Bistability and steady-state spin squeezing in diamond nanostructures controlled by a nanomechanical resonator

Annals of Physics 369 (2016) 36–44

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Bistability and steady-state spin squeezing in diamond nanostructures controlled by a nanomechanical resonator Yong-Hong Ma a,b,∗ , Xue-Feng Zhang a,b , Jie Song c , E Wu b a

Key Laboratory of Integrated Exploitation of Bayan Obo Multi-Metal Resources, Inner Mongolia University of Science and Technology, Baotou 014010, China b

School of Science, Inner Mongolia University of Science and Technology, Baotou 014010, China

c

Department of Physics, Harbin Institute of Technology, Harbin 150001, China

article

info

Article history: Received 25 October 2015 Accepted 2 March 2016 Available online 10 March 2016 Keywords: NV center Spin squeezing Bistability

∗

abstract As the quantum states of nitrogen vacancy (NV) center can be coherently manipulated and obtained at room temperature, it is important to generate steady-state spin squeezing in spin qubits associated with NV impurities in diamond. With this task we consider a new type of a hybrid magneto-nano-electromechanical resonator, the functionality of which is based on a magnetic-field induced deflection of an appropriate cantilever that oscillates between NV spins in diamond. We show that there is bistability and spin squeezing state due to the presence of the microwave field, despite the damping from mechanical damping. Moreover, we find that bistability and spin squeezing can be controlled by the microwave field and the parameter Vz . Our scheme may have the potential application on spin clocks, magnetometers, and other measurements based on spin–spin system in diamond nanostructures. © 2016 Elsevier Inc. All rights reserved.

Corresponding author at: School of Science, Inner Mongolia University of Science and Technology, Baotou 014010, China. E-mail addresses: [email protected] (Y.-H. Ma), [email protected] (X.-F. Zhang).

http://dx.doi.org/10.1016/j.aop.2016.03.001 0003-4916/© 2016 Elsevier Inc. All rights reserved.

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1. Introduction A spin squeezing state [1] is a symmetric state of an ensemble of spin particles, whose fluctuation in one collective spin direction to the mean spin direction is smaller than the classical limit. The purpose of studying the spin squeezing [1] has arisen mainly from exploring the correlation and many-body entanglement [2–4] of particles, especially improving the measurement precision in practice [5–10]. Spin squeezing of many atoms is prepared using atom–light or atom–atom interactions [11–17]. Recently, a scheme for achieving coherent spin squeezing of nuclear spin states has been proposed in semiconductor quantum dots [18]. Very recently, spin-squeezed states are produced and characterized at a temperature of 26 °C in nuclear magnetic resonance quadrupolar system [19]. In recent years, in view of the tremendous progress of localized electron spins in solids which show long relaxation [20] and coherence times, especially their states can be easily controlled and manipulated via microwave or radio frequency pulses. Detection and manipulation of single electron spin states in solids have recently received much attention [21–25]. In the last decade, the negatively charged NV center in diamond has become a promising resource for future quantum technology [26–29]. NV centers, which are consisted by a nitrogen impurity atom with an adjacent vacancy, are naturally generated in bulk diamond or diamond nanocrystals and can be operated even at room temperature [30,31]. In numerous experiments, the coherent coupling and entanglement to nuclear spins of nitrogen [32,33] and carbon-13 [34,35] has been demonstrated. An important condition is to realize the controlling interactions between the NV centers, required for quantum gates or to produce entangled spin states in quantum-enhanced sensing [36]. In light of this challenge, a useful approach toward this goal is to couple NV centers to a resonant optical [37,38] or mechanical [39–41] mode. In this paper, we consider the coherent coupling between an ensemble of NV centers and the quantized motion of the magnetized nanomechanical resonator tips, and externally driven by a microwave field. With the help of this microwave field, bistability and steady state spin squeezing can be obtained in our system. We already know that the master equation is rather difficult to calculate with for large N. However, in our system we give average value for the spin squeezing. Since the fluctuations scale as 1/N, mean-field theory becomes selfconsistent again when N is very large. In the limit of large N, the spin squeezing is calculated analytically by considering fluctuations around the mean-field steady states. Using this method, we prove the present system can produce steady-state spin squeezing. Especially, spin squeezing can be controlled by microwave field and the parameter VZ . 2. Model We consider the coherent coupling between an ensemble of NV centers embedded in a single crystal diamond nanobeam and the quantized motion of the magnetized nanomechanical resonator tips. At the free end of the cantilever, an ensemble of magnetic tips is mounted as illustrated in Fig. 1. On the other hand, the same number of NV centers are positioned at a distance. The ground state of the NV centers is known to have an electron spin triplet structure with a zerofield splitting of 2.88 GHz between the ms = 0 and the degenerate ms = ±1 states. Microwaves are applied to drive the two states of the spins. If the Rabi frequency field contains only one frequency resonant to the transition between the levels |0⟩ and | − 1⟩, the system can be reduced to the S = 1/2 pseudospin model which describes transitions between the states | − 1⟩ and |0⟩ only. Since the transition between the levels | + 1⟩ and |0⟩ is off resonance and therefore forbidden [42]. Thus the Hamiltonian of the system in the frame rotating with the frequency of the rf field has the form [42–44]

 HNV = −δ

  n

| − 1⟩n ⟨−1| −

 n

|0⟩n ⟨0| +

Ω (| − 1⟩n ⟨0| + |0⟩n ⟨−1|), 2

(1)

n

where δ denotes the detuning between the microwave frequency and the intrinsic frequency of the spins. The Rabi frequency Ω can be described as Ω = Ω0 − ∆Ω , with ∆Ω = µB (B0 + Bms ). Here B0 is the amplitude of the external constant magnetic field applied on the system, Bms is the magnetic field produced by the magnetic film on the cantilever at the position of the spins.

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a

b

Fig. 1. (a) The magnetic tips, attached to the end of a nanomechanical resonator of dimensions (l, w, h), are fixed at a distance h above ensemble of NV centers. The distances h between the NV spins and magnetic tips can be adjusted by the field-induced deflection of the cantilever. Therefore, a strong coupling thereby can be created between the electronic spins of the defect center and the motion of the resonators. (b) Level diagram of the driven NV centers in the electronic ground states. Rabi frequencies Ω01 (Ω02 ) of microwaves are used to drive the spin states.

In Fig. 1, the nanomechanical resonator can be described by the Hamiltonian Hm = h¯ ωm aĎ a, where a(aĎ ) is the destruction (creation) operator of the mechanical mode with frequency ωm . The motion of the magnetic tips produces a field, which is proportional to the position operator z = a0 (a + aĎ ). We consider the resonant case between the microwave frequency and the intrinsic frequency of the spins, thus the total Hamiltonian of the system is H = ωm aĎ a + Ω Jx + λ(a + aĎ )Jz ,

(2)

for simplicity, we have assumed the same coupling of each spin to the mechanical mode. λ is the coupling strength. To understand what the parameters mean, one can rewrite  Eq. (2) as we introduce 1 | − 1 ⟩ ⟨− 1 | − | 0 ⟩ ⟨ 0 | , J = collective spin operators, such as Jz = 21 n n x n n | − 1⟩n ⟨0| + |0⟩n ⟨−1|. 2 It is clear that each NV center is driven by a laser with Rabi frequency Ω . In the limit of the regime λ, Ω ≪ ωm . The Hamiltonian can be diagonalized by the transformation eR He−R with R = ωλ (aĎ − a)Jz − ωΩ Jy , it will yield an effective Hamiltonian m

m

Heff = ωm aĎ a + Ω Jx +

λ2 2 Ω2 Jz − J , ωm ωm z

(3)

where the collective NV operators obey the commutation relations: [Jx , Jy ] = iJz . To include spontaneous emission in the model, we consider the master equation for the density matrix ρ

  Vz ρ˙ = −i ωm aĎ a + Ω Jx + mJz − Jz2 , ρ N   1 + γ (nth + 1) σn− ρσn+ − (σn+ σn− ρ + ρσn+ σn− ) 2

n

  1 +γ σn+ ρσn− − (σn− σn+ ρ + ρσn− σn+ ) , 2

n 2 where ωλ

2

(4)

, m = Ω , nth = h¯ ωm /κ1 B T is the equilibrium phonon occupation number at ωm e −1 temperature T . γ is the mechanical dissipation induced by collective spin relaxation. We choose this kind of dissipation because it leads to significant decoherence, and we want to see whether spin squeezing survives under such pessimistic conditions. For convenience, using the similar method in Ref. [45], we introduce the new variables, such as X = ⟨Jx ⟩/j, Y = ⟨Jy ⟩/j, Z = ⟨Jz ⟩/j, where j = N /2, apparently, X , Y , Z ∈ [−1, 1]. Using the master m

=

Vz N

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b

a

Fig. 2. Bifurcation diagram for mean-field equations as the function of the parameter Vz . The parameters are Ω = γ ; m = 6γ with different parameter nth (a) Green line (nth = 0.1γ ); red line (nth = 0.3γ ); blue line (nth = 0.5γ ); magenta line (nth = 0.7γ ). (b) nth = 0.1γ with different parameter m (a) Green line (m = 2γ ); red line (m = 4γ ); blue line (m = 6γ ); magenta line (m = 8γ ). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

equation (4), the mean field equations are given by X˙ = −mY + Vz YZ +

γ 2

(nth + 2)XZ ,

Y˙ = mX − Ω Z − Vz XZ + Z˙ = Ω Y +

γ 2

nth Z 2 − γ

γ

(nth + 2)YZ ,  1 nth + . 2

(5) (6)



2

(7)

3. Bistability Optical bistability has recently received much more attention both as a model physical system and for practical application. The motivation for constructing a bistable device is for potential application as a switching element in an optical communications system. We find that our system can generate bistability including two steady states with different amounts of excitation. This steady state can be obtained in the frame rotating at the frequency of the microwave field oscillation. It also means that the microwave field induces a forced oscillation of the system, meanwhile the system accompanies the microwave field oscillation with a phase shift and an amplitude which vary with the frequency of microwave field. The bistability can be obtained when the variations of the Bloch vector owing to relaxation are compensated by the variations due to precession in the microwave field. The steady-state solutions are calculated by solving Eqs. (5)–(7) for the parameters X , Y and Z . From Eq. (6), if the microwave field is absent corresponding to Ω = 0 (that is m = 0) and Vz = 0, we can see X¯ = Y¯ = 0, it indicates that there are no transition between the two levels. This means that the requirement of X¯ ̸= 0 is m ̸= 0, that is to say, the microwave field must be added. The nonzero solution for X¯ ̸= 0 means the non-zero steady-state spin states dipole moment which has dramatic effect on the resonance fluorescence spectrum [46] in the frequency-dependent environment reservoirs. Even for heat reservoirs but very strong fields X¯ ̸= 0 can be non-zero due to the dependence of the relaxation rates on the strength of the microwave field (Rabi frequency). Fig. 2 displays the dependence of the bistability on the effective control-field intensity m, equilibrium phonon occupation number nth , and the parameter Vz respectively. From Fig. 2 we find bistable threshold for the present parameters, the upper and the lower branch with a positive (negative) slope for the phase (amplitude) are stable, whereas the middle branch is unstable. The hysteresis loop for the parameter Vz is shown in Fig. 2(a), it can be easily seen that increasing the

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intensity of the parameter nth results in a significant decreasing of the bistable threshold. Consider

(1 + ⟨Jz ⟩)/2 initially on the lower stable branch (corresponding to the minimal root). As Vz increases, (1 +⟨Jz ⟩)/2 reaches the end of that branch. At this point slightly increasing for the parameter Vz results in a switch of (1 + ⟨Jz ⟩)/2 to the second stable branch (corresponding to the smallest root).

Fig. 2(b) shows that enhancing the intensity of the control field results in a significant increase of the bistable threshold. This might be useful to control the threshold value and the hysteresis cycle width of the bistable curve simply by adjusting the intensity of the control field. Moreover, we find increasing the pump strength causes the system to approach the bistability region however produces the effect of spin–spin interaction more invisible. When the pump strength has been increased to 8γ , the curve are completely resolved from each other and their bistability regions have become wider enough. Overall, when equilibrium phonon occupation number nth is large enough or the effective microwave field is too week, it leads to the system to be near the low-energy state corresponding to the lower branch. However, if the system is already in the upper branch, the equilibrium phonon occupation number nth become less or the effective microwave field is large, so the system remains excited. This is an example of ‘‘intrinsic optical bistability’’, which means that the bistability is due to the interaction between NV spins instead of the interaction with the resonators. The extent to which bistability is observed in practice will depend on the fluctuations, which in turn determine the time for random switching from one branch to the other. The driving field must be ramped in time intervals shorter than this random switching time in order for bistability to be observed [47]. From the cyan line in Fig. 2, it can be clearly seen that the system is far below the bistability when the value nth exceeds 0.7γ or the parameter m is less than 2γ . Apparently, the mean-field equations [Eqs. (5)–(7)] are equivalent to that of the optical Bloch equations for a two-level atom driven by a laser, where we identify X¯ = ⟨σx ⟩, Y¯ = ⟨σy ⟩, and the inversion Z¯ = ⟨σz ⟩. The equations are 2

aĎ a − δ depends on the parameter Z¯ . This actually nonlinear, since the effective detuning VZ Z¯ − Ω ωm renormalization of the detuning is understood from Eq. (2), where the excitation of one NV center shifts the effective detuning of another. 4. Spin squeezing Spin squeezing parameter ξ 2 is employed [48,49] to ascertain spin squeezing. There is spin squeezing when ξ 2 < 1. Squeezing parameter ξ 2 can be defined as

ξ 2 = min

N (∆Jn⃗⊥ )2

|⟨⃗J ⟩|2

,

(8)

if the Bloch vector has the maximal length |⟨⃗J ⟩| = N /2. Eq. (8) will reduce to the definition proposed by Kitagawa and Ueda [1]. In our system, to calculate the spin squeezing, using the method in Ref. [45], we can rotate the spin operators J by the angle θ around the axis nˆ = (− sin φ, cos φ, 0)

⃗J ′ = e−iθ nˆ ·⃗J ⃗Jeiθ nˆ ·⃗J , θ θ ⃗ e−iθ nˆ ·J = cos I − 2i sin (− sin φ Jx + cos φ Jy ). 2

2

(9) (10)

For the new operators, the mean values in the steady state can be expressed as ⟨⃗Jx′ ⟩ = ⟨⃗Jy′ ⟩ = 0,

⟨⃗Jz′ ⟩ =

N . 2

Thus, in the new spin frame, the spin squeezing parameter can be expressed as [45,49]

ξ2 =

⟨Jx′2 + Jy′2 ⟩ −



⟨Jx′2 − Jy′2 ⟩2 + ⟨Jx′ Jy′ + Jy′ Jx′ ⟩2 N /2

,

(11)

obviously the Bloch vector has the maximal length, so the definition in Eq. (11) are identical to Refs. [1,49] Fig. 3(a) shows ξ 2 as the function of Vz with different nth . When nth = 0.1γ , we can find two curves appear in a broad range of spin squeezing for Vz from 1.8γ to 10γ , the maximum squeezing approaches about 0.1. With the increasing of nth , the curves become narrowed, and maximum

Y.-H. Ma et al. / Annals of Physics 369 (2016) 36–44

a

41

b

Fig. 3. Spin squeezing parameter ξ 2 as the function of the parameters Vz . The parameters are Ω = 0.8γ ; m = γ with different parameter nth (a) green line (nth = 0.1γ ); red line (nth = 0.3γ ); blue line (nth = 0.5γ ); magenta line (nth = 0.7γ ). (b) nth = 0.1γ ; with different parameter m (a) green line (m = 0.5γ ); red line (m = γ ); blue line (m = 1.5γ ); magenta line (m = 2γ ). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

squeezing decreases. Spin squeezing finally dies away when nth exceed 0.7γ . Fig. 3(b) shows there is no steady-state spin squeezing for m = 0 corresponding to the absence of the microwave field. The effect of the microwave field is to re-excite the NVs after they decay, so that the interaction can re-squeeze them. With the microwave field m increasing, the curves denote spin squeezing appears, period is prolonged effectively and the maximum squeezing is increased due to the increase in the intensity of the microwave field. Note that when m approaches to 2γ , it shows the curves appear in a broad range of spin squeezing for Vz from 2γ to 25γ , the maximum squeezing can be less than 0.1. The reason of generating the spin squeezing in our system is simple. For an ensemble of NV, described by a collective spin S, the squeezing is generated by an ensemble phonon interaction 2

Hamiltonian of the form mJz (m = Ω aĎ a − δ) that represents the differential energy shift between ωm the two-level states due to the resonant mechanical mode with phonon number aĎ a. This phonon number depends on the population difference Jz between the spin states because the precise tuning of the resonator mode is relative to the microwave field. In particular, we initially rotate the spin operators J by the angle θ around the axis nˆ = (− sin φ, cos φ, 0) to circular uncertainty region of a coherent spin state (CSS), it is then sheared into an ellipse that is narrower in one direction than the original CSS, corresponding to spin squeezing. The spin correlations between different NV, which is due to the phase between the two states in any individual NV, now depends on the state population difference Jz of the entire ensemble. In addition, the spin squeezing comes from the interaction of the NV spins, different with the direct interaction of the NV centers in our previous paper [50], here with the help of magnetic tips the interaction of the NV spins leads to an indirect interaction between the NV spins. 5. Discussion and conclusion Spin squeezing in our system does no longer depend on the initial state of the system. For the convenience of calculation, we rotate the spin operators ⃗J = (Jx , Jy , Jz ) by the angle θ to a special state, it is reasonable and practical. In quantum mechanics, transformations in space and time are implemented by unitary transformations on the Hilbert space for the system. The idea is that if we apply some transformation to a physical system in 3D, the state of the system is changed and this should be mathematically represented as a transformation of the state vector for the system. The operator ⃗J generates transformations on the Hilbert space corresponding to rotations of the system about the corresponding axis (x, y, z). We identify the operators Ji with the angular momentum observables for the system. Of course, the physical justification of this mathematical model of angular

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Y.-H. Ma et al. / Annals of Physics 369 (2016) 36–44

momentum relies upon the unequivocal success of this strategy in describing physical systems. In particular, Ji will (under appropriate circumstances) be conserved. Superficially, bistability and spin squeezing are independent of the specific number of NV in our paper. We use the mean-field approach, which is useful for the dissipative transverse-field Ising model, because it provides a systematic way of including fluctuations. In fact, it is basically a way of doing perturbation theory in 1/N. We consider N is finite (see from ωλ = NZ ), there are fluctuations m due to the finite sample size; for example, whenever NV spontaneously damping, the mean field instantaneously changes. When N is large but finite, one can think of the system as evolving mostly according to the deterministic mean-field equations but with some noise, called ‘‘quantum noise’’ [51, 52]. These fluctuations are responsible for the correlations that we have calculated. 2

V

According to the expression of the parameter ωλ = NZ , we can find that Vz = N ωλ is determined m m by the number of NV centers N, the frequency of the mechanical mode ωm and the coupling strength λ. From the numerical results (see Figs. 2 and 3), in order to produce the effective bistability and spin squeezing state in our system, the parameter Vz should be a proper range, that is, with the mechanical mode ωm fixed, the number of NV centers N and the coupling strength λ cannot be too large or too small. In experiment, to control the parameter Vz , we can change the coupling strength λ and the number of NV centers. Moreover, the concept of superposition of macroscopically distinct quantum states, i.e., the Schrödinger cat states, plays an important role in understanding the conceptual foundations of quantum mechanics. The methods for generation of such superposition states are, therefore, of fundamental interest [53]. The Schrödinger cat states for the present system can also be generated. The Hamiltonian (3) is quadratic in the population inversion operator Jz [1,53]. It is, therefore, analogous to the Hamiltonian quadratic in the number operator of the single mode field propagating through a Kerr medium like an optical fiber. Consider an ensemble of NV centers prepared in the spin coherent state, an spin coherent state evolves to a superposition of the spin coherent states; i.e., it becomes a cat state of spins. Finally, we discuss the application of spin squeezing generated in our system. One of the most useful applications of spin squeezing is to detect entanglement for many-qubit systems. To determine whether a state is entangled, we just need to measure the collective operators, which in many cases are particle populations. By employing a positive partial transpose method, they found generalized spin squeezing inequalities as the criteria for two- and three-qubit entanglements [54,55]. In our system, NV centers are not accessed individually, and the spin-squeezing parameter is easier to obtain than the concurrence and the entanglement entropy. Different kinds of spin-squeezing inequalities may be used to detect various types of entanglement [48]. To summarize, we report on an investigation of coherent coupling between an ensemble of NV centers and the quantized motion of the magnetized nanomechanical resonator tips at room temperature. In the limit of large N, bistability and spin squeezing are calculated analytically by considering fluctuations around the mean-field steady states. With the help of microwave field, we prove bistability and steady state spin squeezing can be obtained in the present system. Especially, spin squeezing can be controlled by microwave field and the parameter VZ . 2

V

2

Acknowledgments The project was supported by NSFC (Grant Nos.11305085 and 51571126) and Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (Grant No. NJYT-13-B13) and the Youth academic backbone of inner university of science and technology special project (Grants No. 2014QNGG04). Appendix. Expectation of collective operators If the expression meets 3(8Vz mnth )2 = 8(4n2th r 2 + n3th r 2 + 4nth r 2 + 4nth Vz2 )(−8Vx2 − 4nth Vz2 + 4nth m2 − 12nth r 2

− r 2 n3th + 4nth Ω 2 + 8Ω 2 − 8r 2 − 6n2th r 2 ),

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43

the steady-state solution of Eqs. (4)–(6) is the fixed point, such as

(2nth + 1)m4 − 4nth m2 V 2 − 2(2nth + 1)V 2 m2 + 8nV 4 = sin θ cos φ, 2(nth + 2)m2 V Ω γ (2nth + 1)m2 − 4γ nV 2 Y¯ = = sin θ sin φ, 2Ω m2

X¯ =

Z¯ =

2Vm 4nth γ 2 + n2th γ 2 + 4γ 2 + 4V 2

(A.1) (A.2)

= cos θ .

(A.3)

To calculate spin squeezing, in the limit of large N, we use the Holstein–Primakoff transformation to write the operator ⃗J ′ as the bosonic annihilation and creation operator

⃗J+′ =

√

Nb,

⃗J−′ =

√

⃗Jz′ = N − bĎ b

NbĎ ,

(A.4) 2 Ď where the creation and annihilation operators satisfy the relation [a, a ] = 1. After the inverting to Eq. (9), we can get

√ √ ⃗J+ = cos2 θ Nb − e2iφ sin2 θ NbĎ + eiφ sin θ

 − bĎ b , 2 2 2   √ √ θ θ N ⃗J− = cos2 NbĎ − e−2iφ sin2 Nb + e−iφ sin θ − bĎ b . 2



2

N

(A.5) (A.6)

2

According to the definition of ⃗J ′ (Eq. (9)), the terms linear in the operators b, bĎ cancel out, which result in the leading order being quadratic. In the limit of large N, we can only keep quadratic terms. Substituting Eq. (A.5) and (A6) into Eq. (6), we rewrite the master equation as

γ



θ

θ



ρ˙ = −i [H , ρ ] + cos + (nth + 1) sin (2bĎ ρ b − bbĎ ρ − ρ bbĎ ) 2 2 2   γ θ θ + sin4 + (nth + 1) cos4 (2bρ bĎ − bĎ bρ − ρ bĎ b) 2 2 2   γ − sin2 θ e−2iφ + (nth + 1)e2iφ (2bρ b − b2 ρ − ρ b2 ) 8   γ − sin2 θ e2iφ + (nth + 1)e−2iφ (2bĎ ρ bĎ − bĎ2 ρ − ρ bĎ2 ), 4

4

(A.7)

8

where H = c1 b2 + c1∗ bĎ2 + c2 bĎ b,

(A.8)

with c1 =

Vz 4

sin2 θ e−2iφ ,

(A.9)

Vz

sin2 θ − Ω sin θ cos φ − m sin2 θ sin φ. 2 The equations of motion for the fluctuations can be written as c2 = −

d⟨b2 ⟩ dt

(A.10)

= −2ic2 ⟨b2 ⟩ − 2ic1∗ (2⟨bĎ b⟩ + 1) − γ cos θ nth ⟨b2 ⟩  γ  2iφ + e + (nth + 1)e−2iφ sin2 θ ,

(A.11)

4

d⟨bĎ2 ⟩

= 2ic2 ⟨bĎ2 ⟩ − 2ic1 (2⟨bĎ b⟩ + 1) − γ cos θ nth ⟨bĎ2 ⟩  γ  −2iφ + e + (nth + 1)e2iφ sin2 θ , 4   d⟨bĎ b⟩ 2 ∗ Ď2 Ď 4 θ 4 θ = 2i(c1 ⟨b ⟩ − c1 ⟨b ⟩) − γ cos θ nth ⟨b b⟩ + γ cos + (nth + 1) sin . dt

dt

2

2

(A.12) (A.13)

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Y.-H. Ma et al. / Annals of Physics 369 (2016) 36–44

These equations for the steady state can be solved. Here, we do not write the complicated expressions out here. We need only outline the mean values that enter the variance of Eq. (11) N ⟨⃗Jx′2 ⟩ = (⟨a2 ⟩ + ⟨aĎ2 ⟩ + 2⟨aĎ a⟩ + 1),

(A.14)

N ⟨⃗Jy′2 ⟩ = − (⟨a2 ⟩ + ⟨aĎ2 ⟩ − 2⟨aĎ a⟩ − 1),

(A.15)

iN ⟨⃗Jx′⃗Jy′ + ⃗Jy′ ⃗Jx′ ⟩ = (⟨aĎ2 ⟩ − ⟨a2 ⟩).

(A.16)

4

4

2

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