spins.
A. When | s In | ≪ is violated? Before introducing a condition where | s In | ≪ | s In | ≪ p s ( s In ). Forthis case, it is pointed out that a peak of p (ss)s ( s In ) canusually be approximated by a Gaussian. The position s In = s In ,m of such a Gaussian, i.e., a local maximum of p (ss)s ( s In ) in Eq. (16) is given by v In ( s In ,m ) D In ( s In ,m ) = 0 , ∂∂s ′ In v In ( s ′ In ) D In ( s ′ In ) (cid:12)(cid:12)(cid:12)(cid:12) s In ,m < . (17)From Eq. (15) we have D In ( s In ) ≈ ( I +1)Γ In,p NI , hence wecan study v In / Γ In,p to find the solution of Eq. (17). Weplot v In / Γ In,p for four different nominal detunings ∆ , = − . , , .
675 and 2 GHz in Fig. 7(a)-(d), i.e., the sameas those in Fig. 3(a)-(d). From Fig. 7, one finds thatthere are two solutions (denoted by the small circles) toEq. (17) for each case, hence there should be two peaksof p (ss)s ( s In ). Nevertheless, we note that for the two localmaxima of p (ss)s ( s In ) for ∆ , = 0 .
675 or 2 GHz, one peakis much lower than the other peak in Fig. 3(c) or (d) thatit is not visible in Fig. 3 [mathematically, taking Fig. 3(d)as an example, for the two deformed ‘triangles’ formedby the solid curve and the horizontal dashed line, thearea of the triangle pointing down is larger than that ofthe one which points up]. All local maxima of p (ss)s ( s In )given by Fig. 3 occur with | s In | ≪
1, indicating that onlysmall nuclear spin polarization can be generated. Thisis consistent with the condition for deriving the Fokker-Planck equation.Now we consider the case where | s In | ≪ assume | s In | ≪
1, we canstill derive the same Fokker-Planck equation Eq. (10). Inthis formalism, we can also assume that the position fora local maximum of p (ss)s ( s In ) is given by a solution toEq. (17). Again, we plot v In ( s In ) / Γ In, p in Fig. 8. Onefinds that there is always a solution to Eq. (17) at s In ∼ p (ss)s ( s In ) in any of the fourcases in Fig. 8, the peak ∼ s In is almost 1. On the other hand, if we use Eq. (11)to study the evolution of the mean of nuclear spin polar-ization, strong polarization results. This contradicts thecondition | s In | ≪ = Ω and choose weak pumping, e.g.,Ω = Ω = 0 .
24 GHz, the solution to Eq. (17) at s In ∼− .
2, as seen from Fig. 9(d), is already much farther from0 than the corresponding solution shown in Fig. 7(d).When both lasers are strong, e.g., Ω = Ω = 1 .
35 GHz,Fig. 10 shows that the solutions to Eq. (17) can be evenfurther from zero in Fig. 10(c) and (d), thus indicatingthe failure of the Fokker-Planck equation.In conclusion, the dynamic nuclear spin polarization ina system illustrated in Fig. 1 will obey (violate) | s In(As) | ≪ ≪ Ω (Ω ≫ Ω ) and ∆ , = 0. In anyother case the condition | s In(As) | ≪
B. Numerical study when | s In | ∼ Here, the condition for the derivation of a Fokker-Planck equation breaks down, but we can still solveEq. (3) for the dynamics of the nuclear spin Overhauserfield when | s In | ∼
1. We study an artificial QD withonly 40 nuclear spin J = by the Green’s functionMonte Carlo (GFMC). We assume that there are20 gallium and 20 arsenic nuclear spins. The gyromag-netic ratio g j µ N = 0 . / T and hyperfine constant A j = 10 . Ga, Ga, As (the natural abundances forthese three are 60 . , .
9% and 1, respectively ) forthe gyromagnetic ratios and hyperfine constants, wherethe hyperfine constants for gallium and arsenic nuclearspins are taken from Refs. 4 and 52.A detailed review of the numerical method used herecan be found in Ref. 59. In order to implement GFMCsimulation, we convert Eq. (3) into the form of a first-order differential equation, ddt p n ( t ) = Ap n ( t ) , (18)where p n ( t ) is a vector, each of whose elements repre-senting the population of nuclear spins on one of the d = (2 J + 1) different states. The d × d matrix A determines the dynamics of the nuclear spin population.From Eq. (3), one can prove that each off-diagonal el-ement of A is non-negative. Nonetheless, the diagonalelement A kk is given by A kk = − * k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j ( J + J − ˆ J x j − ˆ J xj ) ˆ W j, + + X j ( J + J − ˆ J x j + ˆ J xj ) ˆ W j, − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k + , (19)which is non-positive. Here, ˆ W j, ± is ˆ W α j , ± defined inEq. (4), and | k i represents one of the d nuclear spinstates. Most of the eigenvalues of the matrix A are nega-tive (except one being equal to zero, which we tested byexactly diagonizing A for smaller systems). The eigen-vector of A corresponding to the biggest eigenvalue of A gives the steady population of the nuclear spins. InGFMC, we can study properties of the eigenvector cor-responding to the biggest eigenvalue of the matrix A .To do this, we first choose a positive constant and addthis constant to every diagonal element of A , shifting itsspectrum up by making each of its diagonal matrix el-ements positive. This does not alter the result of theGFMC simulation. At the start of the simulation, werandomly set the state of each nuclear spin, which isequivalent to assuming that the temperature for the nu-clear spins is infinitely high. During the simulation, werun M independent sets of simulations simultaneously for i M Monte Carlo sampling. After each step of sampling,we record the distribution p s ( s ) of the nuclear spin polar-ization s ≡ NJ P j h ˆ J xj i calculated from the M differentnuclear spin states. After a certain step i ′ M the nuclearspin ensemble of these M states reaches equilibrium, andthe distribution of s is calculated by analyzing the datafrom the i ′ M step to the i M step.In order to test the GFMC in our model, we can firstconsider the case Ω = 0 . .
35) GHz, in which casethe nuclear spins polarize weakly when N = 9500, asshown in Fig. 7. We assume that the condition | s | ≪ N = 40nuclear spins. In this case, we can use both the Fokker-Planck equation and the Monte Carlo simulation to cal-culate the steady distribution of the nuclear spin polar-ization s . The reason that we may still use the Fokker-Planck equation is that the condition NJ ≪ N = 40. In Fig. 11, we present the steady dis-tribution p ( ss )s ( s ) of s for a system containing 40 spin- for ∆ , = − p ( ss )s ( s ) from both GFMC and Fokker-Planckequation. For the former, we set M = 200 , i M = 10 and i ′ M = 10 in the GFMC study. We also use Eq. (16) tocalculate p ( ss )s ( s ) for a comparison. From Fig. 11, onecan see that the results from GFMC and Fokker-Planck −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 101234567 s p s ( ss ) ( s ) ∆ = 1 GHz(b) ∆ = −1 GHz(a) GFMCFokker−PlanckThermal
FIG. 11. (Color online) Steady distribution of the nuclearspin polarization s for a system with 40 spin- , calculatedfrom the Green’s function Monte Carlo simulation (solid curvewith circles) and Fokker-Planck equation (dashed curve), re-spectively. Here the nuclear spin gyromagnetic ratio is g j µ N =0 . / T, and the hyperfine constant is A j = 10 . , = − ∆ = −1 GHz(a) GFMCThermal−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10123456 s p s ( ss ) ( s ) ∆ = 1 GHz(b) FIG. 12. (Color online) The solid curves with circles and thedash-dot curves mean the same as the counterparts in Fig. 11except that they are calculated with Ω = 1 .
35 GHz andΩ = 0 .
24 GHz. equation agree very well with each other. Another fea-ture in Fig. 11 is that the distributions of s for ∆ , = − s = 0.We then consider the case Ω = 1 .
35 (0 .
24) GHz.From Fig. 8 [the curve v In / Γ In,p for the present case isdifferent from that in Fig. 8, yet also indicates that s ≪ M = 200simulations simultaneously. However, we found that thestate represented by these samples does not reach equi-librium [i.e., p s ( s ) is going up or down for a given s ifwe continue the sampling] when we run only i ′ M = 10 simulations, as above. So we set i M = 10 here. Now s isanalyzed by the data generated from the i ′ M = 91 × thto the i M th step. In this case, we found that the MonteCarlo simulation reaches equilibrium (we do not meanthat at least 91 × sampling is needed to reach equilib-rium). The distribution of s from this Monte Carlo simu-lation is shown in Fig. 12. One can see that the distribu-tion of the nuclear spin polarization is peaked at s ≈ ± . , = ± s for ∆ , = − s = 0, similar to that in Fig. 11. Because the peak ofthe nuclear spin polarization distribution is shifted from0 to ∼ ± .
5, we conclude that strong nuclear spin po-larization can be generated by optical pumping on singleQDs.
VI. CONCLUSIONS
We have studied dynamic nuclear spin polarization ina strained self-assembled QD pumped by two narrow-linewith continuous wave lasers, where the noncollinearhole spin-nuclear spin hyperfine interaction is assumedto be responsible for nuclear spin polarization. We foundthat the nuclear spins can be polarized to a degree thatis either small or large, depending on the intensities andcentral frequencies of the lasers.We first study the experiments in Ref. 13, where onlyweak nuclear spin polarizations were observed. In thiscase, we derive a Fokker-Planck equation for the timeevolution of the probability of nuclear spin polarization,based on which we show that the distribution of the nu-clear spin Overhauser field can be narrowed, as well asother phenomena, including that the absorption curve ofthe probe laser shifts when we change the probe laserfrequency. When large nuclear spin polarization can begenerated by optical pumping, the condition to derive aFokker-Planck equation breaks down. Then we use theGFMC simulation to directly solve the steady distribu-tion of the nuclear spin Overhauser field. Indeed, we findthat a large nuclear spin polarization up to 50% can begenerated for a small ‘QD’ containing 40 spin- . Note0that a 50% nuclear spin polarization is close to the largenuclear spin polarization observed in experiments by op-tical pumping or other controls in QDs.Throughout this paper we have assumed that the dy-namic nuclear spin polarization results from the holespin-nuclear spin noncollinear hyperfine interaction in aself-assembled QD under optical pumping. When thequadrupole interaction between the electron and nu-clear spins is strong compared to the Fermi-contact hy-perfine interaction in the QD, there will be an effectivenoncollinear interaction between the electron spinand nuclear spins which can play a similar role in po-larizing the nuclear spins as studied in this paper. Wenote that our method is applicable when nuclear spinpolarization is mainly controlled by the effective electronspin-nuclear spin noncollinear interaction. ACKNOWLEDGMENTS
This work was supported by the US ARO-MURI GrantNo. W911NF0910406 and NSF Grant No. PHY1104446. The author thanks L. J. Sham and Xiaodong Xu for help-ful discussions, and Tiamhock Tay for introducing him tothe Green’s Function Monte Carlo method.
Appendix A: Steady state of the electron-holesystem
This appendix gives the steady-state solution to thereduced density matrix of the electron-hole system in anadiabatic nuclear spin Overhauser field h . This steadystate is determined by ˆ H eh and the longitudinal part ofthe hyperfine interaction ˆ H HI in Eq. (1).First of all, it is helpful to use a rotating frame to showthat the optically pumped electron-hole system in Fig. 1is an effective three-level system. The Hamiltonian forthe electron-hole system includes three parts, namely theZeeman energy ˆ H eh-b in an external magnetic field, thecontrol ˆ H eh-l by the electromagnetic fields and the chan-nel of photon exchange ˆ H eh-q with the electromagneticvacuum, ˆ H eh = ˆ H eh-b + ˆ H eh-l + ˆ H eh-q . (A1)The Zeeman term isˆ H eh-b = E x | x + ih x + | + ( E x + g e µ B B ) | x −ih x − | + E T | T + ih T + | + ( E T + g h µ B B ) | T −ih T − | , (A2)where E x and E T are the eigenenergies of the eigenstates | x + i and | T + i . For the system in Fig. 1, the semiclassicalHamiltonian of the control by the electromagnetic fields on the electron-hole system in dipole approximation isˆ H eh-l = (cid:18) Ω e iω t + e − iω t ) | x −i + Ω e iω t + e − iω t ) | x + i (cid:19) h T − | + (cid:18) Ω e iω t + e − iω t ) | x + i + Ω e iω t + e − iω t ) | x −i (cid:19) h T + | + H.c , (A3)where ω is the central frequency of the two coherent laser beams. Here, we assume that the matrix element of theelectric dipole moment e h x + | x ( y ) | T ±i is equal to e h x − | x ( y ) | T ∓i ( e is the elementary charge), hence the Rabifrequency for the transition | x ∓i → | T + i is equal to that for | x ±i → | T −i . Because the two laser beams are almostresonant with the two transitions | x ±i → | T −i , we use a rotating frame with R = ( E T + g h µ B B − ω ) | x −ih x − | + ( E T + g h µ B B − ω ) | x + ih x + | + ( E T + g h µ B B ) | T −ih T − | + E T | T + ih T + | , (A4)to eliminate the obvious time dependence in ˆ H eh for the transition | x ±i → | T −i (we use the subscript r to denote therotating frame),ˆ H eh,r = e iRt ˆ H eh e − iRt − R ≈ − ∆ , | x −ih x − | − ∆ , | x + ih x + | + (cid:18) Ω | x −ih T − | + Ω | x + ih T − | + H.c (cid:19) + e iRt ˆ H eh-q e − iRt , (A5)∆ , = E T + g h µ B B − ( E x + g e µ B B + ω ) , ∆ , = E T + g h µ B B − ( E x + ω ) , where we have ignored the rapidly oscillating terms withphase term exp[ ± i ( ω + ω ) t ] or exp[ ± i ( g e ± g h ) µ B Bt ]since ω ∼ × GHz, and | ( g e ± g h ) µ B B | >
13 GHz [with B = − .
64 T and g e(h) = 0 .
49 ( − . | ∆ | used in the main text and otherparameters in the Hamiltonian. From Eq. (A5), we can1see that the effective electron-hole system involves | x ±i and | T −i , hence can be called a Λ system (ΛS).When we apply the rotating frame transformationto the transverse part of ˆ H HI , we have a e ,j ˆ S ± e ˆ I ∓ j → a e ,j ˆ S ± e ˆ I ∓ j e ± it (∆ , − ∆ , − g e µ B B ) . Here g e µ B B ≈
18 GHz,hence e ± it (∆ , − ∆ , − g e µ B B ) is rapidly oscillating when∆ , and ∆ , are set as in the main text. So thetransverse part of the electron spin-nuclear spin hy-perfine interaction can be neglected. We note thata Schrieffer-Wolff transformation on the trans-verse part of ˆ H HI may give rise to a nonlinear termˆ S x e P j,j ′ = j a e ,j a e ,j ′ g e µ B B ˆ I + j ˆ I − j ′ , which also contributes to nu-clear spin dynamics. However, one can evaluate andshow that the pumping of nuclear spins through this non-linear term is much slower than that through the non-collinear term ∼ ˆ S x h ˆ I yj for the specific system studied inthis paper. For such case, we shall assume that the trans-verse part of electron spin-nuclear spin hyperfine interac-tion can be neglected. Finally, since a h ,j /a e ,j ∼ − . | T + i and | T −i given by a mean field treatment of | β | a h ,j √ ˆ S y h ˆ I xj sin δ is weak compared to the Rabi frequen-cies and decay rate of the trion in this paper, we furtherneglect the transverse part of the hole spin-nuclear spinhyperfine interaction. Equation (2) then becomesˆ H HI ≈ X j (cid:20) a e ,j ˆ S x e ˆ I xj + 2 | β | a h ,j √ S x h ( ˆ I xj cos δ + ˆ I yj sin δ ) (cid:21) , (A6)where we have used 1 + | β | ≈ | β | = 0 . h = X j a e ,j h ˆ I xj i , (A7) h h ≈ X j | β | a h ,j √ h ˆ I xj i cos δ. (A8)Since | β | ∼ . a h ,j /a e ,j ∼ − . h h but keep the nuclear spin Overhauser field h via the electron spin-nuclear spin hyperfine interaction.As a result, the actual detuning between the photon en- ergy of the laser and the energy cost in relevant transi-tions is ∆ = ∆ , + h , ∆ = ∆ , − h . (A9)Next, we solve the steady-state reduced density matrixof the ΛS in an adiabatic nuclear spin Overhauser field h .The last term in Eq. (A5) induces spontaneous decay ofthe trion level, which we incorporate in the optical Blochequation in the Lindblad form, d ˆ ρ Λ dt = − i [ ˆ H ΛS , r , ˆ ρ Λ ] + Γ X k =1 (cid:20) ˆ G k ˆ ρ Λ ˆ G † k − n ˆ G † k ˆ G k , ˆ ρ Λ o(cid:21) + γ s X k =3 (cid:20) ˆ G k ˆ ρ Λ ˆ G † k − n ˆ G † k ˆ G k , ˆ ρ Λ o(cid:21) , (A10)where { a, b } = ab + ba , andˆ H ΛS , r = − ∆ | x −ih x − | − ∆ | x + ih x + | + (cid:18) Ω | x −ih T − | + Ω | x + ih T − | + H.c (cid:19) , ˆ G , = | x ±ih T − | , ˆ G , = | x ±ih x ± | , (A11)and the Λ-system reduced density matrix ˆ ρ Λ in the basisconstructed from the vectors | T −i , | x + i and | x −i isˆ ρ Λ = ρ T ,T ρ T ,x ρ T ,x ρ x,T ρ x,x ρ x,x ρ x,T ρ x,x ρ x,x . (A12)The condition Tr ˆ ρ Λ = 1 allows us to eliminate one matrixelement, say, ρ x,x | x −ih x − | in ˆ ρ Λ and rearrange its othereight matrix elements into one column, ρ Λ = ( ρ T ,T , ρ x,x , ρ x,T , ρ x,T , ρ x,x , ρ
T ,x , ρ
T ,x , ρ x,x ) T , where T denotes transposing a matrix. Then Eq. (A10)becomes, i ddt ρ Λ ( t ) = M ρ Λ ( t ) + X. (A13)Here, X = (0 , , , − Ω , , , Ω , , (A14)and M = − i Γ Ω − Ω − Ω i Γ − Ω Ω Ω − Ω − ∆ − i Γ − Ω − ∆ − i Γ − Ω − Ω − ∆ − iγ s Ω − Ω − i Γ Ω − Ω − Ω Ω − i Γ
00 0 0 − Ω Ω − ∆ − iγ s , (A15)2where Γ ≡
2Γ is the relaxation rate of the trion state,Γ = Γ + γ s is the decay rate of the coherence betweenthe trion and one of the electron spin eigenstates, Γ isthe spontaneous decay rate from the trion to each of thetwo electron spin states, and γ s is the energy-conservingdephasing rate of the electron spin in the presence of fluc-tuating nuclear spins. Note that we have ignored the elec-tron spin relaxation process which is much slower thanthe electron spin decoherence. In the steady state of the ΛS, we obtain M ρ Λ ( t → + ∞ ) + X = 0 , (A16) whose solution, together with the condition Tr ˆ ρ Λ = 1,gives us the reduced density matrix ˆ ρ ( ss )Λ for the steadystate of the ΛS. Appendix B: Fokker-Planck equation
In this appendix, we derive the Fokker-Planck equationstarting from Eq. (3). For the derivation of an analogousequation involving only one species of nuclear spins, seeRef. 31. We first multiply the Kronecker delta function δ ˆ s In ,s In δ ˆ s As ,s As on both sides of Eq. (3), and then traceover the nuclear spin degrees of freedom in it, giving ddt Tr[ ˆ P ( t ) δ ˆ s In ,s In δ ˆ s As ,s As ] ≈ − X j Tr nh ˆ I − j , ˆ I + j ˆ W α j , + ˆ P ( t ) i δ ˆ s In ,s In δ ˆ s As ,s As + h ˆ I + j , ˆ I − j ˆ W α j , − ˆ P ( t ) i δ ˆ s In ,s In δ ˆ s As ,s As o , (B1)where Tr denotes tracing over the nuclear spin degrees of freedom. Using Eq. (9), the left hand side of Eq. (B1)becomes ddt p s ( s In , s As ) [we suppress the variable t in p s for brevity]. Taking the sum in the first commutator on theright hand side of Eq. (B1) as an example, the trace evaluates to − X j Tr nh ˆ I − j , ˆ I + j ˆ W α j , + ˆ P ( t ) i δ ˆ s In ,s In δ ˆ s As ,s As o = − X j X k h k | nh ˆ I − j , ˆ I + j ˆ W α j , + ˆ P ( t ) i δ ˆ s In ,s In δ ˆ s As ,s As o | k i , (B2)where k labels a specific nuclear spin state and runs over the whole Hilbert space of the nuclear spin states.First, we evaluate the sum for ˆ I − j ˆ I + j ˆ W α j , + ˆ P ( t ) on the right hand side of Eq. (B2): − X j X k h k | nh ˆ I − j ˆ I + j ˆ W α j , + ˆ P ( t ) i δ ˆ s In ,s In δ ˆ s As ,s As o | k i = − X j X k h k | h I j ( I j + 1) − ˆ I x j − ˆ I xj i ˆ W α j , + ˆ P ( t ) δ ˆ s In ,s In × δ ˆ s As ,s As | k i . (B3)Below, we label I In = I = and I As = J = unless otherwise specified. Now assume that the following approximationis valid when both polarizations of the indium and arsenic nuclear spins are small, * X j ∈ In(As) ( ˆ I xj ) + ≈ N I
In(As) ( I In(As) + 1)3 , (B4)where h . . . i denotes the respective expectation value, then Eq. (B3) becomes, − X j X k h k | nh ˆ I − j ˆ I + j ˆ W α j , + ˆ P ( t ) i δ ˆ s In ,s In δ ˆ s As ,s As o | k i ≈ − X k X α =In,As h k | (cid:20) N I α ( I α + 1)3 − N I α ˆ s α (cid:21) ˆ W α, + ˆ P ( t ) × δ ˆ s In ,s In δ ˆ s As ,s As | k i = − X α =In,As (cid:20) N I α ( I α + 1)3 − N I α ˆ s α (cid:21) W α, + ( s In , s As ) p s ( s In , s As ) , (B5)where ˆ s α , with α = In (As), is defined in Eq. (6).3Next, we evaluate the remaining sum on the right hand side of Eq. (B2): X j X k h k | nh ˆ I + j ˆ W α j , + ˆ P ( t ) ˆ I − j i δ ˆ s In ,s In δ ˆ s As ,s As o | k i = X j ∈ In X k h k | h I ( I + 1) − ˆ I x j − ˆ I xj i ˆ W α j , + ˆ P ( t ) δ ˆ s In ,s In − a δ ˆ s As ,s As | k i + (similar term with sum over As) ≈ N I (cid:20) I + 1)3 − s In + a (cid:21) W In , + ( s In − a, s As ) p s ( s In − a, s As )+ N J (cid:20) J + 1)3 − s As + b (cid:21) W As , + ( s In , s As − b ) p s ( s In , s As − b ) , (B6)where a = NI , and b = NJ .Similar to Eqs. (B5) and (B6), the sum over the second commutator on the right hand side of Eq. (B1) gives, − X j Tr nh ˆ I + j , ˆ I − j ˆ W α j , − ˆ P ( t ) i δ ˆ s In ,s In δ ˆ s As ,s As o = − X α =In,As N I α (cid:20) I α + 1)3 + ˆ s α (cid:21) W α, − ( s In , s As ) p s ( s In , s As )+ N I (cid:20) I + 1)3 + s In + a (cid:21) W In , − ( s In + a, s As ) p s ( s In + a, s As )+ N J (cid:20) J + 1)3 + s As + b (cid:21) W As , − ( s In , s As + b ) p s ( s In , s As + b ) . (B7)To simplify Eq. (B7), we perform series expansion up to the second order in a and b , and obtain − X j Tr nh ˆ I + j , ˆ I − j ˆ W α j , − ˆ P ( t ) i δ ˆ s In ,s In δ ˆ s As ,s As o ≈ N IaW In , − ( s In , s As ) p s ( s In , s As )+ N JbW As , − ( s In , s As ) p s ( s In , s As ) . + N I (cid:20) I + 1)3 + s In + a (cid:21) (cid:20) a ∂W In , − p s ∂s In + a ∂ W In , − p s ∂s In (cid:21) + N J (cid:20) J + 1)3 + s As + b (cid:21) (cid:20) a ∂W As , − p s ∂s As + b ∂ W As , − p s ∂s As (cid:21) . (B8)Defining Γ In(As),p ( s In , s As ) = W In(As) , + ( s In , s As ) + W In(As) , − ( s In , s As ) ,s In(As) , ( s In , s As ) = (cid:2) W In(As) , + ( s In , s As ) − W In(As) , − ( s In , s As ) (cid:3) / Γ In(As),p ( s In , s As ) , (B9)and using Eqs. (B5), (B8) and series expansion from Eq. (B6) [similar to Eq. (B8)], Eq. (B1) becomes, ddt p s ≈ X α =In,As (cid:26) Γ α, p p s + 2( I α + 1)3 (cid:20) N I α ∂ Γ α, p p s ∂s α − ∂s α, Γ α, p p s ∂s α (cid:21) + s α (cid:20) − N I α ∂ s α, Γ α, p p s ∂s α + ∂ Γ α, p p s ∂s α (cid:21)(cid:27) − X α =In,As N I α ∂s α, Γ α, p p s ∂s α = X α =In,As ∂∂s α (cid:20) ∂∂s α D α p s ( s In , s As ) − v α p s ( s In , s As ) (cid:21) , (B10)where we have dropped terms ∼ /N . The drift and diffusion coefficients in Eq. (B10) are given by v In(As) = − Γ In(As),p (cid:20) s In(As) − I In(As) + 1)3 s In(As) , (cid:21) ,D In(As) = Γ
In(As),p N I
In(As) (cid:20) I In(As) + 1)3 − s In(As) s In(As) , (cid:21) . (B11)4In the definition of v In(As) and D In(As) in Eq. (B11), the factor I In(As) +1)3 comes from the sum of individual nuclearspin fluctuations in Eq. (B4), and Γ
In(As),p is the rate of nuclear spin flip due to the noncollinear interaction in ourmodel. If Γ
In(As),p is large, the distribution of the nuclear spin polarization evolves rapidly. The nonlinear function s In(As) , plays a key role in our feedback loop since the solution to s In(As) − I In(As) +1)3 s In(As) , = 0 gives us the stablenuclear spin polarization (see Sec. V A). Because | s In(As) s In(As) , | ≪
1, the diffusion coefficient can be approximated as D In(As) ≈ Γ In(As),p N I
In(As) I In(As) + 1)3 = σ Γ In(As),p , (B12)where σ In(As),th ≡ q I In(As) +13 NI In(As) is the standard deviation of nuclear spin polarization distribution for N indium (arsenic)nuclear spins at infinite temperature. The appearance of σ In(As),th in the definition of D In(As) is not surprisingsince D In(As) is associated with the diffusion process. Defining the means of the nuclear spin polarization s In and s As by s In = Z Z s In p s ds In ds As ,s As = Z Z s As p s ds In ds As , (B13)and further assuming that the probability for the nuclear spins to be totally polarized is negligible, i.e., p s ( s In = ± , s As ) = p s ( s In , s As = ±
1) = 0, we obtain ddt s In = Z Z v In p s ( s In , s As ) ds In ds As ,ddt s As = Z Z v As p s ( s As , s As ) ds In ds As . (B14) Appendix C: Inhomogeneous broadening
The nuclear spin Overhauser field obeys a Gaus-sian distribution when the nuclear spins are in thermalequilibrium, p h ( h ) ≈ √ π Γ ∗ e − ( h − h ) / (2Γ ∗ ) , (C1) h = A In Is In + A As Js As , Γ ∗ ≈ A I ( I + 1) + A J ( J + 1)3 N , when | s In(As) | ≪ ∼ Here h and s α are the averages of h and s α , respectively. The electron spin decoherence L ( t )is dominated by the inhomogeneous broadening of the nuclear spin Overhauser field, L ( t ) = Z p h ( h ) e − i ( g e µ B B + h ) t dh. (C2)For a thermal nuclear spin bath of Eq. (C1), we have apure Gaussian decay, L ( t ) = e − i ( g e µ B B + h ) t − ( t/T ∗ ) , (C3)where T ∗ = √ / Γ ∗ . Both indium and arsenic nuclearspins contribute to Γ ∗ in a QD with N indium and N arsenic nuclear spins. Under the condition of | s In(As) | ≪ N indiumnuclear spins gives a Γ ∗ that is the same as in the originalInAs system. This is achieved by writing A ′ I ( I + 1) = A I ( I + 1) + A J ( J + 1) , (C4)which gives a 5% increase in the hyperfine constant forindium nuclear spins. However, this does not mean thatthe new system with only N indium nuclear spins retainsall the properties of the initial system. ∗ E-mail: [email protected] Optical Orientation , edited by F. Meier and B. P. Za-karchenya (Elsevier, New York, 1984). D. Gammon, A. L. Efros, T. A. Kennedy, M. Rosen, D. S.Katzer, D. Park, S. W. Brown, V. L. Korenev, and I. A.Merkulov, Phys. Rev. Lett. , 5176 (2001). A. S. Bracker, E. A. Stinaff, D. Gammon, M. E. Ware,J. G. Tischler, A. Shabaev, A. L. Efros, D. Park, D. Ger-shoni, V. L. Korenev, and I. A. Merkulov, Phys. Rev. Lett. , 047402 (2005). P.-F. Braun, B. Urbaszek, T. Amand, X. Marie, O. Krebs,B. Eble, A. Lemaitre, and P. Voisin, Phys. Rev. B ,245306 (2006). C. W. Lai, P. Maletinsky, A. Badolato, and A. Imamoglu,Phys. Rev. Lett. , 167403 (2006). V. L. Korenev, Phys. Rev. Lett. , 256405 (2007). A. Greilich, A. Shabaev, D. R. Yakovlev, A. L. Efros, I. A.Yugova, D. Reuter, A. D. Wieck, and M. Bayer, Science(New York, N.Y.) , 1896 (2007). A. I. Tartakovskii, T. Wright, A. Russell, V. I. Fal’ko,A. B. Van’kov, J. Skiba-Szymanska, I. Drouzas,R. S. Kolodka, M. S. Skolnick, P. W. Fry, A. Tahraoui,H.-Y. Liu, and M. Hopkinson, Phys. Rev. Lett. , 026806(2007). P. Maletinsky, A. Badolato, and A. Imamoglu, Phys. Rev.Lett. , 056804 (2007). B. Eble, O. Krebs, A. Lemaˆıtre, K. Kowalik, A. Kudelski,P. Voisin, B. Urbaszek, X. Marie, and T. Amand, Phys.Rev. B , 081306(R) (2006). J. Skiba-Szymanska, E. A. Chekhovich, A. V. Nikolaenko,A. I. Tartakovskii, M. N. Makhonin, I. Drouzas, M. S. Skol-nick, and A. B. Krysa, Phys. Rev. B , 165338 (2008). J. Danon and Y. V. Nazarov, Phys. Rev. Lett. , 056603(2008). X. Xu, W. Yao, B. Sun, D. G. Steel, A. S. Bracker, D. Gam-mon, and L. J. Sham, Nature (London) , 1105 (2009). C. Latta, A. H¨ogele, Y. Zhao, A. N. Vamivakas,P. Maletinsky, M. Kroner, J. Dreiser, I. Carusotto,A. Badolato, D. Schuh, W. Wegscheider, M. Atature, andA. Imamoglu, Nature Phys. , 758 (2009). A. E. Nikolaenko, E. A. Chekhovich, M. N. Makhonin,I. W. Drouzas, A. B. Van’kov, J. Skiba-Szymanska,M. S. Skolnick, P. Senellart, D. Martrou, A. Lemaˆıtre, andA. I. Tartakovskii, Phys. Rev. B , 081303(R) (2009). C.-W. Huang and X. Hu, Phys. Rev. B , 205304 (2010). M. N. Makhonin, E. A. Chekhovich, P. Senellart,A. Lemaˆıtre, M. S. Skolnick, and A. I. Tartakovskii, Phys.Rev. B , 161309(R) (2010). M. Issler, E. M. Kessler, G. Giedke, S. Yelin, I. Cirac,M. D. Lukin, and A. Imamoglu, Phys. Rev. Lett. ,267202 (2010). E. A. Chekhovich, M. N. Makhonin, J. Skiba-Szymanska,A. B. Krysa, V. D. Kulakovskii, M. S. Skolnick, and A. I.Tartakovskii, Phys. Rev. B , 245308 (2010). E. A. Chekhovich, M. N. Makhonin, K. V. Kavokin, A. B.Krysa, M. S. Skolnick, and A. I. Tartakovskii, Phys. Rev.Lett. , 066804 (2010). E. A. Chekhovich, A. B. Krysa, M. S. Skolnick, andA. I. Tartakovskii, Phys. Rev. Lett. , 027402 (2011). E. Barnes and S. E. Economou, Phys. Rev. Lett. ,047601 (2011). V. L. Korenev, Phys. Rev. B. , 235429 (2011). M. N. Makhonin, K. V. Kavokin, P. Senellart, A. Lemaˆıtre,A. J. Ramsay, M. S. Skolnick, and A. I. Tartakovskii, Na-ture Mater. , 844 (2011). B. Sun, C. M. E. Chow, D. G. Steel, A. S. Bracker, D. Gam-mon, and L. J. Sham, Phys. Rev. Lett. , 187401 (2012). A. H¨ogele, M. Kroner, C. Latta, M. Claassen, I. Caru-sotto, C. Bulutay, and A. Imamoglu, Phys. Rev. Lett. ,197403 (2012). C. Le Gall, A. Brunetti, H. Boukari, and L. Besombes,Phys. Rev. B , 195312 (2012). M. M. Glazov, I. A. Yugova, and A. L. Efros, Phys. Rev.B , 041303(R) (2012). R. Kaji, S. Adachi, H. Sasakura, and S. Muto, Phys. Rev.B , 155315 (2012). B. Urbaszek, X. Marie, T. Amand, O. Krebs, P. Voisin,P. Maletinsky, A. H¨ogele, and A. Imamoglu, Rev. Mod.Phys. , 79 (2013). W. Yang and L. J. Sham, Phys. Rev. B , 235319 (2012). W. Yang and L. J. Sham, (to be published) H. J. Carmichael,
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