Optomechanical state transfer between two distant membranes in the presence of non-Markovian environments^*

Cavity optomechanical system is a well-developed tool to exhibit the squeezing and entanglement phenomena,^[1–4] and has recently received much attention^[5–7] due to its potential ability to realize the purpose of quantum information processing. The strong coupling between mechanical and optical modes opens up great avenues in efficiently transferring a quantum state between photons and phonons.^[8,9] When transferring a quantum state from one site to another,^[10] the mechanical oscillator with small decay rate serves as a potential quantum network node for quantum information storage and processing,^[11,12] which is also the central goal in quantum networking schemes. Therefore, state transfer in optomechanics is an important task, and it has been extensively studied.^[8,13–17] Recently, theoretical research of state transfer between distant mechanical oscillators^[18,19] or in optomechanical arrays^[20] has also been carried out.

The above-mentioned studies are all focussing on the scenario of memoryless environment, and the corresponding Markov approximation is used to derive the quantum Langevin equations.^[21] However, a recent experiment showed that the dynamics of microresonators are non-Markovian.^[22] Therefore, it is necessary to explore the non-Markovian effect in optomechanical systems. In recent years, a lot of theoretical research has been carried out on this topic, and it is proved that the backflow of information from the environments into the system known as the memory effect^[23] is helpful for entanglement protection,^[24,25] ground state cooling,^[26,27] and force detection^[28] in optomechanical system.

Previous studies have shown that the non-Markovian environments appear to be beneficial to quantum state transfer in distant cavities^[29] or qubits.^[30] However, it is still unknown whether the non-Markovian memory effect is helpful for state transfer between coupled optomechanical systems. In this paper, the quantum state transfer between two distant membranes is investigated in non-Markovian regime. The experimental mechanical spectral density^[22] is used and the exact state transfer efficiency is derived based on the modified Laplace transformation.^[31] Our results show that the quantum state transfer between distant membranes can be implemented with high efficiency in the presence of non-Markovian environments. We also find that if the tunneling strength between the two cavities gets smaller, the transfer time required for reaching the maximum value of efficiency increases. At this point, the long-time non-Markovian memory effect becomes significant. Thus, the performance of state transfer in non-Markovian environments will be much better than the Markovian case.

This article is organized as follows. Section 2 describes the model and gives the theoretical description of the dynamics. In Section 3, we investigate in detail the numerical results of quantum state transfer. Finally, the conclusion is given in Section 4.

We consider a cavity optomechanical system consisting of two membranes, which are placed separately in two distant cavities. The cavities are coupled to each other via an optical fiber. The Hamiltonian of this system can be written as

$$\begin{eqnarray}\begin{array}{lll}{\hat{H}}_{{\rm{S}}} & = & \displaystyle \sum _{i=1,2}\hslash [{\Omega }_{i}{\hat{a}}_{i}^{\dagger }{\hat{a}}_{i}+{\omega }_{i}{\hat{b}}_{i}^{\dagger }{\hat{b}}_{i}+{g}_{i}{\hat{a}}_{i}^{\dagger }{\hat{a}}_{i}({\hat{b}}_{i}^{\dagger }+{\hat{b}}_{i})\\ & & +E({{\rm{e}}}^{-{\rm{i}}{\omega }_{{\rm{L}}}t}{\hat{a}}_{i}^{\dagger }+{{\rm{e}}}^{{\rm{i}}{\omega }_{{\rm{L}}}t}{\hat{a}}_{i})]+\hslash J({\hat{a}}_{1}^{\dagger }{\hat{a}}_{2}+{\hat{a}}_{2}^{\dagger }{\hat{a}}_{1}),\end{array}\end{eqnarray} \tag{ 1 }$$

where â_i and ${\hat{b}}_{i}$ are annihilation operators of the i-th cavity field and the membrane, respectively. The cavity fields are coupled to the membranes via radiation pressure with coupling coefficients g_i, and the two cavities are both driven by coherent laser with driving strength E and center frequency ω_L. J gives the strength of the tunneling interaction between the two cavities. In the rotating frame with the laser frequency, and when the intracavity field is strong enough, the Hamiltonian of the system can be linearized as^[7,25,26]

$$\begin{eqnarray}\begin{array}{lll}{\hat{H}}_{{\rm{S}}} & = & \displaystyle \sum _{i=1,2}\hslash [{\Delta }_{i}{\hat{a}}_{i}^{\dagger }{\hat{a}}_{i}+{\omega }_{i}{\hat{b}}_{i}^{\dagger }{\hat{b}}_{i}+({G}_{i}{\hat{a}}_{i}^{\dagger }+{G}_{i}^{* }{\hat{a}}_{i})({\hat{b}}_{i}^{\dagger }+{\hat{b}}_{i})]\\ & & +\hslash J({\hat{a}}_{1}^{\dagger }{\hat{a}}_{2}+{\hat{a}}_{2}^{\dagger }{\hat{a}}_{1}).\end{array}\end{eqnarray} \tag{ 2 }$$

Here, G_i = α_igi is the effective coupling rate, where α_i = 〈â_i〈 is the mean value of the cavity field, and Δ_i ≈ Ω_i–ω_L – 2|G_i|²/ω_i is the optomechanical-coupling modified detuning.^[25] In the Born–Markovian approximation, the side-band condition requires that the cavity linewidth is smaller than the mechanical frequency, i.e., κ < ω_m. However, in the presence of non-Markovian environment, due to the interaction with the environment, the corresponding optical dissipation rates κ_i and the frequencies of the mechanical modes ω_i should be replaced by the modulated parameters, i.e., the effective optical dissipation rate κ_i → κ_eff and the effective mechanical frequency ω_i → ω_eff. Here, for simplicity, we assume ω₁ = ω₂ = ω_m and κ₁ = κ₂ = κ. Thus, the side-band condition can be rewritten as κ_eff < ω_eff. To be specific, the effective optical dissipation rate κ_eff is determined by the poles localized at the lower half of the complex plane corresponding to the branch cuts B_k.^[31] If the non-Markovian effect is not strong enough, then the coupling between the system and environment is much smaller than ω_m, we have ω_eff ≈ ω_m.^[27,32] Meanwhile, the effective optical dissipation is approximately equal to the dissipation rate under Markovian approximation, namely, ${\kappa }_{{\rm{eff}}}\approx {{\mathscr{J}}}_{{\rm{c}}}({\omega }_{{\rm{c}}})/2$. Thus, this model is valid under the red-detuned regime Δ_i = ω_m when the effective side-band condition κ_eff < ω_m is satisfied. In the red-detuned regime, by applying the rotating-wave approximation, the nonresonant term in Eq. (2) can be ignored. The Hamiltonian of the system then reads

$$\begin{eqnarray}\begin{array}{lll}{\hat{H}}_{{\rm{S}}} & = & \displaystyle \sum _{i=1,2}\hslash [\Delta {\hat{a}}_{i}^{\dagger }{\hat{a}}_{i}+{\omega }_{{\rm{m}}}{\hat{b}}_{i}^{\dagger }{\hat{b}}_{i}+G{\hat{a}}_{i}^{\dagger }{\hat{b}}_{i}+{G}^{* }{\hat{a}}_{i}{\hat{b}}_{i}^{\dagger }]\\ & & +\hslash J({\hat{a}}_{1}^{\dagger }{\hat{a}}_{2}+{\hat{a}}_{2}^{\dagger }{\hat{a}}_{1}).\end{array}\end{eqnarray} \tag{ 3 }$$

Here, we assume Δ₁ = Δ₂ = Δ and G₁ = G₂ = G. The environments can be described by a collection of independent harmonic oscillators, so the Hamiltonian of the environments can be expressed as

$$\begin{eqnarray}&&{\hat{H}}_{{\rm{E}}}=\displaystyle \sum _{i,k}\hslash {\Omega }_{ik}{\hat{a}}_{ik}^{\dagger }{\hat{a}}_{ik}+\hslash {\omega }_{ik}{\hat{b}}_{ik}^{\dagger }{\hat{b}}_{ik},\end{eqnarray} \tag{ 4 }$$

where Ω_ik and ω_ik are the reservoir frequency of the k-th optical and mechanical mode interacting with the i-th optomechanical system. The interaction between the system and the environment is given by

$$\begin{eqnarray}&&{\hat{H}}_{{\rm{I}}}=\displaystyle \sum _{i,k}\hslash {g}_{ik}({\hat{a}}_{ik}^{\dagger }{\hat{a}}_{i}+{\hat{a}}_{i}^{\dagger }{\hat{a}}_{ik})+\hslash {{g}^{^{\prime} }}_{ik}({\hat{b}}_{ik}^{\dagger }{\hat{b}}_{i}+{\hat{b}}_{i}^{\dagger }{\hat{b}}_{ik}),\end{eqnarray} \tag{ 5 }$$

where g_ik and ${g}_{ik}^{^{\prime} }$ are the system–bath coupling strength. Here, the interaction is written in rotating-wave-approximation form, it is valid^[33] when the coupling strength is weak compared to the system, i.e., ${g}_{ik},{g}_{ik}^{^{\prime} }\ll {\omega }_{{\rm{m}}}$.

The Heisenberg equations of the system can be derived from the Hamiltonian given above, which are

$$\begin{eqnarray}\begin{array}{lll}{\dot{\hat{a}}}_{1} & = & -{\rm{i}}\Delta {\hat{a}}_{1}-{\rm{i}}G{\hat{b}}_{1}-{\rm{i}}J{\hat{a}}_{2}-{\rm{i}}\displaystyle \sum _{k}{g}_{1k}{\hat{a}}_{1k},\\ {\dot{\hat{a}}}_{2} & = & -{\rm{i}}\Delta {\hat{a}}_{2}-{\rm{i}}G{\hat{b}}_{2}-{\rm{i}}J{\hat{a}}_{1}-{\rm{i}}\displaystyle \sum _{k}{g}_{2k}{\hat{a}}_{2k},\\ {\dot{\hat{b}}}_{1} & = & =-{\rm{i}}{\omega }_{{\rm{m}}}{\hat{b}}_{1}-{\rm{i}}G{\hat{a}}_{1}-{\rm{i}}\displaystyle \sum _{k}{g}_{1k}^{^{\prime} }{\hat{b}}_{1k},\\ {\dot{\hat{b}}}_{2} & = & -{\rm{i}}{\omega }_{{\rm{m}}}{\hat{b}}_{2}-{\rm{i}}G{\hat{a}}_{2}-{\rm{i}}\displaystyle \sum _{k}{g}_{2k}^{^{\prime} }{\hat{b}}_{2k},\end{array}\end{eqnarray} \tag{ 6 }$$

and the Heisenberg equations of the environments read

$$\begin{eqnarray}\begin{array}{lll}{\dot{\hat{a}}}_{1k} & = & -{\rm{i}}{\Omega }_{1k}{\hat{a}}_{1k}-{\rm{i}}{g}_{1k}{\hat{a}}_{1},\\ {\dot{\hat{a}}}_{2k} & = & -{\rm{i}}{\Omega }_{2k}{\hat{a}}_{2k}-{\rm{i}}{g}_{2k}{\hat{a}}_{2},\\ {\dot{\hat{b}}}_{1k} & = & -{\rm{i}}{\omega }_{1k}{\hat{b}}_{1k}-{\rm{i}}{g}_{1k}^{^{\prime} }{\hat{b}}_{1},\\ {\dot{\hat{b}}}_{2k} & = & -{\rm{i}}{\omega }_{2k}{\hat{b}}_{2k}-{\rm{i}}{g}_{2k}^{^{\prime} }{\hat{b}}_{2}.\end{array}\end{eqnarray} \tag{ 7 }$$

Solving Eq. (7) for â_ik and ${\hat{b}}_{\mathrm{ik}}$, then substituting their solutions into Eq. (6), we can obtain the following integro-differential Heisenberg equations

$$\begin{eqnarray}\begin{array}{lll}{\dot{\hat{a}}}_{1} & = & -{\rm{i}}\Delta {\hat{a}}_{1}-{\rm{i}}G{\hat{b}}_{1}-{\rm{i}}J{\hat{a}}_{2}-\displaystyle {\int }_{0}^{t}{\rm{d}}\tau {f}_{{\rm{c}}}(t-\tau ){\hat{a}}_{1}(\tau )+{\hat{a}}_{{\rm{in}}},\\ {\dot{\hat{a}}}_{2} & = & -{\rm{i}}\Delta {\hat{a}}_{2}-{\rm{i}}G{\hat{b}}_{2}-{\rm{i}}J{\hat{a}}_{1}-\displaystyle {\int }_{0}^{t}{\rm{d}}\tau {f}_{{\rm{c}}}(t-\tau ){\hat{a}}_{2}(\tau )+{\hat{a}}_{{\rm{in}}},\\ {\dot{\hat{b}}}_{1} & = & -{\rm{i}}{\omega }_{{\rm{m}}}{\hat{b}}_{1}-{\rm{i}}G{\hat{a}}_{1}-\displaystyle {\int }_{0}^{t}{\rm{d}}\tau {f}_{{\rm{m}}}(t-\tau ){\hat{b}}_{1}(\tau )+{\hat{b}}_{{\rm{in}}},\\ {\dot{\hat{b}}}_{2} & = & -{\rm{i}}{\omega }_{{\rm{m}}}{\hat{b}}_{2}-{\rm{i}}G{\hat{a}}_{2}-\displaystyle {\int }_{0}^{t}{\rm{d}}\tau {f}_{{\rm{m}}}(t-\tau ){\hat{b}}_{2}(\tau )+{\hat{b}}_{{\rm{in}}}.\end{array}\end{eqnarray} \tag{ 8 }$$

When deriving Eq. (8), for simplicity, we also assume the identical environments of the cavities and membranes respectively, namely, the cavity–bath/membrane–bath coupling strength and the cavity/membrane environmental density of states are the same. Therefore, we have Ω_ik = Ω_k, ω_ik = ω_k, g_ik = g_k, and ${g}_{ik}^{^{\prime} }={g}_{k}^{^{\prime} }$. In Eq. (8), â_in = i∑_kg_ke^-iΩ_ktâ_ik(0) and ${\hat{b}}_{{\rm{in}}}=-{\rm{i}}\displaystyle \sum _{k}{g}_{k}^{^{\prime} }{{\rm{e}}}^{-{\rm{i}}{\omega }_{k}t}{\hat{b}}_{ik}(0)$ are the input noise operators which depend on the initial condition of the environment. The non-Markovian effect is fully manifested in Eq. (8) through the non-local time correlation functions ${f}_{{\rm{c}}}(t)=\displaystyle \sum _{k}{g}_{k}^{2}{{\rm{e}}}^{-{\rm{i}}{\Omega }_{k}t}$ and ${f}_{{\rm{m}}}(t)=\displaystyle \sum _{k}{{g}^{^{\prime} }}_{k}^{2}{{\rm{e}}}^{-{\rm{i}}{\omega }_{k}t}$,^[24] where g_k and ${g}_{k}^{^{\prime} }$ are assumed to be real. By introducing the spectral density of the reservoir, one can rewrite the correlation functions,

$$\begin{eqnarray*}&&{f}_{{\rm{c}}}(t)=\displaystyle \int \frac{{\rm{d}}\omega }{2\pi }{J}_{{\rm{c}}}(\omega ){{\rm{e}}}^{-{\rm{i}}\omega t}\end{eqnarray*}$$

and

$$\begin{eqnarray*}&&{f}_{{\rm{m}}}(t)=\displaystyle \int \frac{{\rm{d}}\omega }{2\pi }{J}_{{\rm{m}}}(\omega ){{\rm{e}}}^{-{\rm{i}}\omega t},\end{eqnarray*}$$

where J_c(ω) and J_m(ω) are the spectral density of the optical and mechanical environment, respectively.

For general bosonic environment, the spectral density can be an Ohmic spectrum with a Poisson-type distribution function.^[34] For the cavity, usually the spectral density could be ${J}_{{\rm{c}}}(\omega )=2\pi {\eta }_{{\rm{c}}}\omega {(\omega /{\mathop{\omega }\limits^{\sim }}_{{\rm{c}}})}^{{s}_{{\rm{c}}}-1}{{\rm{e}}}^{-\omega /{\mathop{\omega }\limits^{\sim }}_{{\rm{c}}}}$, where η_c is a dimensionless coupling strength between system and environment, and ${\mathop{\omega }\limits^{\sim }}_{{\rm{c}}}$ is a high-frequency cutoff.^[34,35] The parameter s_c classifies the environment as sub-Ohmic (0 < s_c < 1), Ohmic (s_c = 1), and super-Ohmic (s_c > 1). Recently, the spectral density of the mechanical environment is measured through experiment.^[22] We therefore use this experimental spectral density as the membrane's environment, which can be described by J_m(ω) = 2πCω^k, where C is the coupling constant and k ≈ –2.3±1.05. The bandwidth of the reservoir is Γ ≈ 0.07ω_m, where ω_m = 914 kHz is the mechanical resonance frequency. So the frequency region is ω ∈ [ω_min,ω_max], with ω_min = 885 kHz and ω_max = 945 kHz.

Now we rewrite Eq. (8) in a more compact form by defining $\hat{O}(t)\equiv {[{\hat{a}}_{1}(t),{\hat{a}}_{2}(t),{\hat{b}}_{1}(t),{\hat{a}}_{2}(t)]}^{{\rm{T}}}$ which are

$$\begin{eqnarray}&&\dot{\hat{O}}(t)=-{\rm{i}}M\hat{O}(t)-\displaystyle {\int }_{0}^{t}{\rm{d}}\tau \bar{F}(t-\tau )\hat{O}(\tau )+{\hat{\xi }}_{{\rm{in}}}(t),\end{eqnarray} \tag{ 9 }$$

where the noise term ${\hat{\xi }}_{{\rm{in}}}(t)={[{\hat{a}}_{{\rm{in}}},{\hat{a}}_{{\rm{in}}},{\hat{b}}_{{\rm{in}}},{\hat{b}}_{{\rm{in}}}]}^{{\rm{T}}}$, and the matrix M and $\bar{F}(t)$ is given by

$$\begin{eqnarray}M=\left(\begin{array}{cccc}\Delta & J & G & 0\\ J & \Delta & 0 & G\\ G & 0 & {\omega }_{{\rm{m}}} & 0\\ 0 & G & 0 & {\omega }_{{\rm{m}}}\end{array}\right),\\ \bar{F}(t)=\left(\begin{array}{llll}{f}_{{\rm{c}}}(t) & 0 & 0 & 0\\ 0 & {f}_{{\rm{c}}}(t) & 0 & 0\\ 0 & 0 & {f}_{{\rm{m}}}(t) & 0\\ 0 & 0 & 0 & {f}_{{\rm{m}}}(t)\end{array}\right).\end{eqnarray} \tag{ 10 }$$

Equation (9) can be formally solved by assuming $\hat{O}(t)={\mathscr{U}}(t)\hat{O}(0)+\hat{{\mathscr{V}}}(t)$, where the 4 × 4 matrix functions ${\mathscr{U}}(t)$ and $\hat{{\mathscr{V}}}(t)$ are related to the non-equilibrium Green's functions^[36,37] of the system. These Green's functions obey the following Dyson equations:

$$\begin{eqnarray}\begin{array}{lll}\mathop{{\mathscr{U}}}\limits^{.}(t) & = & -{\rm{i}}M{\mathscr{U}}(t)-\displaystyle {\int }_{0}^{t}{\rm{d}}\tau \bar{F}(t-\tau ){\mathscr{U}}(\tau ),\\ \mathop{\hat{{\mathscr{V}}}}\limits^{.}(t) & = & -{\rm{i}}M\hat{{\mathscr{V}}}(t)-\displaystyle {\int }_{0}^{t}{\rm{d}}\tau \bar{F}(t-\tau )\hat{{\mathscr{V}}}(\tau )+{\hat{\xi }}_{{\rm{in}}}(t).\end{array}\end{eqnarray} \tag{ 11 }$$

The integro-differential equations (11) can be solved by utilizing the modified Laplace transformation.^[31] Note that $\hat{{\mathscr{V}}}(t)$ satisfies the initial condition $\hat{{\mathscr{V}}}(0)=0$, therefore, we can first give the following formal solution:

$$\begin{eqnarray}&&\hat{{\mathscr{V}}}(t)=\displaystyle {\int }_{0}^{t}{\rm{d}}\tau {\mathscr{U}}(t-\tau ){\hat{\xi }}_{{\rm{in}}}(\tau ).\end{eqnarray} \tag{ 12 }$$

Obviously, ${\mathscr{U}}(t)$ reveals the general properties of non-Markovian dynamics. Using the modified Laplace transform ${\mathscr{U}}(z)= {\mathcal L} \{{\mathscr{U}}(t)\}=\displaystyle {\int }_{0}^{\infty }{\mathscr{U}}(t){{\rm{e}}}^{{\rm{i}}zt}{\rm{d}}t$ and in view of the initial condition ${\mathscr{U}}(0)=1$, we have

$$\begin{eqnarray}&&-{\rm{i}}z{\mathscr{U}}(z)-I=-{\rm{i}}M{\mathscr{U}}(z)-\Sigma (z){\mathscr{U}}(z),\end{eqnarray} \tag{ 13 }$$

where I is the identity, Σ(z) is the Laplace transform of the self-energy correction,^[31]

$$\begin{eqnarray}\Sigma (z)= {\mathcal L} \{\bar{F}(t)\}=\displaystyle \int \frac{{\rm{d}}\omega }{2\pi }\left(\begin{array}{cccc}{J}_{{\rm{c}}}(\omega )/(z-\omega ) & 0 & 0 & 0\\ 0 & {J}_{{\rm{c}}}(\omega )/(z-\omega ) & 0 & 0\\ 0 & 0 & {J}_{{\rm{m}}}(\omega )/(z-\omega ) & 0\\ 0 & 0 & 0 & {J}_{{\rm{m}}}(\omega )/(z-\omega )\end{array}\right).\end{eqnarray} \tag{ 14 }$$

Solving Eq. (13) and applying the inverse Laplace transformation, we obtain the following Bromwich integral:

$$\begin{eqnarray}&&{\mathscr{U}}(t)={ {\mathcal L} }^{-1}\{{\mathscr{U}}(z)\}=\frac{1}{2\pi }\displaystyle {\int }_{-\infty +{\rm{i}}\lambda }^{\infty +{\rm{i}}\lambda }{\rm{d}}z\frac{{{\rm{ie}}}^{-{\rm{i}}zt}}{z-M-\Sigma (z)},\end{eqnarray} \tag{ 15 }$$

where λ is an arbitrary positive number greater than the imaginary part of all the poles of i/(z–M–Σ(z)).

With the solution ${\mathscr{U}}(t)$, we can exactly describe the dynamical evolution of ${\hat{b}}_{2}(t)$ , namely, ${\hat{b}}_{2}(t)={{\mathscr{U}}}_{41}(t){\hat{a}}_{1}(0)+{{\mathscr{U}}}_{42}(t){\hat{a}}_{2}(0)+{{\mathscr{U}}}_{43}(t){\hat{b}}_{1}(0)+{{\mathscr{U}}}_{44}(t){\hat{b}}_{2}(0)+{\hat{{\mathscr{V}}}}_{4}(t)$. If the state is initially encoded onto the first membrane ${\hat{b}}_{1}(0)$ and the second membrane is initially in a vacuum state, the state-independent efficiency^[38,39] can be defined as the coefficient in front of ${\hat{b}}_{1}(0)$. In order to characterize the performance of the state transfer from the first membrane to the second membrane, we can take ${{\mathscr{U}}}_{43}$ as a criterion, and assume the efficiency $E=|{{\mathscr{U}}}_{43}|$, where ${{\mathscr{U}}}_{43}$ is given by

$$\begin{eqnarray}&&{{\mathscr{U}}}_{43}(t)=\frac{1}{2\pi }\displaystyle {\int }_{-\infty +{\rm{i}}\lambda }^{\infty +{\rm{i}}\lambda }{\rm{d}}z\frac{{\rm{i}}J{G}^{2}{{\rm{e}}}^{-{\rm{i}}zt}}{{({G}^{2}-{K}_{{\rm{m}}}{K}_{{\rm{c}}})}^{2}-{J}^{2}{K}_{{\rm{m}}}^{2}},\end{eqnarray} \tag{ 16 }$$

where

$$\begin{eqnarray*}\begin{array}{lll}{K}_{{\rm{m}}} & = & z-{\omega }_{{\rm{m}}}-\displaystyle \int \frac{{\rm{d}}\omega }{2\pi }\frac{{J}_{{\rm{m}}}(\omega )}{z-\omega },\\ {K}_{{\rm{c}}} & = & z-\Delta -\displaystyle \int \frac{{\rm{d}}\omega }{2\pi }\frac{{J}_{{\rm{c}}}(\omega )}{z-\omega }.\end{array}\end{eqnarray*}$$

In this section, we numerically solve Eq. (16) to show the performance of the state transfer. First, we discuss the comparison between Markovian and non-Markovian dynamics. The Born–Markovian approximation is usually used when the coupling strength between system and environment is very weak, and the characteristic correlation time of the environment is sufficiently shorter than that of the system.^[31,40] Therefore, no memory effect remains. In optomechanical system, the quantum Langevin equations^[21] was developed under the Born–Markovian approximation, in which the spectral density of the environment is assumed to be a flat spectrum, as it varies little around the system's frequency. In this case, according to Weisskopf–Wigner approximation,^[41] one can replace J_i(ω) by J_i(ω_i), where i = c,m, and extend the limit of frequency integration to infinite. Then the time integral in Eq. (11) reduces to

$$\begin{eqnarray}\begin{array}{ccc}\displaystyle {\int }_{0}^{t}{\rm{d}}\tau \bar{F}(t-\tau ){\mathscr{U}}(\tau ) & = & \displaystyle \int \frac{{\rm{d}}\omega }{2\pi }\left(\begin{array}{cccc}{J}_{{\rm{c}}}(\omega ) & 0 & 0 & 0\\ 0 & {J}_{{\rm{c}}}(\omega ) & 0 & 0\\ 0 & 0 & {J}_{{\rm{m}}}(\omega ) & 0\\ 0 & 0 & 0 & {J}_{{\rm{m}}}(\omega )\end{array}\right)\,\displaystyle {\int }_{0}^{t}{\rm{d}}\tau {{\rm{e}}}^{-{\rm{i}}\omega (t-\tau )}{\mathscr{U}}(\tau )\\ & \approx & \left(\begin{array}{cccc}{J}_{{\rm{c}}}({\omega }_{{\rm{c}}})/2 & 0 & 0 & 0\\ 0 & {J}_{{\rm{c}}}({\omega }_{{\rm{c}}})/2 & 0 & 0\\ 0 & 0 & {J}_{{\rm{m}}}({\omega }_{{\rm{m}}})/2 & 0\\ 0 & 0 & 0 & {J}_{{\rm{m}}}({\omega }_{{\rm{m}}})/2\end{array}\right){\mathscr{U}}(t).\end{array}\end{eqnarray} \tag{ 17 }$$

Thus, for comparison, we choose optical decay rate κ = J_c(ω_c/2 and mechanical decay rate γ_m = J_m(ω_m/2. The corresponding result is shown in Fig. 1. Clearly, the non-Markovian case has better performance of the state transfer than the Markovian case. Thus the non-Markovian memory effect is helpful for state transfer.

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In Fig. 2, we explore the dynamical evolution of the efficiency by varying the tunneling interaction strength J. The maximum value of efficiency increases with the increase of J. By comparing Figs. 2(a), 2(b), and 2(c), we can also find that the transfer time for reaching the maximum value of efficiency becomes shorter. Meanwhile, in Fig. 2(c), when ω_mt ≈ 18.6, the efficiency may reach 99.3%.

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In order to have a better understanding of the properties of the efficiency, we explore the maximum value of efficiency via the tunneling interaction strength J and the dimensionless coupling strength η_c. The result is plotted in Fig. 3. Not surprisingly, η_c plays a negative role for achieving large values of E_max, which agrees with the Markovian case. In the presence of non-Markovian environment, as shown in Fig. 3, the high-efficiency state transfer requires an appropriate value of J, and the oscillation becomes apparent when the value of J increases. To show clearly the relation between E_max and J, we plot Fig. 4, where we keep η_c = 10^–4. In Fig. 4, we also compare the Markovian and non-Markovian cases. It shows that for both cases, E_max non-monotonically increases with the increase of J. However, when J is not very large, the performance of state transfer marked by E_max obtained in non-Markovian regime is much better than the one obtained in Markovian regime. If J gets smaller, the transfer time required for E_max increases. Then the long-time memory effect plays an important role. Therefore, the results in non-Markovian environments will be much better than the Markovian case. In addition, in the inset of Fig.4, our numerical result shows that the efficiency can reach 99.7% when J ≈ 0.258ω_m.

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In this paper, we have investigated the quantum state transfer between two distant membranes in non-Markovian regime. We have derived the analytical expression of the efficiency by using the method of modified Laplace transformation. Our results show that quantum state transfer between two membranes can be implemented with high efficiency by using the recently measured experimental mechanical spectral density. Our study might open up a promising perspective for manipulating and transferring the quantum state of optomechanical arrays in memory environments.