Nuclear spin cooling by electric dipole spin resonance and coherent population trapping

Author’s Accepted Manuscript Nuclear spin cooling by electric dipole spin resonance and coherent population trapping Ai-Xian Li, Su-Qing Duan, Wei Zhang

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S1386-9477(17)30196-0 http://dx.doi.org/10.1016/j.physe.2017.06.001 PHYSE12822

To appear in: Physica E: Low-dimensional Systems and Nanostructures Received date: 7 February 2017 Revised date: 1 June 2017 Accepted date: 2 June 2017 Cite this article as: Ai-Xian Li, Su-Qing Duan and Wei Zhang, Nuclear spin cooling by electric dipole spin resonance and coherent population trapping, Physica E: Low-dimensional Systems and Nanostructures, http://dx.doi.org/10.1016/j.physe.2017.06.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Nuclear spin cooling by electric dipole spin resonance and coherent population trapping Ai-Xian Li, Su-Qing Duan, and Wei Zhang∗ Institute of Applied Physics and Computational Mathematics, P. O. Box 8009(28), Beijing 100088, China Nuclear spin fluctuation suppression is a key issue in preserving electron coherence for quantum information/computation. We propose an efficient way of nuclear spin cooling in semiconductor quantum dots (QDs) by the coherent population trapping (CPT) and the electric dipole spin resonance (EDSR) induced by optical fields and ac electric fields. The EDSR can enhance the spin flip-flop rate and may bring out bistability under certain conditions. By tuning the optical fields, we can avoid the EDSR induced bistability and obtain highly polarized nuclear spin state, which results in long electron coherence time. With the help of CPT and EDSR, an enhancement of 1500 times of the electron coherence time can been obtained after a 500 ns preparation time. PACS numbers: 85.75.-d, 76.70.Fz, 78.67.Hc

I.

INTRODUCTION

Single electron spin in artificial structures, such as semiconductor quantum dots (QDs), has become available and to a large extent controlable.1–7 Spin degrees of freedom are widely considered to be promising candidates for storing information. The interaction between localized electron spins and the environment is of particular interest both from a fundamental physics point of view and their potential use as quantum bits (qubits) for quantum information processing. Minimizing decoherence due to coupling of a quantum system to its fluctuating environment is a critical challenge in the quantum information and quantum computation.7–9 A number of experimental and theoretical studies reveal that the predominant decoherence mechanism in QD is the hyperfine coupling between the electron spin and the nuclear spins in the host material1–6,10–12 . The electron-nuclear spin dynamics has been studied extensively.4–6,8–16 To suppress the decoherence effect due to the fluctuation of nuclear spins, one may either generate a state of highly polarized nuclear spins or squeeze the distribution of nuclear spin polarization.8–10 One basic approach to suppress the nuclear spin fluctuation is the feedback control by coupling the nuclear spins to a “background” of electron spin in specific configuration. One effective way of preparing such electron “background” for cooling the nuclear spins is to use coherent population trapping (CPT), which is at the heart of a number of key advances in quantum optics.16–20 Two circularly polarized laser fields/lights tuned to satisfy two-photon resonance could be used to deterministically

∗ Author

to whom any correspondence should be addressed, Email: zhang [email protected]

prepare a nuclear-spin environment with ultranarrow Overhauser-field (OF) distribution by using CPT.21–24 In turn, polarized nuclear spins, which can be described as an effective magnetic (Overhauser) field, induce an energy shift of the electron spin states. Under a large (effective) magnetic field, the direct electron-nuclear flipflop processes are strongly suppressed due to the large mismatch in the electronic and nuclear Zeeman splitting, which reduces the “cooling” efficiency. In this paper we propose to use two optical fields to realize CPT with highly polarized electron spin state, which leads to highly polarized nuclear spin. In addition, we add an ac electric field to realize electric dipole spin resonance (EDSR),25–29 which may increase the cooling efficiency by compensating the energy difference of electron states with different spins and enhancing electronnuclear spin flip-flop rate. Compared with the electronspin-resonance with magnetic field, the EDSR promises higher efficiency and strong local electric fields are easier to obtain than strong local magnetic fields. The EDSR enhanced feedback control of the nuclear spin bath (dynamical nuclear spin polarization) leads to interesting physics, such as bistable electronic states and the existence of an optimal ac electric field for nuclear spin fluctuation suppression. By the combination effects of the optical fields and the ac electric fields, we are able to greatly prolong the electron spin decoherence time of 1500 times.

II.

THEORETICAL MODEL AND APPROACH

We explore the feedback control of the nuclear spin bath in a semiconductor QD, which includes the nuclear spin states, the Zeeman-split electron ground states and the negatively charged exciton (trion) |ti = | ↑↓⇑i (| ↑i, | ↓i, | ⇑i the electron spin-up,-down states and

2

FIG. 1: (Color online) The energy level diagram of the system with a magnetic field B. Two laser fields with frequencies ωp and ωc are applied. The detunings of the two fields are ∆1 and ∆2 , respectively. The Rabi frequency of the transition between the electronic spin states due to the ac electric field (with frequency ω) is G.

heavy hole state with Jz0 = 3/2, the quantization axis z 0 normal to the quantum dot plane). We manipulate the electron-nuclear dynamics by the optical fields and an ac electric field. The lowest electronic states in a QD formed in GaAs semiconductor are optically active under σ+ circularly polarized excitation. |ti = | ↑↓⇑i is the only optically excited state (transitions to Jz0 = −3/2 are forbidden because of the optical selection rules due to the conservation of angular momentum30 ) (FIG. 1). The axis ~ tot , and the axes z is parallel to the total magnetic field B 0 z and z enclose an angle θ. The electron spin-up and down states are then | ↑i = | ↑iz = cos( θ2 )| ↑iz0 −sin( θ2 )| ↓ iz0 and | ↓i = | ↓iz = cos( θ2 )| ↓iz0 + sin( θ2 )| ↑iz0 . The laser with frequency ωp (ωc ) leads to the transition | ↑i ↔ | ↑↓⇑i (| ↓i ↔ | ↑↓⇑i) with Rabi frequency Ωp (Ωc ) and detuning ∆1 (∆2 ). The Rabi frequencies Ωp and Ωc can be respectively tuned by the intensities of two independent laser fields. Nuclear spins can be coupled to each other by an indirect interaction mediated by the electron in the dot. A large magnetic field of the order of Tesla is applied to suppress the electron induced indirect dipole interaction among nuclear spins.9,30 The large magnetic field ~ tot (including the external magnetic field and the OF) B results in inefficient direct electron-nuclear spin flip-flop process due to the large energy mismatch in the electronic and nuclear Zeeman splitting (µB >> µN ). In other words, for large magnetic field, the coherent coupling between the electron spin states induced by the transverse components of the (quasi-static) nuclear OF is suppressed.23 Yet, the ac electric field is helpful for the energy compensation and may increase the electronnuclear spin flip-flop rate. In the absence of the ac electric

field, the laser fields satisfying the two-photon resonance condition (δ = ∆2 − ∆1 = 0) drive the electron spin into dark state |Di ∝ Ωc | ↑i − Ωp | ↓i with the population of state | ↑↓⇑i being zero.21,22 The Hamiltonian for the system of coupled electron spin-nuclear spin is H = H0 + Hlaser + Hhf + Hel . Here H0 = ~ω2 z σ↓↓ − ~ω2 z σ↑↑ + ~ωt σtt , with the operators σij defined as σij = |iihj|, i, j =↑, ↓, t. The laser fields are described by Hlaser = ~(Ωc σ↓t eiωc t + Ωp σ↑t eiωp t ) + H.c.. The standard expression for the hyperfine(HF) Hamiltonian between the electron spin s and the surrounding P nuclear spin bath Ik is Hhf = A k δ(r − rk )(Ik · s), where the summation is performed over all nuclear spin Ik at position rk (for simplicity, the individual nuclear spin is assumed to be spin- 21 ), A is the average HF coupling constant [A=90 µeV for GaAs31 ]. The last term ˜ · r = d cos ωt describes the potential energy Hel = eE(t) of the electron, with a charge (−e), in an in-plane driving ˜ z. The last two terms can be simplified electric field E⊥ˆ by a series of transformations. After a time-dependent canonical transformation ψ → e−ik·R(t) ψ (i.e., change ˜ 27 ), we the operator as r → r + R(t) with R(t) ∝ −eE(t) perform a standard Schrieffer-Wolff type canonical trans˜ = eSˆ He−Sˆ ) to eliminate Hel by choosing formation (H ˆ ˆ H0 ] = 0. Then, after S from the condition Hel + [S, averaging Hhf [r + R(t)] over the ground state ψ(rk ) of the dot, expending the average in R(t) and keeping two leading terms of the expansion, results in Hel + Hhf = P z H ac + Hhf = k gk (ˆ s+ Iˆk− e−iωt + h.c.) + AIPnuc , where P P d 2 k gk ≡ k A 2 · ∇|ψ(rk )| ≡ ~G (ψ(rk ) is the electron envelope wave function at the k-th nucleus),27,32 ↑ −N ↓ , N ↑(↓) is the number of nuclei with spin Pnuc = N N ↑ +N ↓ 31 up(down). The back action of nuclear spins on the electron spin can be treated in the mean field frame work, since there are about 105 nuclear spins around an electron spin. In the rotating frame with the transformation ˜ Ψ(t) = U (t)Ψ(t), U (t) = e−iωc t σ↓↓ + e−iωp t σ↑↑ + σtt , we neglect off-resonant terms and find   δ − AIPnuc Ge−i(ω+ωc −ωp )t Ωc ~ ˜ = − Ge(iω+ωc −ωp )t −δ + AIPnuc Ωp  , (1) H 2 Ωc Ωp −∆ where ∆ = ∆1 + ∆2 . If we set Pnuc = 0 (the average nuclear spin polarization is zero), ω 0 = ω − ωp + ωc = 0 and δ = δ0 with δ0 =

G (Ω2 − Ω2p ), 2Ωc Ωp c

(2)

then, as can be easily verified, the dark state |Di ∝ Ωc | ↑ i − Ωp | ↓i is an eigenstate of Hamiltonian (1).33 The dynamics of our system may be described by the master

3 0.6

equation

0.8 (b)

(a)

Pnuc P˙nuc = R↓,↑ ρ↑ − R↑,↓ ρ↓ − , τrelax

(4)

where τrelax ≈ 100µs is the phenomenological nuclear spin relaxation time,30 and the spin-flip scattering rates Ri,j depend on the different processes:31 1 1 − Pnuc ][ ], τsf 2 1 1 + Pnuc ][ ]. =[ τsf 2

| ↑i → | ↓i ⇒ R↓,↑ = [

(5)

| ↓i → | ↑i ⇒ R↑,↓

(6)

π~ 2 1 γ G 2 , ]= τsf 2 γ + (−δ + AIPnuc − ω 0 )2

(7)

where the width γ ' 5µeV is the electronic state lifetime broadening, which is of the order of the phonon scattering rate.31 The time evolution of the system is calculated by the Runge-Kutta method. In short, we use the laser fields to generate the electron dark states as the cooling background. Moreover, the ac electric field is applied to compensate the energy mismatch of electron spin state. Thus the spin-flip rate (cooling efficiency) is increased due to EDSR.

III.

0.4

RESULTS AND DISCUSSIONS

In this section, we present our results on the dynamics of electron spin/nuclear spin and the optical fields/ac electric field modulation of nuclear spin fluctuation. In our system, the two laser fields may generate electron dark state of fixed electron spin components, which provides the nuclear spin cooling background. The ac electric field leads to EDSR, which may enhance the cooling efficiency by increasing the effective spin flip-flop rate. At the same time, the nuclear spins induce OF and have impact on the electronic dynamics, which may bring out electronic state bistability under certain conditions.

0.2

P

0.0

0.5

nuc

0.2

nuc

0.6

-0.2

0.0

-0.4

-0.2

-0.6 0

10

20

30

40

G( eV)

50

-0.4

0.0 0

0

10

20

G(

30

eV)

1

40

G( eV)

FIG. 2: (Color online) The nuclear spin polarization versus the Rabi frequency G of the transition between the electronic spin states due to the ac electric fields. (a) Ωp /Ωc = 1.0 and (b) Ωp /Ωc = 1.6 with ~Ωc = 1.0µeV . The line with black squares is the result from numerical simulation. The line with red dots is the approximate analytical result. Insert: the zoomed in view of Pnuc for small ~G. Other parameters are: Γt↓ = Γt↑ = 0.5ns−1 , γt = γ↓ = 0.001ns−1 , ∆ = 0.

A.

The transition rate can be calculated by Fermi Golden rule: [

0.4

P

˜ in equation (1) and the dissipawith the Hamiltonian H P tive term Wρ = α=↑,↓ Γtα (2σαt ρσtα − σtt ρ − ρσtt )/2 + P β=↓,t γβ (2σββ ρσββ − σββ ρ − ρσββ )/2. The rate Γtα describes the radiative decay of state |ti into state α = | ↑i, | ↓i, while γβ is the pure dephasing rate of state β = | ↓i, |ti with respect to | ↑i. The dynamics of the nuclear spins in QD is determined by the flip-flop interaction and the nuclear spin relaxation due to the scattering between nuclei. The time evolution of the nuclear spin polarization is described by the equation

nuc

(3)

P

1 ˜ ρ˙ = Lρ ≡ [H, ρ] + Wρ, i~

Electronic state bistability

We first explore the electronic state bistability. Under the condition of dark state, the electron spin arrives the dark state and the nuclear spin polarization reaches a fixed value in the long time limit. Both the laser fields and the ac electric field can control the electronic spin state and nuclear spin polarization. • Dependence of the electron spin polarization on the ac electric field with fixed Ωp /Ωc = 1.0 and 1.6 In the long time limit, the nuclear spin polarization is completely determined by the cooling background, i.e., the electron spin state population. In FIG. 2, we present the nuclear spin polarization in the long time limit. The electron spin state population shows similar dynamic information. It is clearly seen that the system shows bistability for both the cases of Ωp = Ωc (FIG. 2(a)) and Ωp = 1.6Ωc (FIG. 2(b)). Here we perform the steady state (in the long time limit) analysis for the case of Ωp = Ωc ≡ Ω to gain more physical insight. The coupling between electronic spin degrees of freedom and nuclear spin degrees of freedom leads to nonlinear master equations, which in general may have up to three solutions in certain parameter range. For simplicity, we neglect the higher-order of decay and dephasing effects in this analysis. In static limit, the three solutions for the nonlinear equations are ρ↓ = 1/2, 1/2 ± q ~2 (Ω2 −G2 ) A2

2

γG + ~G/A( 2ΓΩ 2 + 2); ρ↑ = 1−ρ↓ ; ρt = 0; (i.e., q 2 2 2 ) γG2 Pnuc = 0, ±2 ~ (ΩA−G + ~G/A( 2ΓΩ Two of 2 2 + 2)). the solutions are polarized in opposite directions and are

50

4 1.0

2.0

0.8 1.8

0.6

c

1.6

Ω /Ω

0.2

p

Pnuc

0.4

0.0

1.4

Bistable Region

-0.2

1.2

-0.4 1.0

2.0

/ p

2.5

3.0

0

5

10

stable. The other one is unpolarized and unstable. Several important features on the bistablity can been seen from the analysis and FIG. 2. Firstly, the bistability is mainly due to the ac electric field induced effective interaction between electron spin and nuclear spin. Secondly, a non-monotonic dependence of nuclear spin polarization on ~G is found and there is an optimal ~G for the maximum nuclear spin polarization in the bistable regime. Thirdly, the bistability exists in the range 0 < ~G < ~Gc and it disappears for large value of ~G. Gc increases with the increase of Ω. FIG. 2(b) shows the dependence of nuclear spin polarization on ~G for the case of Ωp /Ωc = 1.6, where bistability also appears. Unlike the case of Ωp /Ωc = 1.0, a smaller bistability region was observed and some amount of minimum ~G is needed for the appearance of bistability. More discussions of the dependence of Pnuc on Ωp /Ωc are given below. • Dependence of the electron spin polarization on the laser fields with fixed ~G = 7µeV We consider the case with fixed ~G = 7µeV as an example to explore the laser fields dependence. As shown in FIG. 3, the systems show bistable dynamics for certain parameter regime with small Ωp /Ωc . The bistability disappears when Ωp /Ωc exceeds a critical value. Overall, the bistability of nuclear spin polarization/electron spin state not only depends on the ac electric field, but also depends on the laser fields (optical fields). The full phase diagram is presented in FIG. 4. In general, the bistability disappears for large ~G and/or Ωp /Ωc , which corresponds to large electron-spin-nuclearspin interaction (ESNSI) and/or highly polarized electron state, i.e., very “cold” electron background. In the nonlinear process of feedback control of nuclear spins,

20

25

30

µ

c

FIG. 3: The nuclear spin polarization versus the ratio of laser Rabi frequencies Ωp /Ωc . ~G = 7µeV . Other parameters are the same as those in FIG. 2.

15

G( eV)

FIG. 4: The phase diagram of bistable region with parameters Ωp /Ωc and ~G. Other parameters are the same as those in FIG. 2.

150

probability density

1.5

probability density

1.0

100

100

50

0 0.9531

P

0.9532

nuc

50

0 0.00

0.02

0.950

Pnuc

0.952

0.954

FIG. 5: (Color online) The probability distribution of the nuclear spin polarization for Ωp /Ωc = 5.0 and ~G = 7µeV . The black thick line represents the initial Gaussian distribution. The line with red dots is the probability distribution after a preparation time of 400ns based on the numerical simulation. The line with blue triangle is the Gaussian distribution with width σ = σ0 /500. Insert: the zoomed in view of the distribution around Pnuc =0.953. Other parameters are the same as those in FIG. 2.

the electronic state bistability may appear, which reduces the nuclear spin cooling efficiency. Therefore, one should choose appropriate parameters to avoid the bistable region and effectively control the nuclear spins.

5 The suppression of nuclear spin fluctuation

Under experimentally achievable temperature, the nuclear spin orientation in a thermal distribution is highly random because of its small energy scale. Thus, the configurations of the nuclear spin bath has large statistical fluctuations, which give rise to a large inhomogeneous broadening of statistical distribution of the nuclear spins. In a large external magnetic field, the statistical fluctuations of the nuclear spins give rise to pure dephasing of the electron spin. We assume a Gaussian distribution of the nuclear spins f (Pnuc ) = √ −1 ( 2πσ) exp[−(Pnuc )2 /2σ 2 ], with the width σ = σ0 ∼ √ 1/I N (plotted as the black thick line in FIG. 5). The electron spin coherence time is quantified using the width ∗ of nuclear spin polarization distribution (T2,0 ∼ 1/σ).30 To have longer spin coherence time, we would like to achieve high value of mean nuclear spin polarization and using CPT to reduce intrinsic nuclear spin fluctuations. Importantly, one should avoid the bistability to obtain high cooling efficiency. Usually, the distributions of nuclear spin polarization obtained from the feedback model with nonlinear response are still Gaussian (except for the bistability region) after some evolution time. FIG. 5 presents the probability distribution of Pnuc for Ωp /Ωc = 5.0 and ~G = 7µeV . It is clearly seen that the nuclear spin has been highly polarized (with the mean value of Pnuc ∼ 0.953) after a preparation time of 400ns. The probability density of the distribution does conform well to the Gaussian distribution with the width 1/500 of σ0 (the line with blue triangle in FIG. 5). An enhancement of 500 times of the electron coherence time has been obtained after 400 ns preparation time. In Ref. [21] a narrowing effect has been demonstrated for a QD, where the electron spin coherence time is increased by 100 times after a preparation time of 10µs. Two important factors in cooling an object are a cold background and an efficient heat transfer, i.e., the interaction between the object and the background. Here the cold background is the dark state with fixed and highly polarized electron spin, which can be established by using laser fields with large ratio of Ωp /Ωc . The effective ESNSI can be enhanced by applying a strong ac electric field. FIG. 6 shows the dependence of dephasing time on the nuclear spin polarization, which is closely related to the electron spin polarization (Pnuc (∞) = ρ↓ − ρ↑ ) and can be tuned by the laser fields. One can see from FIG. 6 that with the increase of nuclear spin polarization, the reinforce of electron spin coherence is more effective and the electron coherence time becomes much longer. In a word, high nuclear spin polarization is much helpful for maintaining the electron spin coherence. We then discuss the other factor in the suppression

1000

800

T2 /T2,0

B.

600

400

200

0 0.6

0.7

0.8

Pnuc

0.9

1.0

∗ FIG. 6: The electron spin dephasing time T2∗ (T2,0 is the bare dephasing time in the absence of laser fields and ac electric field) as a function of nuclear spin polarization Pnuc after a preparation time of 500ns when ~G = 7µeV . Other parameters are the same as those in FIG. 2.

of nuclear spin fluctuation, i.e., the effective ESNSI. To avoid the bistability and have high nuclear spin polarization, we choose a large ratio of laser field Rabi frequencies Ωp /Ωc = 5.0. As shown in FIG. 7, with turning on the ESNSI, the electron dephasing time could be significantly prolonged. There is a non-monotonic dependence of T2∗ on G and there is an optimal value of ~G, with which the enhancement of the dephasing time could be as large as 1500 times. This interesting G dependence can be understood in the following way. The effective electron spin-nuclear spin coupling constant ∝ G determines the 1 spin-flip rate [ τsf ]. One can see from equation (7) that 1 that there is a non-monotonic dependence of [ τsf ] on G, which is a consequence of the back action of nuclear spin on the electron spin. The OF field affects the energy of electron spin states and thus influences the ESNSI. The 1 ] has similar deinsert of FIG. 7 clearly shows that [ τsf ∗ pendence on ~G as that for T2 and the optimal value of 1 ~G = 9µeV for [ τsf ] is the same as that for T2∗ . The preparation time also affects the electron spin coherence time. FIG. 8 shows the dependence of electron spin coherence time on the preparation time. Three orders of magnitude enhancement of coherent time (from ns to µs) can be achieved after a preparation time of 500ns. As seen in the insert of Fig.8, the electron spin coherence time has an exponential growth with the preparation time until it reaches the nuclear spin relaxation time scale. The upper limit of the electron-spin coherence time T2∗ is set by the nuclear spin relaxation time due to the dipole-dipole interaction of nuclear spins, which does not conserve the total nuclear spin. Finally, we give a set of parameters achievable in ex-

6

](GHz)

1600

20 10

[1/

800

2

T /T

2,0

sf

1200

0 0

5 10 15 20 25 30 G(

eV)

400 0

5

10

15

eV)

G(

20

25

30 IV.

FIG. 7: The electron spin dephasing time as a function of ~G after a preparation time of 500ns when Ωp /Ωc = 5.0. Insert: 1 the transition rate [ τsf ] (calculated by Eq. (7)) verses ~G. Other parameters are the same as those in FIG. 2.

6

ln(T

600

4

T

2

/T

2,0

2

/T

2,0

)

1000 800

400

2

100

200

300

400

500

t(ns)

200 0

100

200

300

the experiment.19 The frequencies and Rabi frequencies of two laser fields are ωp = 319086GHz, ~Ωp = 5.0µeV and ωc = 319082GHz, ~Ωc = 1.0µeV , respectively.18,19 The frequency and Rabi frequency of the ac electric field could be ω = 4GHz and ~G = 7µeV (the corresponding ˜ ∼ 3×105 V m−1 for a QD of 40 electric field intensity is E 34 nm). This set of parameters can be implemented easily in the experiment. Under such condition, more than 1000 times enhancement of the electron coherent time can be obtained after a preparation time of 500ns.

400

500

t(ns)

FIG. 8: The preparation time t versus the electron spin dephasing time for ~G = 7µeV and Ωp /Ωc = 5.0. Other pa∗ rameters are the same as those in FIG. 2. Insert: ln(T2∗ /T2,0 ) versus t.

We have studied the suppression of nuclear spin fluctuation in semiconductor QD by using CPT (induced by two laser fields) and EDSR (induced by an ac electric field). We can use the laser fields to tune the electron spin configuration in the dark state and obtain a cooling background of highly polarized electron spin state, which leads to highly polarized nuclear spin. The ac electric field can compensate the energy mismatch of electron spin state and greatly accelerate the flip rate of electron and nuclear spins. Making use of the feedback controls of the nuclear spin bath and avoiding the bistability at the same time, one can obtain highly polarized nuclear spin state with much narrower distribution width. The competition between the ac electric field enhanced ESNSI and back action leads to an optimal ac electric field intensity and more than one thousand times enhancement of the electron coherent time. This cooled nuclear spin background (with almost fully polarized nuclear spins and small fluctuation) is very useful in the storage of an electron spin state in the presence of collective excitations of nuclear spins.35

V.

periment. To avoid the transverse components of the OF and external magnetic field32 and eliminate/minimize the effect of the nuclear spin dipole-dipole interaction,9,30 we may choose a large magnetic field Bext = 1.29T like in

1 2 3 4

5

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CONLUSION

ACKNOWLEDGMENTS

This work was partially supported by the National Natural Science Foundations of China (11374039); National Basic Research Program of China (973 Program) (2013CB632805, 2011CB922204).

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