1. Introduction
In quantum information science, the generation of controlled spin–spin interactions is an outstanding challenge. Traditionally, single-photons have served as a universal bus interface to mediate interactions and information transfer between remote quantum systems, due to their excellent coherence and controllability [1]. However, it still remains challenging for single photons to mediate interactions between remote quantum systems of disparate energy scale. On the other hand, with the advent of quantum acoustics, phonon, the quanta of mechanical vibration, offers a promising solution to this challenge because mechanical motion can couple a wide range of quantum systems through various interactions [2–4]. For example, a mechanical system can couple to photons through radiation pressure or solid spins through magnetic dipole force. To date, a variety of hybrid mechanical systems have been realized [5–25] in order to deal with issues in a broad range of fields, from fundamental physics to quantum information processing. Recently, much attention has been paid to the hybrid mechanical system based on the silicon-vacancy (SiV) colour centers in diamond. These devices are promising since in terms of present nanofabrication techniques, it is capable of fabricating high-quality mechanical modes at GHz frequency [26–30], while SiV centers in diamond, owing to their long coherence time and the highly favourable spectral properties, have become one of the most promising solid-state quantum emitters [19–22]. So far, the strain coupling between solid spins and mechanical modes has been regarded as a new type of coupling interface to realize a phonon network in the strong coupling regime [31–45].
At present, phonon-mediated spin–spin interactions have been used to entangle SiV centers, utilizing strain as coupling interfaces [43, 44]. However, the fidelity of the entangling gate is always limited by the inevitable spin inhomogeneous dephasing (also called ${T}_{2}^{* }$ process in literature). There have been several strategies proposed for protecting quantum gates of solid spins from decoherence, including dynamical decoupling protocols [46–49], optimal control [50–53], and quantum error-correcting codes [54–56]. Among those proposals, dynamical decoupling is the most effective method to suppress inhomogeneous dephasing and extend the spin coherence time ${T}_{2}^{* }$ to homogeneous dephasing time T2. However, this technique is not easily incorporated into other operations and requires a lot of power. Alternatively, continuous dynamical decoupling (CDD), has attracted lots of interest in recent years and is an advantageous approach to suppress spin decoherence [57–61]. The CDD procedure is more experimentally friendly than the dynamical decoupling procedure. By using the CDD procedure, the spin can be continuously and coherently addressed with an external control field, creating new ‘dressed'spin states, which are insensitive to environmental fluctuations. Most importantly, the spin dephasing time can be extended in these dressed states.
In this paper, by using the CDD method, we provide a feasible proposal to realize a high-fidelity entangling gate in a hybrid mechanical system based on SiV centers in diamond. In this scheme, each spin is driven by a mechanical driving field, which can introduce new dressed spin states. Within the dispersive regime, we show that a controllable spin–spin interaction can be obtained in the dressed basis. By directly calculating the entangling gate fidelity in the dressed basis, we demonstrate that the quality of the entangling gate is more robust than that in the unprotected case. In particular, by optimizing the detuning between the spins and phonons, we find that the two-qubit gate is capable of reaching fidelity exceeding 0.99. In addition, a more realistic condition is considered, that is, before directly performing the gate in the dressed basis, the initial spin state should be transferred from the bare basis to the dressed basis. Then, the protected mechanism can be implemented and the high-quality entangled state can be produced in the dressed basis. In addition, another adiabatic process is also required for sending this entangled state back to the bare basis. Therefore, compared with the case of simulating the gate fidelity directly in the dressed basis, the two adiabatic processes should be also considered in the calculation of the full gate fidelity. Here, we employ appropriate driving modulation to implement the two adiabatic processes and show that even under the more realistic condition, the full gate fidelity is still possible to reach as high as 0.99. These results provide a promising approach for realizing high-fidelity quantum information processing.
The rest of the paper is organized as follows. In section 2 , we present the phonon-mediated entanglement model with an external mechanical driving. In section 3 , by introducing the dressed spin states, we derive the effectively dressed spin–spin interaction and the corresponding master equation in the dispersive regime. In section 4 , by directly simulating the gate fidelity in the dressed basis, we show that the mechanical driving can significantly suppress the influence of the environment fluctuation on the gate, leading to a robust entangling gate. In section 5 , we propose a protocol to realize a high-fidelity full gate with appropriate driving modulation. Finally, we provide a short summary and outlook in section 6 .
2. Model and method
As illustrated in figure 1, we consider two SiV center spins that are strain-coupled to a nanomechanical resonator. The Hamiltonian of the spin-mechanical hybrid system is given by [17, 31, 32, 43–46]1 ) is the free Hamiltonian of the single phonon mode. The second term is the Hamiltonian of spin system, in which every spin is protected by an external driving field with Rabi frequency ${{\rm{\Omega }}}_{\perp }^{j}$ and frequency ${\omega }_{d}^{j}$. ${\omega }_{s}^{j}$ is the resonance frequency of the jth spin, ${\sigma }_{z}^{j}$ and ${\sigma }_{\pm }^{j}$ are the usual Pauli operators of the jth spin. The third term describes the interaction between the spins and phonons with coupling strength gj being the spin-phonon coupling strength.
$\begin{eqnarray}\begin{array}{rcl}H & = & {\omega }_{m}{a}^{\dagger }a+\displaystyle \sum _{j=1}^{2}\left[\displaystyle \frac{{\omega }_{s}^{j}}{2}{\sigma }_{z}^{j}+\displaystyle \frac{{{\rm{\Omega }}}_{\perp }^{j}}{2}\right.\\ & & \left.\times \left({{\rm{e}}}^{-{\rm{i}}{\omega }_{d}^{j}t}{\sigma }_{+}^{j}+{{\rm{e}}}^{{\rm{i}}{\omega }_{d}^{j}t}{\sigma }_{-}^{j}\right)\right]\\ & & +\sum _{j=1}^{2}{g}_{j}({a}^{\dagger }{\sigma }_{-}^{j}+a{\sigma }_{+}^{j}).\end{array}\end{eqnarray}$
The first term in equation (
Figure 1. Schematics of the hybrid mechanical system in this work. Two SiV centers implanted separately in a diamond mechanical resonator couple to each other through a common mechanical mode. |
For simplicity, hereafter, we assume gj = g, ${\omega }_{s}^{j}={\omega }_{s}$, ${{\rm{\Omega }}}_{\perp }^{j}={{\rm{\Omega }}}_{\perp }$ and ${\omega }_{d}^{j}={\omega }_{d}$. In the rotating frame with respect to the driving frequency ωd, the Hamiltonian in equation (1 ) becomes
$\begin{eqnarray}H^{\prime} ={\rm{\Delta }}{a}^{\dagger }a+{H}_{s}+g\sum _{j}({a}^{\dagger }{\sigma }_{-}^{j}+a{\sigma }_{+}^{j}),\end{eqnarray}$
$\begin{eqnarray}{H}_{s}=\displaystyle \frac{{{\rm{\Delta }}}_{s}}{2}\sum _{j}{\sigma }_{z}^{j}+\displaystyle \frac{{{\rm{\Omega }}}_{\perp }}{2}\sum _{j}({\sigma }_{+}^{j}+{\sigma }_{-}^{j}),\end{eqnarray}$
where Δ = ωm − ωd and Δs = ωs − ωd.In order to capture the full system dynamics, including the damping of the phonon mode and spin decoherence, we model the system using the following master equation.
$\begin{eqnarray}\begin{array}{rcl}\dot{\rho } & = & -{\rm{i}}[H^{\prime} +{H}_{\mathrm{noise}},\rho ]+\displaystyle \frac{{\gamma }_{m}}{2}{ \mathcal D }[o]\rho \\ & & +\displaystyle \sum _{j=1,2}{\gamma }_{s}{ \mathcal D }[{\sigma }_{z}^{j}]\rho ,\end{array}\end{eqnarray}$
where ${H}_{\mathrm{noise}}={\sum }_{j}\tfrac{{\xi }_{j}}{2}{\sigma }_{z}^{j}$ with ξj being the inhomogeneous broadening in transition frequencies and ${ \mathcal D }[o]\rho =2o\rho {o}^{\dagger }-{o}^{\dagger }o\rho -\rho {o}^{\dagger }o$ for a given operator o. Numerically, we simulate Hnoise by randomly assigning values ξj in a Gaussian distribution $P({\xi }_{j})=\tfrac{1}{\sqrt{2\pi {\sigma }^{2}}}{{\rm{e}}}^{-{\xi }_{j}^{2}/(2{\sigma }^{2})}$ with the width $\sigma =\sqrt{2}/{T}_{2}^{* }$. γm = ωm/Q and γs represent the dissipation of the mechanical mode and the spin dephasing rate of the SiV centers, respectively. Here, we focus on the regime of large mechanical frequency (in the GHz range) and at cryogenic temperature, the thermal phonon number is nearly zero, i.e. ${N}_{\mathrm{th}}=1/({{\rm{e}}}^{\tfrac{{\hslash }{\omega }_{m}}{{k}_{{\rm{B}}}T}}-1)\simeq 0$, which corresponds to coupling with the vacuum bath for the mechanical oscillator.3. Effective spin–spin Hamiltonian modified by the mechanical driving field
In this section, we show that the effective spin–spin Hamiltonian can be highly modified by the external control field. As shown in equation (4 ), if we take the fluctuation noise ξj into consideration, the total spin Hamiltonian becomes Hs + Hnoise, which can be diagonalized by introducing the following dressed spin states5 ) can be also expressed as $\left(\begin{array}{c}| \tilde{0}{\rangle }_{j}\\ | \tilde{1}{\rangle }_{j}\end{array}\right)$=Uj$\left(\begin{array}{c}| 0{\rangle }_{j}\\ | 1{\rangle }_{j}\end{array}\right)$. Correspondingly, we use ${\tilde{\sigma }}_{k}^{j}={U}_{j}{\sigma }_{k}^{j}{U}_{j}^{\dagger }$ to represent the Pauli operator in the dressed states. Based on these dressed spins states, the total spin Hamiltonian is then given by
$\begin{eqnarray}\begin{array}{rcl}| \tilde{0}{\rangle }_{j} & = & \cos ({\theta }_{j}/2)| 0{\rangle }_{j}-\sin ({\theta }_{j}/2)| 1{\rangle }_{j},\\ | \tilde{1}{\rangle }_{j} & = & \cos ({\theta }_{j}/2)| 1{\rangle }_{j}+\sin ({\theta }_{j}/2)| 0{\rangle }_{j},\end{array}\end{eqnarray}$
with $\tan ({\theta }_{j})={{\rm{\Omega }}}_{\perp }/({{\rm{\Delta }}}_{s}+{\xi }_{j})$. The dressed (bare) states $| \tilde{0}{\rangle }_{j}$ (∣0⟩j) and $| \tilde{1}{\rangle }_{j}$ (∣1⟩j) represent that the spin j is in spin down and up states. By defining a unitary matrix ${U}_{j}=\left(\begin{array}{cc}\cos ({\theta }_{j}/2) & -\sin ({\theta }_{j}/2)\\ \sin ({\theta }_{j}/2) & \cos ({\theta }_{j}/2)\end{array}\right)$, equation ( $\begin{eqnarray}\begin{array}{rcl}{\tilde{H}}_{s} & = & \sum _{j}\displaystyle \frac{{{\rm{\Omega }}}_{j}}{2}{\tilde{\sigma }}_{z}^{j},\\ {{\rm{\Omega }}}_{j} & = & \sqrt{{\left({{\rm{\Delta }}}_{s}+{\xi }_{j}\right)}^{2}+{{\rm{\Omega }}}_{\perp }^{2}}.\end{array}\end{eqnarray}$
Next, by using the relation ${\sigma }_{+}^{j}={U}_{j}^{\dagger }{\tilde{\sigma }}_{+}^{j}{U}_{j}=\tfrac{1}{2}\sin ({\theta }_{j}){\tilde{\sigma }}_{z}^{j}+(\cos ({\theta }_{j})+1){\tilde{\sigma }}_{+}^{j}+(\cos ({\theta }_{j})-1){\tilde{\sigma }}_{-}^{j}$, we can rewrite the interaction Hamiltonian in the dressed basis7 ) are ${\zeta }_{j}=g\sin ({\theta }_{j})/2$, ${\lambda }_{j}=g(\cos ({\theta }_{j})+1)/2$ and ${\eta }_{j}=g(\cos ({\theta }_{j})-1)/2$, respectively. Therefore, the whole Hamiltonian of the system in the dressed basis becomes
$\begin{eqnarray}\begin{array}{rcl}{\tilde{H}}_{I} & = & \displaystyle \sum _{j}[{\zeta }_{j}{\tilde{\sigma }}_{z}^{j}({a}^{\dagger }+a)+{\lambda }_{j}({\tilde{\sigma }}_{+}^{j}a\\ & & +{\tilde{\sigma }}_{-}^{j}{a}^{\dagger })+{\eta }_{j}({\tilde{\sigma }}_{+}^{j}{a}^{\dagger }+{\tilde{\sigma }}_{-}^{j}a)].\end{array}\end{eqnarray}$
The corresponding coefficients in equation ( $\begin{eqnarray}\tilde{H}={\rm{\Delta }}{a}^{\dagger }a+{\tilde{H}}_{s}+{\tilde{H}}_{I}.\end{eqnarray}$
In the dispersive coupling regime Ωj, ∣Δ∣ and ${{\rm{\Delta }}}_{\pm }^{j}=| {\rm{\Delta }}\pm {{\rm{\Omega }}}_{j}| \gg g$, the spin–spin effective interaction are mediated by a virtual exchange of phonons. In this limit, an effective Hamiltonian can be obtained by performing a Schrieffer–Wolff transformation ${\tilde{H}}_{\mathrm{eff}}={{\rm{e}}}^{{\rm{i}}S}\tilde{H}{{\rm{e}}}^{-{\rm{i}}S}$ with $S\,={\rm{i}}{\sum }_{j}[\tfrac{{\zeta }_{j}}{{\rm{\Delta }}}{\tilde{\sigma }}_{z}^{j}({a}^{\dagger }-a)\,+\,\tfrac{{\lambda }_{j}}{{{\rm{\Delta }}}_{-}^{j}}({\tilde{\sigma }}_{+}^{j}a-{a}^{\dagger }{\tilde{\sigma }}_{-}^{j})+\tfrac{{\eta }_{j}}{{{\rm{\Delta }}}_{+}^{j}}({\tilde{\sigma }}_{+}^{j}{a}^{\dagger }-a{\tilde{\sigma }}_{-}^{j})]$. Up to the second order in λj, the effective Hamiltonian can be written as9 ) is the Stake shifts Hamiltonian, which can be expressed as
$\begin{eqnarray}{\tilde{H}}_{\mathrm{eff}}={\rm{\Delta }}{a}^{\dagger }a+{\tilde{H}}_{s}+{\tilde{H}}_{\mathrm{stark}}+{\tilde{H}}_{s-s}.\end{eqnarray}$
The third term in equation ( $\begin{eqnarray}{\tilde{H}}_{\mathrm{stark}}=-\sum _{j}\delta {{\rm{\Omega }}}_{j}({a}^{\dagger }a+\displaystyle \frac{1}{2}){\tilde{\sigma }}_{z}^{j},\end{eqnarray}$
with $\delta {{\rm{\Omega }}}_{j}=\tfrac{{\lambda }_{j}^{2}}{{{\rm{\Delta }}}_{-}^{j}}-\tfrac{{\eta }_{j}^{2}}{{{\rm{\Delta }}}_{+}^{j}}$. The fourth term is the effective spin–spin interaction Hamiltonian, which is $\begin{eqnarray}{\tilde{H}}_{s-s}=-\sum _{i\ne j}\left[{J}_{z}^{{ij}}{\tilde{\sigma }}_{z}^{i}{\tilde{\sigma }}_{z}^{j}+{J}_{\mathrm{eff}}^{{ij}}({\tilde{\sigma }}_{-}^{i}{\tilde{\sigma }}_{+}^{j}+{\tilde{\sigma }}_{+}^{i}{\tilde{\sigma }}_{-}^{j})\right],\end{eqnarray}$
where ${J}_{z}^{{ij}}=\tfrac{{\zeta }_{i}{\zeta }_{j}}{4{\rm{\Delta }}}$ and ${J}_{\mathrm{eff}}^{{ij}}=\tfrac{{\lambda }_{i}{\lambda }_{j}}{4}\left(\tfrac{1}{{{\rm{\Delta }}}_{-}^{i}}+\tfrac{1}{{{\rm{\Delta }}}_{-}^{j}}\right)+\tfrac{{\eta }_{i}{\eta }_{j}}{4}\left(\tfrac{1}{{{\rm{\Delta }}}_{+}^{i}}+\tfrac{1}{{{\rm{\Delta }}}_{+}^{j}}\right)$.By tracing out the phonons, we obtain an approximate master equation for the dressed spins11 ) and (12 ), respectively, are presented in the appendix .
$\begin{eqnarray}\begin{array}{l}{\dot{\tilde{\rho }}}_{s}\simeq -{\rm{i}}[{\tilde{H}}_{\mathrm{eff}}-({\rm{\Delta }}-\displaystyle \sum _{j}\delta {{\rm{\Omega }}}_{j}{\tilde{\sigma }}_{z}^{j}){a}^{\dagger }a,{\tilde{\rho }}_{s}]\\ \quad +\displaystyle \sum _{i,j}\displaystyle \frac{{{\rm{\Gamma }}}_{\perp }^{{ij}}}{2}(2{\tilde{\sigma }}_{-}^{i}{\tilde{\rho }}_{s}{\tilde{\sigma }}_{+}^{j}-{\tilde{\sigma }}_{+}^{j}{\tilde{\sigma }}_{-}^{i}{\tilde{\rho }}_{s}-{\tilde{\rho }}_{s}{\tilde{\sigma }}_{+}^{j}{\tilde{\sigma }}_{-}^{i})\\ \quad +\displaystyle \sum _{i,j}\displaystyle \frac{{{\rm{\Gamma }}}_{\parallel }^{{ij}}}{2}(2{\tilde{\sigma }}_{z}^{i}{\tilde{\rho }}_{s}{\tilde{\sigma }}_{z}^{j}-{\tilde{\sigma }}_{z}^{j}{\tilde{\sigma }}_{z}^{i}{\tilde{\rho }}_{s}-{\tilde{\rho }}_{s}{\tilde{\sigma }}_{z}^{j}{\tilde{\sigma }}_{z}^{i}),\end{array}\end{eqnarray}$
where ${{\rm{\Gamma }}}_{\perp }^{{ij}}={\gamma }_{m}\tfrac{{\lambda }_{i}{\lambda }_{j}}{{{\rm{\Delta }}}_{-}^{i}{{\rm{\Delta }}}_{-}^{j}}$ and ${{\rm{\Gamma }}}_{\parallel }^{{ij}}={\gamma }_{m}\tfrac{{\zeta }_{i}{\zeta }_{j}}{{{\rm{\Delta }}}^{2}}$. The last two terms are the dephasing rates of the qubits that are associated with the relaxation of the phonon mode. The detailed derivation of the effective Hamiltonian and the master equation shown in equations (In the case of ξj = 0, the two spins are identical, hence, for the arbitrary parameters defined in equations (4 )–(12 ), the label j can be removed, such as Ωj = Ω, ${{\rm{\Omega }}}_{\perp }^{j}={{\rm{\Omega }}}_{\perp }$ and ${{\rm{\Delta }}}_{\pm }^{j}={{\rm{\Delta }}}_{\pm }={\rm{\Delta }}-{\rm{\Omega }}$. Then, the effective spin–spin interaction Hamiltonian in equation (11 ) becomes14 ), we plot the effective coupling strength Jeff as a function of the spin-phonon detuning Δ in figure 2. As shown in figure 2, for a certain Ω⊥, Jeff decreases with Δ. Moreover, under the condition Ω⊥ ≠ 0, the coupling strength Jeff can be significantly improved by enhancing Ω⊥. For comparison, we plot the correspondingly effective spin–spin coupling in the unprotected case (Ω⊥ = 0) denoted by the dark-dotted curve shown in figure 2. We find that when Ω⊥ = 50 g and 55 g ≤ Δ ≤ 70 g, Jeff in the protected case can be stronger than that in the unprotected case. Especially, for Δ = 55 g, Jeff is around triple times stronger than that in the unprotected case. It is worth noticing that the spin-phonon detuning Δ cannot be too small, since a smaller Δ will trigger a larger effective decay represented by ${{\rm{\Gamma }}}_{\perp }^{{ij}}$ and ${{\rm{\Gamma }}}_{\parallel }^{{ij}}$ (see equation (12 )). The coupling strength for the design in figure 1 can reach g ∼ 2 MHz, in this work, we use g to normalize other parameters.
$\begin{eqnarray}{\tilde{H}}_{s-s}=-{J}_{z}{\tilde{\sigma }}_{z}^{1}{\tilde{\sigma }}_{z}^{2}-{J}_{\mathrm{eff}}({\tilde{\sigma }}_{-}^{1}{\tilde{\sigma }}_{+}^{2}+{\tilde{\sigma }}_{+}^{1}{\tilde{\sigma }}_{-}^{2}),\end{eqnarray}$
with $\begin{eqnarray}{J}_{z}=\displaystyle \frac{{\zeta }^{2}}{2{\rm{\Delta }}},\qquad {J}_{\mathrm{eff}}=\left[\displaystyle \frac{{\lambda }^{2}}{{{\rm{\Delta }}}_{-}}+\displaystyle \frac{{\eta }^{2}}{{{\rm{\Delta }}}_{+}}\right].\end{eqnarray}$
In order to prepare an optimum entangling gate, here and after, we only consider the resonant driving case with ωd = ωs (Δs = 0), thus the detuning Δ = ωm − ωd = ωm − ωs. Based on the expression of Jeff shown in equation (
Figure 2. The effective spin–spin coupling Jeff as a function of the spin-phonon detuning Δ. Here, we set ωm = 500 g and ωs = ωd. |
In the following, we consider that the initial state of the two-qubit subsystem is $| \tilde{0}{\rangle }_{1}| \tilde{1}{\rangle }_{2}$ and the oscillators are prepared into the ground state ∣0⟩. According to equation (11 ), one can find that there is only the transition, i.e. $| \tilde{0}{\rangle }_{1}| \tilde{1}{\rangle }_{2}\longleftrightarrow | \tilde{1}{\rangle }_{1}| \tilde{0}{\rangle }_{2}$. Under the condition Ωj < Δ, the final entangled state $| {\tilde{\psi }}_{+}\rangle =\tfrac{1}{\sqrt{2}}(| \tilde{0}{\rangle }_{1}| \tilde{1}{\rangle }_{2}+{\rm{i}}| \tilde{1}{\rangle }_{1}| \tilde{0}{\rangle }_{2})$ can be produced after the entangling gate time given byappendix ). In addition, in the unprotected case (Ω⊥ = 0), the initial state and entangled state of the spin system reduce to ∣0⟩1∣1⟩2 and $| {\psi }_{+}\rangle =\tfrac{1}{\sqrt{2}}(| 0{\rangle }_{1}| 1{\rangle }_{2}+{\rm{i}}| 1{\rangle }_{1}| 0{\rangle }_{2})$, respectively.
$\begin{eqnarray}{t}_{g}=\displaystyle \frac{\pi }{4\sqrt{{\chi }^{2}+4{\left({J}_{\mathrm{eff}}^{{ij}}\right)}^{2}}},\end{eqnarray}$
where $\chi =\tfrac{{{\rm{\Omega }}}_{1}-\delta {{\rm{\Omega }}}_{1}}{2}-\tfrac{{{\rm{\Omega }}}_{2}-\delta {{\rm{\Omega }}}_{2}}{2}$. Equivalently, after different operation times, different target states can be produced. For example, when ${J}_{\mathrm{eff}}t=\tfrac{\pi }{2}$ $({J}_{\mathrm{eff}}t=\tfrac{3\pi }{2})$, the state $| \tilde{\psi }(t)\rangle $ evolves to $| \tilde{\psi }(t)\rangle =| \tilde{1}{\rangle }_{1}| \tilde{0}{\rangle }_{2}$ $\left(\tfrac{1}{\sqrt{2}}\left(| \tilde{0}{\rangle }_{1}| \tilde{1}{\rangle }_{2}-{\rm{i}}| \tilde{1}{\rangle }_{1}| \tilde{0}{\rangle }_{2}\right)\right.$. Note that, when we turn off the mechanical control field, the dressed states will disappear, thus, the effective Hamiltonian and the master equation of the spin system are quite simple (see By choosing Δ = 60 g and Ω⊥ = 50 g, we use the master equation shown in equation (12 ) to simulate the populations of the excited state of the two spins versus the operation time t under the ideal conditions. As shown in figure 3, via exchanging virtually phonons, the two spins can be excited and exchange their excitations.
Figure 3. Time Evolution of the populations of the excited state of two spins. The physical parameters are chosen as ωm = 500 g, ωs = ωd = 440 g, Ω⊥ = 50 g, γm = 0, γs = 0 and ξ = 0. |
4. Simulating the entangling gate fidelity in the dressed basis
In this section, we will show that the quality of the entangling gate can be significantly improved by the presence of control field. Under the experiment condition with γm = 3 × 10−2 g, γs = 3 × 10−4 g and ξ1 − ξ2 = ξ = 0.1 g, we simulate the time evolution of the fidelity of the state $| {\tilde{\psi }}_{+}\rangle $ in figure 4 based on equation (4 ) and equation (12 ), respectively. It is shown that for each case (with or without the mechanical driving), there is a good agreement between the numerical and analytical results. Substituting the relevant parameters into equation (11 ) and equation (15 ), we have Jeff ∼ 0.027 g and tg = 14.39 μs (Jeff ∼ 0.017 g and tg = 13.07 μs) for Ω⊥ = 50 g (Ω⊥ = 0). In addition, we find that in the unprotected case, the maximum gate fidelity is only 0.79. However, in the protected case, the maximum gate fidelity can still reach 0.993. This suggests that the mechanical driving field can effectively protect the spins from decoherence. To guarantee the results more precisely, the following numerical simulations are based on the full model with the master equation in equation (4 ).
Figure 4. The time evolution of the fidelity of the bell state $| {\tilde{\psi }}_{+}\rangle $ with different Ω⊥. Here, we set γm = 3 × 10−2 g, γs = 3 × 10−4 g and ξ = 0.1 g. Other relevant parameters are the same as those in figure 3. |
In order to show the effect of ξ on gate fidelity more detail, we plot the maximum gate fidelity as a function of ξ with different Ω⊥ in figure 5. It is clearly shown that once ξ deviates from 0, the maximum gate fidelity in the unprotected case represented by the red solid curve decreases sharply, which means that the entangling gate is very sensitive to ξ; in contrast, when the mechanical field is turned on, according to equation (8 ), under the resonant-driving condition (Δs = 0), the dressed spin frequency becomes ${{\rm{\Omega }}}_{j}=\sqrt{{\xi }_{j}^{2}+{{\rm{\Omega }}}_{\perp }^{2}}$. Since ξj ≪ Ω⊥, Ωj can be approximated as ${{\rm{\Omega }}}_{j}\simeq {{\rm{\Omega }}}_{\perp }+\tfrac{1}{2}\tfrac{{\xi }_{j}^{2}}{{{\rm{\Omega }}}_{\perp }}$, that is to say, in the presence of the mechanical driving field, the fluctuation noise ξj becomes nonlinear. In particular, this nonlinear noise can be significantly suppressed by choosing a stronger Ω⊥. Hence, the gate fidelity in the protected case can be much higher than that in the unprotected case. As shown in figure 5, even for a smaller Ω⊥ (see the olive dashed curve in figure 5), the quality of this entangling gate can be significantly improved. Moreover, if Ω⊥ becomes stronger (see the blue dotted–dashed curve in figure 5), the maximum gate fidelity can be further optimized.
Figure 5. The maximum gate fidelity as a function of the random fluctuations ξ with different Ω⊥. Other relevant parameters are the same as those in figure 4. |
Since ξ obeys the Gaussian distribution, we are more interested in the average maximum fidelity defined as ${\overline{F}}_{\max }=\int {F}_{\max }(\xi )P(\xi ){\rm{d}}\xi $. By fixing Ω⊥ = 50 g and γs = 0, we plot the average maximum gate fidelity as a function of Δ with different ${T}_{2}^{* }$ in figure 6. It is shown that for a certain ${T}_{2}^{* }$, there exists an optimal detuning Δ, which ${\overline{F}}_{\max }$ can reach its maximum value. Moreover, within our expectation, a longer spin coherence time leads to a higher fidelity gate. For SiV centers, under the present experiment technique, ${T}_{2}^{* }$ can reach ∼5 − 10 μs. From figure 6(d), we find that when ${T}_{2}^{* }=5$ μs, the gate fidelity can surpass 0.9995 and the correspondingly optimal detuning Δ can be as large as 90 g. However, Δ = 90 g cannot be regarded as an optimal detuning, since a larger Δ results in a slower gate. To build an entangling gate characterized by the features of high efficiency and high fidelity at the same time, we choose Δ = 75 g as an optimal detuning. Figure 7 displays the dependence of the infidelity $I=1-{\overline{F}}_{\max }$ on ${T}_{2}^{* }$ under the optimal spin-phonon detuning Δ = 75 g. We find that in the protected case (Ω⊥ = 50 g), when the spin dephasing γs is taken into consideration, the gate infidelity in the regime of 2 μs $\lt {T}_{2}^{* }\lt 5$ μs can be significantly suppressed. In particular, once ${T}_{2}^{* }\gt 5$ μs, the gate infidelity can be suppressed to 10−2. In addition, if γs is ignored, the gate infidelity can be further suppressed to 10−3. However, if this mechanical driving field is absent (see olive dashed curve in figure 7), the entangling gate will suffer from a higher infidelity. As shown in figure 7, even when ${T}_{2}^{* }=20$ μs, the minimum infidelity in the unprotected case is never less than 10−1. Those results confirm again that the presence of the mechanical field can make the entangling gate more robust to the realistic experimental environment.
Figure 6. The average maximum gate fidelity as a function of detuning Δ for (a) ${T}_{2}^{* }=2$ μs; (b) ${T}_{2}^{* }=2.5$ μs; (c) ${T}_{2}^{* }=3.3$ μs; (d) ${T}_{2}^{* }=5$ μs. Other relevant parameters are the same as those in figure 4. |
Figure 7. The average infidelity as a function of the spin dephasing ${T}_{2}^{* }$. Other relevant parameters are the same as those in figure 4. |
5. High-fidelity full quantum gate with driving modulation
In above section, we confirm that the dressed spin states created by the presence of the external driving field are insensitive to the environmental fluctuation. However, in a more realistic situation, the initial spin state is prepared in the bare basis rather than in the dressed basis. Moreover, the produced entangled state should be also in the bare basis. Thus, before the protected mechanism is started, we should use an adiabatic process to transfer the spin initial state (which is in the bare basis) to the dressed basis. Additionally, another adiabatic process is also required for sending the entangled sate produced in the dressed basis back to the bare basis. Therefore, compared with the case of simulating the gate fidelity directly in the dressed basis, the two adiabatic processes should be also considered in the calculation of the gate fidelity. Here, the two adiabatic processes are performed by employing the sine-shaped modulation for the external driving field. The detailed expressions of the Ω⊥(t) and ωd(t) are shown in below.
$\begin{eqnarray}{{\rm{\Omega }}}_{\perp }(t)=\left\{\begin{array}{ll}{\omega }_{x}{\sin }^{2}(\pi t/2{t}_{{s}}) & 0\leqslant t\leqslant {t}_{{s}\ },\\ {\omega }_{x} & {t}_{{s}}\lt t\leqslant {t}_{{s}}+{t}_{{g}}={t}_{\mathrm{off}},\\ {\omega }_{x}[1-{\sin }^{2}(\pi (t-{t}_{\mathrm{off}})/2{t}_{{s}})] & {t}_{\mathrm{off}}\lt t\leqslant 2{t}_{{s}}+{t}_{{g}},\end{array}\right.\end{eqnarray}$
$\begin{eqnarray}{\omega }_{d}(t)=\left\{\begin{array}{ll}{\omega }_{x}{\sin }^{2}(\pi t/2{t}_{{\rm{s}}})+{\omega }_{s}-{\omega }_{x} & 0\leqslant t\leqslant {t}_{{\rm{s}}\ },\\ {\omega }_{s} & {t}_{{\rm{s}}}\lt t\leqslant {t}_{{\rm{s}}}+{t}_{{\rm{g}}}={t}_{\mathrm{off}},\\ {\omega }_{x}[1-{\sin }^{2}(\pi (t-{t}_{\mathrm{off}})/2{t}_{{\rm{s}}})]+{\omega }_{s}-{\omega }_{x} & {t}_{\mathrm{off}}\lt t\leqslant 2{t}_{{\rm{s}}}+{t}_{{\rm{g}}},\end{array}\right.\end{eqnarray}$
with ωx = 50 g.Specifically, this full gate includes three steps: step 1, at the beginning we set Ω⊥(t = 0) = 0 and ωd(t = 0) = ωs − ωx, then turning on the switch, after the period of time ts, the initial spin state ∣0⟩1∣1⟩2 in the bare basis can be transferred to the dressed basis and turns to $| \tilde{0}{\rangle }_{1}| \tilde{1}{\rangle }_{2}$. In particular, when t = ts, we have Ω⊥(ts) = ωx and ωd(ts) = ωs, which means that the protected mechanism mentioned in above section starts; step 2, in the whole period of time tg (t changes from ts to ts + tg), we fix Ω⊥(t) = ωx and ωd(t) = ωs, thus, the protected entangled state $| {\tilde{\psi }}_{+}\rangle $ can be produced in time t = ts + tg; step 3, turning off the switch in time t > ts + tg, then the second adiabatic process starts. In time t = 2ts + tg, $| {\tilde{\psi }}_{+}\rangle $ can be totally sent back to the bare basis and the final entangled state becomes ∣ψ+⟩. According to our full gate procedures, ts corresponds to the necessary time to perform the adiabatic process. Thus, to guarantee the high fidelity of the state conversion, the switch speed measured by ts cannot be too fast. Here we choose ts = 5 × 10−2tg, which can be optimized further.
The numerical simulation for the full gate protocol is calculated by solving mater equation (4 ), in which Ω⊥, Δs and Δ should be substituted by Ω⊥(t), Δs(t) and Δ(t). Similarly, as the average fidelity defined in the dressed basis, the average fidelity for the full gate is equal to Ffull(t) = ∫Ff(ξ, t)P(ξ)dξ with Ff being the fidelity of ∣ψ+⟩. The time evolution of Ffull with ${T}_{2}^{* }=5$ μs is plotted in figure 8. It is shown that the full gate fidelity experiences two dramatic changes, which correspond to the two adiabatic processes. Further, around t = tg, the finally entangled state ∣ψ+⟩ is produced. More importantly, according to the numerical result shown in figure 8, we find that even the two additional adiabatical processes are included, with the appropriate driving modulation, high-fidelity 0.998 is still possible to reach. It is worthy to indicate that even we assume that the magnetic noise ξj is dynamic, for example, ξj is modeled as an Ornstein–Uhlenbeck process [61], a high-quality gate with fidelity as high as 0.99 can be still realized. Since the main purpose of this paper is to discuss the protected mechanism of the high-fidelity gate induced by the external driving field, we would not discuss the case of dynamic noise more detail in here.
Figure 8. The time evolution of the full gate fidelity. Here, ωx = 50 g, ts = 5 × 10−2tg, ${T}_{2}^{* }=5$ μs and Δ = 75 g. Other relevant parameters are the same as those in figure 4. |
6. Conclusion
By utilizing the CDD method, we have provided a feasible proposal to realize a high-fidelity entangling gate based on the strain coupling between two SiV centers and a phonon mode. In order to reveal more essential reasons for the entangling gate modified by the mechanical driving field, we derived the reduced spin–spin Hamiltonian and the corresponding master equation in the dispersive regime. We have shown that the effective spin–spin coupling can be easily controlled by adjusting the Rabi frequency of the mechanical field and the spin-phonon detuning. Moreover, by directly simulating the gate fidelity in the dressed basis, we found that in the unprotected case, the gate fidelity is very sensitive to ξ. In particular, once ξ > 0.1 g, the entangling gate can be remarkably damaged. However, in the protected case (with a stronger Ω⊥), even ξ changing from −g to +g, the gate fidelity can be maintained at a high value. In addition, we optimized the gate further by studying the dependence of the average maximum gate fidelity on the spin-phonon detuning. Our numerical results showed that in the realistic experimental condition, the average maximum gate fidelity under the optimal detuning can be optimized to 0.99. Further, a more real situation was considered, in which two additional processes of the adiabatic state transfer were included. Here, we used the sine-shaped modulation for the driving field to implement the two adiabatic processes. The detailed procedures for building the full quantum gate with appropriate driving modulation was well explained. Further, by calculating the fidelity of the full gate, we showed that even though the two adiabatic processes are included, the full gate fidelity was still capable to reach 0.99. Those results indicate that the presence of the external driving field can effectively protect the entangling gate from noise, which has potential applications in quantum computing and quantum simulation.
Appendix: The derivation of the effective spin–spin Hamiltonian
In this appendix , we outline the derivation of the effective Hamiltonian equation (9 ) and the master equation for the reduced density operator of the dressed spins equation (12 ). Starting from equation (8 ), by applying a Schrieffer–Wolff transformation and up to second order in λj, the effective spin–spin interaction Hamiltonian can be written asA1 ) is $\tfrac{{\rm{i}}}{2}[S,{\tilde{H}}_{I}]$. Since, the expressions of S and ${\tilde{H}}_{I}$ are very complicated, for convenience, we rewrite S and ${\tilde{H}}_{I}$ as $S={\sum }_{j}({S}_{1}^{j}+{S}_{2}^{j}+{S}_{3}^{j})$ and ${\tilde{H}}_{I}={\sum }_{j}({\tilde{H}}_{{\rm{I}}1}^{j}+{\tilde{H}}_{{\rm{I}}2}^{j}+{\tilde{H}}_{{\rm{I}}3}^{j})$ with
$\begin{eqnarray}\begin{array}{rcl}{\tilde{H}}_{\mathrm{eff}} & = & {{\rm{e}}}^{{\rm{i}}S}\tilde{H}{{\rm{e}}}^{-{\rm{i}}S}\\ & = & \mathop{\underbrace{{\rm{\Delta }}{a}^{\dagger }a+{\tilde{H}}_{s}}}\limits_{{\tilde{H}}_{0}}+{\tilde{H}}_{I}+{\rm{i}}[S,{\tilde{H}}_{0}]+{\rm{i}}[S,{\tilde{H}}_{I}]\\ & & -\displaystyle \frac{1}{2}[S,[S,{\tilde{H}}_{0}]]+\cdots ,\end{array}\end{eqnarray}$
with $S={\rm{i}}{\sum }_{j}[\tfrac{{\zeta }_{j}}{{\rm{\Delta }}}{\tilde{\sigma }}_{z}^{j}({a}^{\dagger }-a)+\tfrac{{\lambda }_{j}}{{{\rm{\Delta }}}_{-}^{j}}({\tilde{\sigma }}_{+}^{j}a-{a}^{\dagger }{\tilde{\sigma }}_{-}^{j})+\tfrac{{\eta }_{j}}{{{\rm{\Delta }}}_{+}^{j}}({\tilde{\sigma }}_{+}^{j}{a}^{\dagger }-a{\tilde{\sigma }}_{-}^{j})],$ ${{\rm{\Delta }}}_{\pm }^{j}=| {\rm{\Delta }}\pm {{\rm{\Omega }}}_{j}| \gg g$. Meanwhile, the operator S obeys the commutation relation ${\rm{i}}[S,{\tilde{H}}_{0}]=-{\tilde{H}}_{I}$. Thus, the remaining term in equation ( $\begin{eqnarray}\begin{array}{rcl}{S}_{1}^{j} & = & {\rm{i}}\displaystyle \frac{{\zeta }_{j}}{{\rm{\Delta }}}{\tilde{\sigma }}_{z}^{j}({a}^{\dagger }-a),\\ {\tilde{H}}_{{\rm{I}}1}^{j} & = & {\zeta }_{j}{\tilde{\sigma }}_{z}^{j}({a}^{\dagger }+a),\\ {S}_{2}^{j} & = & {\rm{i}}\displaystyle \frac{{\lambda }_{j}}{{{\rm{\Delta }}}_{-}^{j}}({\tilde{\sigma }}_{+}^{j}a-{a}^{\dagger }{\tilde{\sigma }}_{-}^{j}),\\ {\tilde{H}}_{{\rm{I}}2}^{j} & = & {\lambda }_{j}({\tilde{\sigma }}_{+}^{j}a+{\tilde{\sigma }}_{-}^{j}{a}^{\dagger }),\\ {S}_{3}^{j} & = & {\rm{i}}\displaystyle \frac{{\eta }_{j}}{{{\rm{\Delta }}}_{+}^{j}}({\tilde{\sigma }}_{+}^{j}{a}^{\dagger }-a{\tilde{\sigma }}_{-}^{j}),\\ {\tilde{H}}_{{\rm{I}}3}^{j} & = & {\eta }_{j}({\tilde{\sigma }}_{+}^{j}{a}^{\dagger }+{\tilde{\sigma }}_{-}^{j}a).\end{array}\end{eqnarray}$
Then the term $\tfrac{{\rm{i}}}{2}[S,{\tilde{H}}_{I}]$ becomesA3 ) is very cumbersome, but many terms, such as ${a}^{2}{\sigma }_{\pm }^{j}$, ${\sigma }_{\pm }^{j}{\sigma }_{z}^{j}$, etc., describe energy-non-conserving interactions and can be neglected as long as $| {{\rm{\Omega }}}_{j}| ,| {\rm{\Delta }}| ,| {{\rm{\Delta }}}_{\pm }^{j}| \gg g$. Therefore, we keep only resonant contributions, which can be divided into single qubit Stark shifts and qubit–qubit interaction. The Stark shifts Hamiltonian shown in equation (10 ) can be obtained from the term $\tfrac{{\rm{i}}}{2}{\sum }_{j}\{[{S}_{2}^{j},{\tilde{H}}_{{\rm{I}}2}^{j}]+[{S}_{3}^{j},{\tilde{H}}_{{\rm{I}}3}^{j}]\}$, which gives11 ) is obtained from the term $\tfrac{{\rm{i}}}{2}{\sum }_{i\ne j}\{[{S}_{2}^{j},{\tilde{H}}_{{\rm{I}}2}^{i}]+[{S}_{3}^{j},{\tilde{H}}_{{\rm{I}}3}^{i}]\}$, which yields13 ) and (14 ) shown in the main text can be recovered.
$\begin{eqnarray}\displaystyle \frac{{\rm{i}}}{2}[S,{\tilde{H}}_{I}]=\displaystyle \frac{{\rm{i}}}{2}\sum _{j}[{S}_{1}^{j}+{S}_{2}^{j}+{S}_{3}^{j},{\tilde{H}}_{{\rm{I}}1}^{j}+{\tilde{H}}_{{\rm{I}}2}^{j}+{\tilde{H}}_{{\rm{I}}3}^{j}].\end{eqnarray}$
The final full expression in equation ( $\begin{eqnarray}\begin{array}{rcl}{\tilde{H}}_{\mathrm{stark}} & = & \displaystyle \frac{{\rm{i}}}{2}\{[{S}_{2}^{1},{\tilde{H}}_{{\rm{I}}2}^{1}]+[{S}_{2}^{2},{\tilde{H}}_{{\rm{I}}2}^{2}]\\ & & +\left[{S}_{3}^{1},{\tilde{H}}_{{\rm{I}}3}^{1}\right]+[{S}_{3}^{2},{\tilde{H}}_{{\rm{I}}3}^{2}]\}\\ & = & -\delta {{\rm{\Omega }}}_{j}({a}^{\dagger }a+\displaystyle \frac{1}{2}){\tilde{\sigma }}_{z}^{j},\end{array}\end{eqnarray}$
with $\delta {{\rm{\Omega }}}_{j}=\tfrac{{\lambda }_{j}^{2}}{{{\rm{\Delta }}}_{-}^{j}}-\tfrac{{\eta }_{j}^{2}}{{{\rm{\Delta }}}_{+}^{j}}$. Correspondingly, the spin–spin effective Hamiltonian shown in equation ( $\begin{eqnarray}\begin{array}{rcl}{\tilde{H}}_{s-s} & = & \displaystyle \frac{{\rm{i}}}{2}\{[{S}_{2}^{1},{\tilde{H}}_{{\rm{I}}2}^{2}]+[{S}_{2}^{2},{\tilde{H}}_{{\rm{I}}2}^{1}]\\ & & +[{S}_{3}^{1},{\tilde{H}}_{{\rm{I}}3}^{2}]+[{S}_{3}^{2},{\tilde{H}}_{{\rm{I}}3}^{1}]\},\\ & = & -\displaystyle \sum _{{ij}}{J}_{z}^{{ij}}{\tilde{\sigma }}_{z}^{i}{\tilde{\sigma }}_{z}^{j}-\sum _{{ij}}{J}_{\mathrm{eff}}^{{ij}}({\tilde{\sigma }}_{-}^{i}{\tilde{\sigma }}_{+}^{j}+{\tilde{\sigma }}_{+}^{i}{\tilde{\sigma }}_{-}^{j}),\end{array}\end{eqnarray}$
where ${J}_{z}^{{ij}}=\tfrac{{\zeta }_{j}^{2}}{4{\rm{\Delta }}}$ and ${J}_{\mathrm{eff}}^{{ij}}=\tfrac{{\lambda }_{i}{\lambda }_{j}}{4}\left(\tfrac{1}{{{\rm{\Delta }}}_{-}^{i}}+\tfrac{1}{{{\rm{\Delta }}}_{-}^{j}}\right)+\tfrac{{\eta }_{i}{\eta }_{j}}{4}\left(\tfrac{1}{{{\rm{\Delta }}}_{+}^{i}}+\tfrac{1}{{{\rm{\Delta }}}_{+}^{j}}\right)$. Thus, the effective Hamiltonian becomes ${\tilde{H}}_{\mathrm{eff}}={\tilde{H}}_{0}+{\tilde{H}}_{\mathrm{stark}}+{\tilde{H}}_{s-s}$. Moreover, if we ignore the environment fluctuations (ξj = 0), then equations (Since we focus on the regime γm ≫ γs, it is reasonable to ignore the dephasing rates of spins. Under the Schrieffer–Wolff transformation and γs = 0, we can obtain the master equation in the dressed basisA7 ) can be further expressed asA8 ) into (A6 ) and tracing out the phonons, we can obtain the master equation of the spin–spin system
$\begin{eqnarray}\dot{\tilde{\rho }}=-{\rm{i}}[{\tilde{H}}_{\mathrm{eff}},\tilde{\rho }]+\displaystyle \frac{{\gamma }_{m}}{2}(2\tilde{a}\tilde{\rho }{\tilde{a}}^{\dagger }-{\tilde{a}}^{\dagger }\tilde{a}\tilde{\rho }-\tilde{\rho }{\tilde{a}}^{\dagger }\tilde{a}),\end{eqnarray}$
in which $\tilde{a}$ and ${\tilde{a}}^{\dagger }$ can be expanded as $\begin{eqnarray}\begin{array}{rcl}\tilde{a} & = & {{\rm{e}}}^{-{\rm{i}}S}a{{\rm{e}}}^{{\rm{i}}S}=a+\displaystyle \sum _{j}\displaystyle \frac{{\zeta }_{j}}{{\rm{\Delta }}}{\tilde{\sigma }}_{z}^{j}\\ & & -\displaystyle \sum _{j}\displaystyle \frac{{\lambda }_{j}}{{{\rm{\Delta }}}_{-}^{j}}{\tilde{\sigma }}_{-}^{j}-\displaystyle \sum _{j}\displaystyle \frac{{\eta }_{j}}{{{\rm{\Delta }}}_{+}^{j}}{\tilde{\sigma }}_{+}^{j},\\ {\tilde{a}}^{\dagger } & = & {{\rm{e}}}^{-{\rm{i}}S}{a}^{\dagger }{{\rm{e}}}^{{\rm{i}}S}={a}^{\dagger }-\displaystyle \sum _{j}\displaystyle \frac{{\zeta }_{j}}{{\rm{\Delta }}}{\tilde{\sigma }}_{z}^{j}\\ & & -\displaystyle \sum _{j}\displaystyle \frac{{\lambda }_{j}}{{{\rm{\Delta }}}_{-}^{j}}{\tilde{\sigma }}_{+}^{j}-\displaystyle \sum _{j}\displaystyle \frac{{\eta }_{j}}{{{\rm{\Delta }}}_{+}^{j}}{\tilde{\sigma }}_{-}^{j}.\end{array}\end{eqnarray}$
In the dispersive regime (∣Δ∣ and ${{\rm{\Delta }}}_{\pm }^{j}=| {\rm{\Delta }}\pm {{\rm{\Omega }}}_{j}| \gg g$), the term $\tfrac{{\eta }_{j}}{{{\rm{\Delta }}}_{+}^{j}}$ is very small and can be ignored. Hence, equation ( $\begin{eqnarray}\begin{array}{rcl}\tilde{a} & \simeq & a+\displaystyle \sum _{j}\displaystyle \frac{{\zeta }_{j}}{{\rm{\Delta }}}{\tilde{\sigma }}_{z}^{j}-\displaystyle \sum _{j}\displaystyle \frac{{\lambda }_{j}}{{{\rm{\Delta }}}_{-}^{j}}{\tilde{\sigma }}_{-}^{j},\\ {\tilde{a}}^{\dagger } & \simeq & {a}^{\dagger }-\displaystyle \sum _{j}\displaystyle \frac{{\zeta }_{j}}{{\rm{\Delta }}}{\tilde{\sigma }}_{z}^{j}-\displaystyle \sum _{j}\displaystyle \frac{{\lambda }_{j}}{{{\rm{\Delta }}}_{-}^{j}}{\tilde{\sigma }}_{+}^{j}.\end{array}\end{eqnarray}$
Substituting equations ( $\begin{eqnarray}\begin{array}{rcl}{\dot{\tilde{\rho }}}_{s} & = & -{\rm{i}}\left[{\tilde{H}}_{\mathrm{eff}}-({\rm{\Delta }}-\displaystyle \sum _{j}\delta {{\rm{\Omega }}}_{j}{\tilde{\sigma }}_{z}^{j}){a}^{\dagger }a,{\tilde{\rho }}_{s}\right]\\ & & +\displaystyle \sum _{i,j}\displaystyle \frac{{{\rm{\Gamma }}}_{\perp }^{{ij}}}{2}(2{\tilde{\sigma }}_{-}^{i}{\tilde{\rho }}_{s}{\tilde{\sigma }}_{+}^{j}-{\tilde{\sigma }}_{+}^{j}{\tilde{\sigma }}_{-}^{i}{\tilde{\rho }}_{s}-{\tilde{\rho }}_{s}{\tilde{\sigma }}_{+}^{j}{\tilde{\sigma }}_{-}^{i})\\ & & +\displaystyle \sum _{i,j}\displaystyle \frac{{{\rm{\Gamma }}}_{\parallel }^{{ij}}}{2}(2{\tilde{\sigma }}_{z}^{i}{\tilde{\rho }}_{s}{\tilde{\sigma }}_{z}^{j}-{\tilde{\sigma }}_{z}^{j}{\tilde{\sigma }}_{z}^{i}{\tilde{\rho }}_{s}-{\tilde{\rho }}_{s}{\tilde{\sigma }}_{z}^{j}{\tilde{\sigma }}_{z}^{i}),\end{array}\end{eqnarray}$
where ${{\rm{\Gamma }}}_{\perp }^{{ij}}={\gamma }_{m}\tfrac{{\lambda }_{i}{\lambda }_{j}}{{{\rm{\Delta }}}_{-}^{i}{{\rm{\Delta }}}_{-}^{j}}$ and ${{\rm{\Gamma }}}_{\parallel }^{{ij}}={\gamma }_{m}\tfrac{{\zeta }_{i}{\zeta }_{j}}{{{\rm{\Delta }}}^{2}}$.In the absence of the mechanical control field, the effective spin–spin effective Hamiltonian becomes
$\begin{eqnarray}\begin{array}{rcl}{H}_{\mathrm{eff}} & = & \mathop{\underbrace{({\omega }_{m}-\sum _{j}\displaystyle \frac{{g}^{2}}{{{\rm{\Delta }}}_{j}}{\sigma }_{z}^{j}){a}^{\dagger }a}}\limits_{{H}_{a}}+\displaystyle \sum _{j}\displaystyle \frac{{\omega }_{j}}{2}{\sigma }_{z}^{j}\\ & & -\displaystyle \sum _{i,j,i\ne j}\displaystyle \frac{{g}^{2}({{\rm{\Delta }}}_{i}+{{\rm{\Delta }}}_{j})}{2{{\rm{\Delta }}}_{i}{{\rm{\Delta }}}_{j}}({\sigma }_{-}^{i}{\sigma }_{+}^{j}+{\sigma }_{+}^{i}{\sigma }_{-}^{i})\end{array}\end{eqnarray}$
with ${\omega }_{j}={\omega }_{s}^{j}-\tfrac{{g}^{2}}{{{\rm{\Delta }}}_{j}}$ and ${{\rm{\Delta }}}_{j}={\omega }_{m}-({\omega }_{s}^{j}+{\xi }_{j})$. Correspondingly, the master equation for spins becomes $\begin{eqnarray}\begin{array}{rcl}{\dot{\rho }}_{s} & = & -{\rm{i}}[{H}_{\mathrm{eff}}-{H}_{a},{\rho }_{s}]\\ & & +\displaystyle \sum _{i,j}\displaystyle \frac{{{\rm{\Gamma }}}_{{ij}}}{2}(2{\sigma }_{-}^{i}{\rho }_{s}{\sigma }_{+}^{j}-{\sigma }_{+}^{j}{\sigma }_{-}^{i}{\rho }_{s}-{\rho }_{s}{\sigma }_{+}^{j}{\sigma }_{-}^{i}),\end{array}\end{eqnarray}$
where ${{\rm{\Gamma }}}_{{ij}}={\gamma }_{m}\tfrac{{g}^{2}}{{{\rm{\Delta }}}_{i}{{\rm{\Delta }}}_{j}}$.