Macroscopic Schr&#x00F6;dinger cat state swapping in optomechanical system

Ye-Xiong Zeng; Jian Shen; Jian Shen; Ming-Song Ding; Chong Li; Chong Li

doi:10.1364/OE.385814

1. Introduction

The quantum state swapping (QSS) is a crucial task in quantum information processing (QIP) [1–5]. High fidelity QSS has been realized in various hybrid systems including quantum dots [6], ions [7], optical lattices [8], superconducting circuits [9–12], atoms [13–16], quantum mechanical oscillators [17–19], and so on. In particular, hybrid cavity optomechanical systems consisting of optical and mechanical modes are promising platforms for the study of macroscopic quantum effects, e.g., macroscopic quantum state transfer due to the feature of ranging from the nanoscale to macroscopic sizes [19–25].

Recent research has focused on the study of nonclassical states in cavity optomechanical system, for example, mechanical single-phonon Fock state [26–29], macroscopic Schrödinger cat states [30–32], entangled Schrödinger cat state [33] and so on. Especially, Schrödinger cat states, as the superposition of two coherent states $\vert \pm \xi \rangle$ with opposite phases, has been applied to track their decoherence under energy loss, explore the boundary between the classical and quantum world, enhance quantum computation, implement quantum measurement, just to name a few [34–37]. Moreover, cat state has been created by various methods in theory, Rydberg interaction [38], one-dimensional quantum walks [39], using ultrafast laser pulses [40], and optical parametric amplification [41]. In [30], Liao has proposed a scheme to create macroscopic cat states of mechanical mirror in a two-mode cavity system, which not only provides chance to examine macroscopic quantum properties, but also offers a possibility to achieve the transfer of cat states between mechanical oscillators.

Implementing QSS between mechanical oscillators can open the possibility for the study of a variety of novel and interesting phenomena in macroscopic quantum physics. The swapping of cat states is also a useful resource for QIP and quantum computing. As discussed above, nevertheless, all of these previous studies not specifically grasp the nonclassical and non-Gaussian features to study the transfer of macroscopic Schrödinger cat states between mechanical oscillators. Besides, a cavity optomechanical system inevitably interacts with the ambient environment so that the system is negatively affected by the so-called decoherence phenomenon. Schrödinger cats has been theoretically and experimentally confirmed more susceptible to the effects of loss than other state, which makes it difficult to swap them between mechanical oscillators [42–45]. Therefore it is necessary to study a robust transfer scheme for macroscopic cat states theoretically and experimentally. Over the past several years, different strategies have been studied to restrain the decoherence effect and protect quantum features, such as decoherence subspace [46], non-Markovian characters [47], quantum control [48,49], and so on. Especially, recent works declare that squeezing non-Gaussian macroscopic quantum states can restrain the decoherence effect and can help to maintain the negative part of its Wigner distribution in the presence of loss [50–53], which can provide a possibility to engineer an effective and robust transfer scheme.

In this work, we put forward a robust scheme for the swapping of a macroscopic cat state between two mechanical oscillators. Our model consists of two independent mechanical oscillators coupled to an optical cavity via radiation pressure where the cavity mode is driven by a feedback-controlled pump field. After effectively eliminating the cavity mode, we obtain the cavity mode an effective Hamiltonian with parametric amplification terms of the two mechanical modes, respectively. Our scheme is divided into three steps. In the first, we squeeze the initial oscillator to a squeezed cat state. Secondly, transferring the created squeezed state to the target oscillator by indirect beam-split interaction. When the squeezed cat state arrived in the target oscillator, the third step starts. The third step is to antisqueeze the target oscillator to restore the original amplitude of the cat state, and then QSS completed. To reduce the decoherence effect in macroscopic cat states, we simulate the dynamical of the system analytically and numerically. Our results show that our scheme can reduce the decoherence effect in macroscopic cat states, which means that our swapping protocol is effective and feasible.

This paper is organized as follows. In Sec. 2, we present our model of two independent mechanical oscillators coupled to a cavity mode that adjusted by a feedback-controlled pump field. In Sec. 3, we introduce the state transfer scheme in detail. In Sec. 4, we study the influence of squeezed parameters on the dynamics of the system. Conclusions are presented in Sec. 5.

2. Physical model

We consider an optomechanical step comprised of two mechanical modes and a cavity as schematically shown in Fig. 1, where the quadrature of the optical field is detected via homodyne detection and the corresponding detection results are feedback to the optical cavity. In detail, the optical mode with frequency $\omega _c$ is coupled to two mechanical modes, at frequencies $\omega _1$ and $\omega _2$, respectively. The Hamiltonian of this hybrid system is given by

(1)$$\hat{H}=\hat{H}_{0}+\hat{H}_{I}+ \hat{H}_{\mathrm{diss}}.$$

The free energy of optical and mechanical modes is $\hat {H}_{0}$ and its expression is

(2)$$\hat{H}_0=\omega_c\hat { a }^ { {\dagger} } \hat { a }+\sum_{j=1}^{2}\omega_{j} \hat { b }^{{\dagger}}_{j} \hat { b }_{j},$$

where $\hat {a}$ is the annihilation operator of the cavity mode and $\hat {b}_j$ is the annihilation operator of the $i$th mechanical mode. The interaction Hamiltonian is

(3)$$\hat {H}_I=\sum_{j=1,2}g_{j} \hat{a}^{{\dagger}}\hat{a}(\hat{b}^{{\dagger}}_{j} + \hat{b}_{j})^2,$$

where $g_j$ describes the coupling strength between the optical cavity and the $j$th mechanical oscillator. The cavity field is driven by a classical field with strength $\epsilon$ and frequency $\Omega$ and thus the driving Hamiltonian is

(4)$$\hat{H}_{\mathrm{ diss } }=\hat{a}^{{\dagger}}\epsilon e^{{-}i\Omega t}+\hat {a}\epsilon^* e^{i\Omega t}.$$

The driving laser is detuned by $\Delta =\Omega -\omega _c$ from the cavity mode. In a frame rotating with the laser frequency $\Omega$, the Hamiltonian of the system can be written as

(5)$$\begin{aligned} \hat{H}^{\prime}_{diss}= &- \Delta\hat {a}^{{\dagger}}\hat{a}+\sum_{j=1}^{2}\omega_{j} \hat { b } ^ { {\dagger} }_{j} \hat { b }_{j}+g_{j} \hat{a}^{{\dagger}}\hat{a}(\hat{b}^{{\dagger}}_{j} + \hat{b}_{j})^2+\hat{a}^{{\dagger}}\epsilon+\hat {a}\epsilon^*. \end{aligned}$$

According to the above Hamiltonian, we perform a rotating-wave approximation (RWA) by considering the coupling frequency $g_j\ll \omega _j$. Then the effective Hamiltonian has the following form

(6)$$\begin{aligned} \hat{H}_{diss}= &\Delta^{\prime}\hat {a}^{{\dagger}}\hat{a}+\sum_{j=1}^{2}\omega_{j} \hat { b } ^ { {\dagger} }_{j} \hat { b }_{j}+2g_{j} \hat{a}^{{\dagger}}\hat{a}\hat{b}^{{\dagger}}_{j}\hat{b}_{j}+\hat{a}^{{\dagger}}\epsilon+\hat {a}\epsilon^*, \end{aligned}$$

where the effective detune $\Delta ^{\prime }=-\Delta +g_1+g_2$. Considering the strong driven condition, the optomechanical interaction can be linearized in the usual way, i.e., the cavity and mechanical modes are split into an average amplitude and a fluctuation term, i.e., $\hat {a} \rightarrow \alpha +\hat {a}$ and $\hat {b}_j \rightarrow \beta +\hat {b}_j$. The linalized Hamiltonian can be written as

(7)$$\begin{aligned} \hat{H}_{diss}= &\Delta^{\prime}\hat {a}^{{\dagger}}\hat{a}+\sum_{j=1}^{2}\omega^{\prime}_{j} \hat { b } ^ { {\dagger} }_{j} \hat { b }_{j} +2g_{j}\lbrace(\alpha\beta_j\hat{a}^{{\dagger}}\hat{b}^{{\dagger}}_j+\alpha^*\beta_j\hat{a}\hat{b}^{{\dagger}}_j)+H.c\rbrace, \end{aligned}$$

where we have ignored some high frequency oscillation $2g_j(\alpha \hat {a}^{\dagger }+\alpha ^*\hat {a})\hat {b}^{\dagger }\hat {b}$, $2g_j(\beta _j\hat {b}^{\dagger }+\beta ^*_j\hat {b})\hat {a}^{\dagger }\hat {a}$ due to frequency conditions $\Delta ^{\prime }, \omega ^{\prime }_j\gg 2g_j|\alpha |, 2g_j|\beta _j|$. In addition, the $2g_j$ is so small that the cross-Kerr interaction $2g_j\hat {a}^{\dagger }\hat {a}\hat {b}^{\dagger }\hat {b}$ has been dropped. The frequency has a shift $\Delta ^{\prime }=\Delta +2(g_1\vert \beta _1\vert ^2+g_2\vert \beta _2\vert ^2)$ and $\omega ^{\prime }_j=\omega _j+2g_j\vert \alpha \vert ^2$. Considering the feedback photocurrent into the system, we obtain the quantum Langevin equation

(8)$$\begin{aligned} \dot{\hat{a}}=&-(\kappa-i\Delta^{\prime})\hat{a}-i\sum^{2}_{j=1}G_j\alpha(\beta^*_j\hat{b}_{j}+\beta_j\hat{b}_{j}^{{\dagger}})+\sqrt{2\kappa}\hat{a}_{in},\\ \dot{\hat{b}}_j=&-(\gamma_j+i\omega^{\prime}_{j})\hat{b}_j-iG_j\beta_j(\alpha^*\hat{a}+\alpha\hat{a}^{{\dagger}})+\sqrt{2\gamma_j}\hat{b}_{in,j}, \end{aligned}$$

where $\kappa$ and $\gamma _j$ are the decay rates of the cavity and the $j$th mechanical modes, respectively. $G_j=2g_j$. $\hat {b}_{in,j}$ represents the thermal noise with $n_{th,j}$ thermal excitations that satisfy the correlation function $\langle \hat { b } _ { i n,j } ( t ) \hat { b } _ { i n,j } ^ { \dagger } \left ( t ^ { \prime } \right )\rangle = \left ( 1 + n _ { th,j } \right ) \delta \left ( t - t ^ { \prime } \right )$. The input of the cavity mode $\hat {a}_{in}$ can be decomposed in terms of the noise operators $\hat {a}_{in,1}$ and $\hat {a}_{in,2}$ associated with the left and the right decay channel, i.e.,

(9)$$\hat{a}_{in}=\frac{\sqrt{2\kappa_1}\hat{a}_{in,1}+\sqrt{2\kappa_{2}}\hat{a}_{in,2}}{\sqrt{2\kappa}}.$$

Fig. 1. Sketch of the physical setup to transfer the Schrödinger cat state between optomechanical modes. Two mechanical modes are coupled to an optical cavity pumped by a classical laser field. The quadrature of the optical field is detected via homodyne detection at phase $\theta _ { f b }$ and the corresponding detection results are feedback to the optical cavity.

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Here $\kappa _1$ and $\kappa _2$ are the decay rates of the left and the right decay channel, respectively. $\hat {a}_{in,2}$ describe the vacuum fluctuations and satisfy the correlation functions $\langle \hat {a}_{in,2}(t) \hat {a}_{in,2}^{\dagger } \left ( t ^ { \prime } \right ) \rangle = \delta \left ( t - t ^ { \prime } \right )$. The input operator $\hat {a}_{in,1}$ is composed of feedback photocurrent $\Phi _{1}(t)$ and vacuum fluctuation $\hat {a}_{in,0}$, that is [54],

(10)$$\hat{a}_{in}^{(1)}(t)=\hat{a}_{in,0}(t)+\hat{\Phi}_1(t),$$

where the vacuum fluctuation $\hat {a}_{in,0}(t)$ satisfies $\langle \hat { a } _ {in,0}(t)\hat {a}_{in,0}^{\dagger }(t^{ \prime })\rangle =\delta (t-t^{\prime })$. The feedback photocurrent $\Phi _{1}(t)$ can be approximated as $\overline {g}_{fb}\hat {i}_{ fb} \left ( t-\tau _{fb}\right )$ when $\overline { g } _ { f b }$ is constant over a sufficiently large band of frequencies around the mechanical resonance. $t^{\prime }$ represents the feedback time delay time $\tau _ {fb}$. The photocurrent from the results of homodyne detection can be written as

(11)$$\hat {i}_{fb}(t)=\sqrt{\eta_{d}}\hat { X }_{out,f b}^{\left(\theta_{f b}\right) }(t)+\sqrt{1-\eta_{d}} \hat{X}_{v}(t),$$

where $\theta _ { f b }$, $\eta _d$ and $\hat { X } _ { v } ( t )$ describe the additional phase, detection efficiency and additional noise due to the homodyne detection, respectively. The additional noise $\hat {X}_{v}(t)$ satisfies the relation $\langle \hat {X}_{v}(t)\hat {X}_{v}\left (t^{\prime } \right )\rangle =\delta \left (t-t^{\prime }\right )$. Moreover, the detected field quadrature is $\hat {X}_{out, fb}^{\left (\theta _{fb}\right )}(t)=e^{-i\theta _{fb}}\hat {a}_{out,2}(t)+e^{i\theta _{fb}}\hat {a}_{out,2}^{\dagger }(t)$ with the output operator $\hat {a}_{out,2}(t)$ given by the standard input-output relation

(12)$$\hat { a }_{out,2}(t) = \sqrt {2\kappa_{2}} \hat{a}(t)-\hat{a}_{in,2}( t ).$$

Supposing the detuning is greater than the optomechanical coupling constant and cavity decay rate, i.e., $\Delta ^{\prime }\gg G_j, \kappa _{c}$, one can rewrite the output operator $\hat { a }_{out,2}(t)$ as $\hat {a}_{out,2} \left (t-\tau _{fb}\right )=\hat {\overline {a}}_{out,2}\left (t- \tau _{fb}\right )e^{-i\Delta ^{\prime }\left (t-\tau _{fb}\right ) }$ [23]. We can approximately rewrite the output operator as

(13)$$\begin{aligned} \hat {a}_{out,2 } \left( t - \tau_{fb}\right) &\simeq \hat { \overline { a } } _ {out,2} ( t ) e ^ { - i \Delta' t } e ^ { i \Delta' \tau _ { f b } }\\ &= \hat { a } _ {out,2} ( t ) e ^ { i \Delta' \tau _ { f b } }, \end{aligned}$$

when the time delay $\tau _ { f b }$ is much shorter than $G_j$ and $\kappa _{ c}/2$. Therefore the quadrature $\hat { X } _ { \textrm {out} , f b } ^ { \left ( \theta _ { f b } \right ) } ( t )$ can be rewritten as $\hat { X } _ { \textrm {out} , f b } ^ { \left ( \theta _ { f b } \right ) } ( t )=\left ( e ^ { - i \phi } \hat { a } _ { o u t,2} ( t ) + e ^ { i \phi } \hat { a } _ { o u t,2 } ^ { \dagger } ( t ) \right )$, where the phase $\phi$ is defined as $\phi \equiv \theta _ { f b } - \Delta '\tau _ { f b }$. We can obtain a zero phase $\phi =0$ by properly adjusting the value of $\theta _ { f b }$. According to the discussion above, we can simplify the dynamical evolution of the optical mode $\hat {a}$ as

(14)$$\begin{aligned} \dot {\hat{a}}(t)=&- \left( \kappa_{fb}-i \Delta'\right) \hat { a }( t )-i\sum_j G_j\alpha(\beta^*\hat{b}_j(t) + \beta\hat{b}^{{\dagger}}_j(t)) +2\xi\hat {a}^{{\dagger}}(t)+ \sqrt{2\kappa_{fb}}\hat{a}_{in,fb}(t), \end{aligned}$$

with the effective decay rate is $\kappa _{fb}=\kappa -2\overline {g}_{fb}\sqrt {\eta _{d}\kappa _{1} \kappa _{2}}$, the effective parametric amplification coefficient is $\xi =(\kappa -\kappa _{bf})/2$ and the effective input operator is

(15)$$\begin{aligned} \hat { a }_{ i n , f b }(t) = & \frac { 1 }{ \sqrt { 2 \kappa _ { f b } } } \lbrace \sqrt { 2 \kappa _ { 1 } } \hat { a } _ { i n , 0 }( t ) + \sqrt { 2 \kappa _ { 2 } } \hat { a } _ { i n,2} ( t ) - \overline { g } _ { f b } \sqrt { 2 \eta _ {d}\kappa_1} \left[ \hat { a } _ {in,2} ( t ) +\hat { a } _ {in,2}^{{\dagger}} ( t )\right]\\ & + \overline { g } _ { f b } \sqrt { 2 \left( 1 - \eta _ { d } \right) \kappa _ { 1 } } \hat { X } _ { v } ( t ) \rbrace. \end{aligned}$$

The effective noise satisfies the following correlation relations

(16)$$\begin{aligned} &\left\langle \hat{a}_{in,fb}^{{\dagger}}(t)\hat{a}_{in,fb}\left(t^{\prime}\right) \right\rangle = n _ {opt,f b} \delta \left( t - t ^ { \prime } \right), \end{aligned}$$

where the number of thermal excitations is

(17)$$n _ {opt, fb} = \frac { \left( \kappa-\kappa_{ f b } \right) ^ { 2 } } {4\eta_{d}\kappa_{2}\kappa_{fb}}.$$

From the simplified Langevin Eq. (14), we can derive the Hamiltonian with the feedback information, i.e,

(18)$$\begin{aligned} \hat { H } _ {eff} = &\Delta'\hat { a }^{ {\dagger} } \hat { a }+i(\xi\hat{a}^{{\dagger} 2}-\xi^*\hat{a}^{2}) + \sum _ { i = 1 } ^ { 2 } \omega _ {j} ^ {\prime} \hat { b } _ { m,i} ^ { {\dagger} } \hat {b}_{m,i} + G _ { i }\left( \beta\hat { b } _ { i } ^ { {\dagger} } + \beta^*\hat { b } _ { i } \right) \left( \alpha\hat { a } _ { i } ^ { {\dagger} } + \alpha^*\hat { a } _ { i } \right). \end{aligned}$$

It is obvious that a parametric amplification term is obtained in the Eq. (16). This term originates from the feedback information which can be used to impress the decoherence effect of the system. We can delete the cavity mode effectively when the detune $\Delta ^{\prime }$ is very large. The detail process and the corresponding effective Hamiltonian are given in Appendix A, i.e.,

(19)$$\begin{aligned} \hat{H}_{eff}=&\sum_{j=1,2}\frac{\mathcal{G}_{j\bar{j}}}{2}(\beta_j\hat{b}^{{\dagger}}_j+\beta^*_j\hat{b}_j)(\beta_{\bar{j}}\hat{b}^{{\dagger}}_{\bar{j}}+\beta^*_{\bar{j}}\hat{b}_{\bar{j}}) +\omega^{\prime\prime}_j\hat{b}^{{\dagger}}_j\hat{b}_j+ \frac{\mathcal{G}_{jj}}{2}(\beta_j\hat{b}^{{\dagger} 2}_j+\beta^*_j\hat{b}^{2}_j), \end{aligned}$$

where the effective parameters are given in Appendix A and labels $(j=1,2,~\bar {j}=2,1)$. By applying this effective Hamiltonian, we can design a scheme to complete the cat state transfer and suppress the decoherence effect. In Eq. (19), the effective parameters is flexible and adjustable that is due to the controllable classical driving $\epsilon$ and the feedback factor $\bar {g}_{fb}$. Especially, the $\bar {g}_{fb}$ is introduced by feedback loop that is easy to achieve in experiment so that the experiment was greatly reduced in difficulty.

3. Specific scheme of QSS

In this section, we concretely discuss how to achieve a robust QSS specifically for the Schrödinger cat state. Before discussing the transfer in detail, we simplify the effective Hamiltonian given in Eq. (19). This Hamiltonian can be adjusted to different forms by designing the corresponding effective parameters, and the detail processing steps are discussed as follows. As a first step, we consider the frequency conditions $\omega ^{(m)\prime }_1=0$ and $\omega ^{\prime \prime }=\omega ^{\prime \prime }_2\gg 0$ in a period of time $\tau$. In this process, the two mechanical modes are uncoupled and the dynamics of mechanical mode $\hat {b}_1$ is governed by the Hamiltonian

(20)$$\hat{H}_{e1}=\omega^{\prime\prime}_2\hat{b}^{{\dagger}}_2\hat{b}_2+\sum_{j=1,2}-i\gamma_j\hat{b}^{{\dagger}}_j\hat{b}_j+\frac{\mathcal{G}_{11}}{2}(\beta_{1}\hat{b}^{2\dagger}_1+\beta_{1}^*\hat{b}^2_1).$$

Therefore, the mechanical mode $\hat {b}_1$ can be prepared in a squeezed Schrödinger cat state. Then, the squeezed state is transferred from mode $\hat {b}_1$ to mode $\hat {b}_2$ by setting the frequency $\omega ^{\prime \prime }=\omega ^{\prime \prime }_1=\omega ^{\prime \prime }_2\gg 0$. With this condition and considering the rotating wave approximation, one can rewrite the effective Hamiltonian as

(21)$$\begin{aligned} \hat{H}_{e2}=&\sum_{j=1,2}\frac{\mathcal{G}_{j\bar{j}}}{2}(\beta_j\beta^*_{\bar{j}}\hat{b}^{{\dagger}}_j\hat{b}_{\bar{j}}+\beta^*_j\beta_{\bar{j}}\hat{b}_j\hat{b}^{{\dagger}}_{\bar{j}}) +(\omega^{\prime\prime}_j-i\gamma_j)\hat{b}^{{\dagger}}_j\hat{b}_j, ~(j=1,2,~\bar{j}=2,1), \end{aligned}$$

which can execute state exchange between the two mechanical modes. In Eq. (21), we have applied the rotating wave approximation, i.e., ignoring the fast oscillation terms $\hat {b}_{\bar {j}}\hat {b}_j$ and $\hat {b}^{\dagger }_{\bar {j}}\hat {b}^{\dagger }_j$. When the system is governed by Hamiltonian (21), the states of two mechanical oscillators will be exchanged, i.e., the state transfer is completed. Finally, we apply the frequency condition $\omega ^{\prime \prime }=\omega ^{\prime \prime }_1\gg 0$ and $\omega ^{\prime \prime }_2=0$ to antisqueeze the target mechanical mode, where the Hamiltonian can be written as

(22)$$\hat{H}_{e3}=\omega^{\prime\prime}_1\hat{b}^{{\dagger}}_1\hat{b}_1+\sum_{j=1,2}-i\gamma_j\hat{b}^{{\dagger}}_j\hat{b}_j+\frac{\mathcal{G}_{22}}{2}(\beta_{2}\hat{b}^{2\dagger}_2+\beta_{2}^*\hat{b}^2_2).$$

By adjusting the effective parameter $\beta _2$, one can squeeze and antisqueeze the target mechanical mode under dynamics dominated by Hamiltonian (22). In above, three Hamiltonians are obtained by the feedback-controlled information and classical driving $\epsilon e^{i \Omega t}$ that domain the corresponding condition of effective parameters.

The detail steps are shown by sketch Fig. 2. The initial states of the two mechanical oscillators are supposed as vacuum state and macroscopic Schrödinger cat state [55,56]

(23)$$|cat\rangle=\frac{1}{2}\left(\left|\chi\right\rangle+ e^{i \vartheta}\left|\chi^*\right\rangle\right),$$

which can be demonstrated theoretically and experimentally. Our scheme can be departed into three steps. First, we squeeze the cat state of the initial mechanical oscillator in a short time with the Hamiltonian in Eq. (20). Then, we use the Hamiltonian in Eq. (21) to transfer the squeezed cat state to the target mechanical oscillator. When the squeezed cat state arrives at the target mechanical oscillator, we decouple the two mechanical modes and antisqueeze the target mechanical oscillator Hamiltonian in Eq. (22). To measure the transfer, we calculate the fidelity between the target state $|\psi _{t}\rangle =|0\rangle _1|cat\rangle _2$ and the transient state of the system. The fidelity of the state can be written as

(24)$$\begin{aligned} \mathcal{F}=&|\langle\phi_{t}\vert e^{{-}i\hat{H}_{3}\tau}e^{{-}i\hat{H}_{2}t}e^{{-}i\hat{H}_{1}\tau}\vert \phi_i\rangle|\\ =&\frac{1}{4}{\bigg \vert}\langle\left.\chi\right|_{2}\langle\left. 0\right|_{1} e^{{-}i \hat{H}_{3} \tau} e^{{-}i \hat{H}_{2} t} e^{{-}i \hat{H}_{1} \tau} | \chi\rangle_{1} | 0\rangle_{2}\\ &+e^{i\theta}\langle\left.\chi\right|_{2}\langle\left. 0\right|_{1} e^{{-}i \hat{H}_{3} \tau} e^{{-}i \hat{H}_{2} t} e^{{-}i \hat{H}_{1} \tau} | \chi^*\rangle_{1} | 0\rangle_{2}\\ &+e^{{-}i\theta}\langle\left.\chi^*\right|_{2}\langle\left. 0\right|_{1} e^{{-}i \hat{H}_{3} \tau} e^{{-}i \hat{H}_{2} t} e^{{-}i \hat{H}_{1} \tau} | \chi\rangle_{1} | 0\rangle_{2}\\ &+\langle\left.\chi^*\right|_{2}\langle\left. 0\right|_{1} e^{{-}i \hat{H}_{3} \tau} e^{{-}i \hat{H}_{2} t} e^{{-}i \hat{H}_{1} \tau} | \chi^*\rangle_{1} | 0\rangle_{2}{\bigg \vert}. \end{aligned}$$

Its analytical results are also obtained and the details are given in the Appendix B. To simplify the calculation of the fidelity we have approximately ignore the noise in Eq. (24), but we take the influence of noise back into the numerical simulation. $\omega ^{\prime \prime }=\omega ^{\prime \prime }_1~(\omega ^{\prime \prime }\neq 0)$ or $\omega ^{\prime \prime }=\omega ^{\prime \prime }_2~ (\omega ^{\prime \prime }\neq 0)$

Fig. 2. Sketch of the scheme for QSS of the macroscopic Schrödinger cat state between mechanical modes. In a short duration $\tau$, the initial mechanical oscillator is squeezed to a squeezed cat state. Then two mechanical oscillators complete state exchange in the time duration $t$, that is, the target oscillator is prepared in a squeezed cat state. In the last time period $\tau$, the target mechanical oscillator is antisqueezed to the macroscopic Schrödinger cat state.

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To ensure the validity we simulate the result of a complete dynamical process in Fig. 3. It is obvious in Fig. 3(a) that the squeezed cat state transfer from the initial oscillator to the target oscillator at $t=30$. According to the time of state transfer in Fig. 3(a), we draw the whole evolution of fidelity between the target state and instantaneous state of the system in Fig. 3(b). In this process, we take $\tau =2$ to squeeze the initial oscillator and antisqueeze the target oscillator. It is obvious that the numerical result and analytical results are approximately coincidence. Moreover, we draw the Wigner function of the mechanical modes at different times in Fig. 4. From (a) to (e), the state of initial oscillator transfer from cat state and squeezed state to a vacuum state. On the contrary, the target oscillator obtains a cat state after the state transferring and antisqueezing processes, which means a perfect QSS is completed. As discussed above, this scheme has achieve the swapping of macroscopic Schrödinger state between two macroscopic mechanical oscillators. QSS has various applications in QIP, for example achieving a SWAP gate, storing quantum information and completing the teleportation of state. However, our scheme has expend the QSS to a macroscopic quantum system and macroscopic Schrödinger cat state so that it can be developed to achieve some macroscopic quantum applications.

Fig. 3. (a) Fidelity between the instantaneous state of the system between $\vert 0\rangle _1\vert$ squeezed cat state $\rangle _2$, i.e., the squeezed cat state exchange between the two mechanical modes. (b) The dynamical evolution of fidelity between the state of the system and the target $\vert 0\rangle _1\vert \textrm {cat}\rangle _2$. We consider $n^{ef}_1=n^{ef}_2=2$ for the numerical result in (b). The parameters used in panel (a) and (b) are $\gamma _1/\omega ^{\prime \prime }=0.00001$, $\gamma _2/\omega ^{\prime \prime }=0.00001$, $\tau =2$, $\chi =1.5$, $\mathcal {G}_{11}\beta _1/(2\omega ^{\prime \prime })=\exp (i\pi /2)$, $\mathcal {G}_{22}\beta _2/(2\omega ^{\prime \prime })=-\exp (i\pi /2)$, and $\mathcal {G}_{12}\beta _1\beta ^*_2/\omega ^{\prime \prime }=0.05$.

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Fig. 4. (a), (b), (c), (d) and (e) are numerical visualization of the Wigner functions of the initial mechanical oscillator at different time, respectively. (f), (g), (h), (i) and (j) represent the Wigner functions of the initial mechanical oscillator at different time, respectively. (a) and (f): $t_1=0$. (b) and (g): $t_1=2$. (c) and (h): $t_1=17$. (d) and (i): $t_1=32$. (e) and (j): $t_1=34$. The other parameters are the same with Fig. 3.

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4. Analyse of squeezed parameter

In the above section, we have discussed how to use the scheme proposed above achieving a perfect QSS. As mentioned before, the QSS of cat state is sensitive to the decoherence created by the surrounding environment. Therefore, here we study the robustness of the scheme against the decoherence of the surrounding environment.

In order to explain characters of the scheme for suppressing the decoherence of the system, we analyze the effect of squeezed parameters on the state transfer. First, we rewritten the parameter $re^{i\theta }=\frac {\mathcal {G}_{11}}{2}\beta _{1}$, where $r$ and $\theta$ is determined by the feedback coefficient $\bar {g}_{fb}$ and the classical driving strength $\epsilon$. In Fig. 5, we compute the fidelity as a function time $t_1$ and compare the fidelity for the different squeezed strength. It is observable that the fidelity is largely affected by the squeezed strength. Moreover, we can find some suitable strength to obtain a higher fidelity than $r=0$ that directly transfers the state without squeezing. Meanwhile, we also draw the influence of the diverse squeezing angle in Fig. 6. An appropriate theta can be found to enhance the microscopic QSS in this cavity mechanical system and the better angle is $\theta =\frac {pi}{2}$ for our example. The system does not squeeze and antisqueeze processes, i.e., $\tau =0$, and the corresponding simulation is shown in Fig. 7. It is obvious that the max value in Fig. 7 is lower than the max value in Fig. 5 and Fig. 6. Therefore, suitable squeezing parameters can reinforce the robustness of the cat state transfer. For this results, we can give a possible physical explanation that the decay of negativity part the winger function is suppressed by the squeezing. Besides, smaller decay of the winger function means the smaller decoherent effect. This physical description has been used to complete other quantum task [51].

Fig. 5. The fidelity between the target state and the instantaneous density matrix evolves with time $t_1$ under different squeezed strength. The parameters are $\theta =\frac {\pi }{2}$, $\chi =3$, $\tau =1$, $\gamma _{1} / \omega ^{\prime \prime }=0.001$, $\gamma _{2} / \omega ^{\prime \prime }=0.001$, and $\mathcal {G}_{12} \beta _{1} \beta _{2}^{*} / \omega ^{\prime \prime }=0.05$, and $n^{ef}_1=n^{ef}_2=2$.

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Fig. 6. The fidelity between the target state and the instantaneous density matrix evolves with time $t_1$ under the different squeezed angle. The squeezed strength is $r=1$. The other parameters are the same as Fig. 5.

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Fig. 7. The fidelity between the target state and the instantaneous density matrix evolves with time $t_1$, where the squeezed time $\tau =0$. The other parameters are the same as Fig. 6.

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In the above, we have obtained a robustness Schrödinger cat swapping scheme by analyzing the squeezing parameters in a decoherence environment. Then our goal is to verify the decoherence is protected by the squeezed process, that is, the second step of this scheme. Just like a recent experiment phenomenon mentioned by Le Jeannic et al. [51] who showed that squeezing a Schrödinger cat state can help to maintain the negative part of its Wigner distribution in the presence of decoherence effect. To complete this study aim, we rewrote the initial and the target as

(25)$$\begin{aligned} |\textrm{squeezed cat state}\rangle_1\vert 0\rangle_2=S_1(r_1e^{i\theta_1})\vert \textrm{cat}\rangle_1\vert 0\rangle_1,\\ \vert 0\rangle_1|\textrm{squeezed cat state}\rangle_2=S_2({-}r_1e^{i\theta_1})\vert 0\rangle_1\vert \textrm{cat}\rangle_2, \end{aligned}$$

where the squeezed operator is defined as

(26)$$S(x)=\exp{\lbrace-i(x\hat{b}^{{\dagger} 2}_j+x^*\hat{b}_j^2)\rbrace}, ~ ~ j=1,2.$$

In Figs. 8 and 9, we draw the fidelity $\mathcal {F}_1$ that is defined in (24) and can describe the distance between the target state and the instantaneous state. It shows that the fidelity gradually rises with the increase of the squeezed strength and angle. These results further confirm that the higher robustness of our scheme benefited from the squeezed Schrödinger cat state with the features of protecting the negative part of its Wigner distribution in the presence of decoherence effect. Therefore, the squeezing and antisqueezing process has weaker robustness than the swapping process and the robustness can be enhanced by reducing the time $\tau$ and increasing $\beta _1$ to a squeeze a cat state. Due to the characters for suppressing decoherence, the protocol improves the feasibility, reduce experimental costs leaded by decoherence effect, and provides the possibility for experimental realization.

Fig. 8. The fidelity of the target state and the instantaneous density matrix as a function of time. The initial state is $\vert \textrm {squeezed cat state}\rangle _1\vert 0\rangle _2$ and the target is $\vert 0\rangle _1\vert \textrm {squeezed cat state}\rangle _2$. The squeezed angle is $\theta _1=\frac {\pi }{2}$. The other parameters are $\mathcal {G}_{12} \beta _{1} \beta _{2}^{*} / \omega ^{\prime \prime }=0.05$, $\chi =4$, $\gamma _{1} / \omega ^{\prime \prime }=0.005$, $\gamma _{2} / \omega ^{\prime \prime }=0.005$, and $n^{ef}_1=n^{ef}_2=2$.

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Fig. 9. The fidelity of the target state and the instantaneous density matrix as a function of time. The initial state is $\vert \textrm {squeezed cat state}\rangle _1\vert 0\rangle _2$ and the target is $\vert 0\rangle _1\vert \textrm {squeezed cat state}\rangle _2$. The squeezed angle is $r_1=1.5$. The other parameters are the same with Fig. 8.

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5. Conclusion

In conclusion, we have presented a proposal for transferring the macroscopic cat state between two mechanical oscillators. Our model is composed of two mechanical oscillators coupled to an optical cavity that is pumped by a feedback-controlled driving field. By effectively eliminating the cavity mode, we obtain an effective Hamiltonian which can squeeze the two mechanical oscillators and achieve state transfer between the mechanical modes. This scheme can be divided into three processes together with these forms of the Hamiltonian. The three processes are squeezing, followed by transferring, and sntisqueezing. After numerically and analytically analyzing the dynamical of the system, the results show that our scheme is robust for maintaining the coherence character of the system. Moreover, the physical essence of suppressing decoherence is due to the negative part of its Wigner distribution in the presence of loss and decoherence protected by the squeezing cat state. Due to the characters for suppressing decoherence, the protocol improves the feasibility, reduce experimental costs leaded by decoherence effect, and provides the possibility for experimental realization. Our scheme also has many practical applications in macroscopic QIP, for example macroscopic SWAP gate, the storage of macroscopic quantum information, and teleportation of macroscopic Schrödinger state in the nearly feature.

A. Equations for the coefficients

In this appendix, we eliminate the optical cavity mode and obtain effective Hamiltonian. Starting from the Langevin equation Eq. (14), we suppose the cavity mode arrive its steady-state rapidly, i.e.,

(27)$$\begin{aligned} 0=&-(\kappa_{fb}-i\Delta^{\prime})\hat{a}-\sum_jiG_j\alpha(\beta^*_j\hat{b}_j +\beta_j\hat{b}^{{\dagger}}_j)+2\xi\hat{a}^{{\dagger}}+\sqrt{2\kappa_{bf}}\hat{a}_{in,bf}. \end{aligned}$$

which is can be achieved by setting a big detune $\Delta '\gg \vert G_j\alpha \beta _j\vert$. Therefore, the optical mode can be expended by the two mechanical modes effectively together Eq. (27) with its complex conjugate, i.e,

(28)$$\begin{aligned} \hat{a}=&\sum_jiB_j(\beta^*_j\hat{b}_j+\beta_j\hat{b}^{{\dagger}}_j) +\sqrt{2\kappa_{fb}}(U\hat{a}^{{\dagger}}_{in,fb}+V\hat{a}_{in,fb}), \end{aligned}$$

where the corresponding parameters are

(29)$$\begin{aligned} &B_j=G_j(U\alpha^*-V\alpha),\ \ \ \ \ \ \ \ \ \ \ U=\frac{\kappa-\kappa_{bf}}{\Delta^{\prime}-\kappa^2+2\kappa\kappa_{bf}}, \ \ \ \ \ \ \ \ \ \ \ V=\frac{\kappa_{fb}+i\Delta^{\prime}}{\Delta^{\prime}-\kappa^2+2\kappa\kappa_{bf}}. \end{aligned}$$

Substituting Eq. (28) to the Eq. (8), we can obtain the effective Langivan equation which can be simplified as

(30)$$\begin{aligned} \dot{\hat{b}}_j=&-(\gamma_j+i\omega^{\prime\prime}_j)\hat{b}_j-i\mathcal{G}_{j\bar{j}}\beta_j(\beta_{\bar{j}}\hat{b}^{{\dagger}}_{\bar{j}}+\beta^*_{\bar{j}}\hat{b}_{\bar{j}}) -i\mathcal{G}_{jj}\beta_j\hat{b}^{{\dagger}}_j+\sqrt{2\gamma_j}\hat{b}^{ef}_{in,j}, \end{aligned}$$

with the effective parameters

(31)$$\begin{aligned} &\omega^{\prime\prime}_j=\omega^{\prime}_j+\mathcal{G}_{jj}\vert\beta_j\vert^2, \ \ \ \ \ \ \ \ \ \mathcal{G}_{jk}=2G_jG_k(\Im{(\mu)}\vert\alpha\vert^2+\nu \Im{(\alpha^2)}), \ \ \ \ \ \ \ \ \ \mu=\frac{\kappa+i\Delta^{\prime}}{\Delta^{\prime 2}+2\kappa_{bf}\kappa-\kappa_{bf}^2},\\ & \nu=\frac{\kappa-\kappa_{fb}}{\Delta^{\prime 2}+2\kappa_{bf}\kappa-\kappa_{bf}^2}, \ \ \ \ \ \ \hat{b}'_{j,in}=l\hat{a}^{{\dagger}}_{in,fb}+H.c., \ \ \ \ \ \ \hat{b}^{eff}_{j,in}=\hat{b}_{in,j}-iG_j\beta_j\sqrt{\frac{\kappa_{fb}}{\gamma_j}}\hat{b}^{\prime}_{j,in},\\ & l=\nu\alpha^*+\mu^*\alpha. \end{aligned}$$

According to the Langevin Eq. (31), we can derive the effective Hamiltonian

(32)$$\begin{aligned} \hat{H}_{eff}=&\sum_{j=1,2}\omega^{\prime\prime}_j\hat{b}^{{\dagger}}_j\hat{b}_j+\frac{\mathcal{G}_{j\bar{j}}}{2}(\beta_j\hat{b}^{{\dagger}}_j+\beta^*_j\hat{b}_j)(\beta_{\bar{j}}\hat{b}^{{\dagger}}_{\bar{j}}+\beta^*_{\bar{j}}\hat{b}_{\bar{j}}) +\frac{\mathcal{G}_{jj}}{2}(\beta_j\hat{b}^{{\dagger} 2}_j+\beta^*_j\hat{b}^{2}_j). \end{aligned}$$

The dynamics of the two mechanical systems is governed by the effective master

(33)$$\begin{aligned} \dot{\hat{\rho}} ={-}\frac{i}{\hbar}\left[\hat{H}_{f b}(t), \hat{\rho}\right]+\sum_{j=1,2}\mathcal{D}[\hat{b}_j] , \end{aligned}$$

where the Lindblad superoperators, describing the loss of the system, can be written as

(34)$$\begin{aligned} \mathcal{D}[\hat{b}_j]=&\gamma_j\left(n^{ef}_{t h,j}+1\right)\left(2 \hat{b}_j \hat{\rho} \hat{b}^{{\dagger}}_j-\hat{b}^{{\dagger}}_j \hat{b}_j \hat{\rho}-\hat{\rho} \hat{b}^{{\dagger}}_j \hat{b}_j\right) +\gamma_j n^{ef}_{th,j}\left(2 \hat{b}^{{\dagger}}_j \hat{\rho} \hat{b}_j-\hat{b}_j \hat{b}^{{\dagger}}_j \hat{\rho}-\hat{\rho} \hat{b}_j \hat{b}^{{\dagger}}_j\right) , \end{aligned}$$

and the corresponding effective heat phonon numbers of the $j$th mechanical bath is

(35)$$\begin{aligned} n^{ef}_{th,j}=&n_{th,j}+G^2_j\vert \beta_j\vert^2\frac{\kappa_{fb}}{\gamma_j}{\big \lbrace} \vert l\vert^2(2n_{opt,fb}+1) +2\Re{(l^2)}n_{opt,fb}(1-\sqrt{\frac{\eta_d\kappa_2}{\kappa_1}}) {\big \rbrace}. \end{aligned}$$

B. Dynamical of the system

As noted in the main text, the fidelity $\mathcal {F}$ is expended to four terms. In each term, the dynamical evolution is divide into three evolutionary operators act on the state of the system. Besides, Hamiltonians $\hat {H}_1$, $\hat {H}_2$ and $\hat {H}_3$ can depart to the product a series of time-dependent evolution operator that has a clear and simple expression in the phase space. It is due to the three Hamiltonians are composed by the generator elements of three closed group, respectively [57–59]. Here we give the three decomposed the evolutionary operators. The evolutionary operator depending on the Hamiltonian $\hat {H}_1$ is

(36)$$\begin{aligned} e^{{-}i\hat{H}_1\tau}&=e^{\zeta^{(1)}_0}e^{\zeta^{(1)}_1\hat{b}^{{\dagger}}_2\hat{b}_2}e^{\zeta^{(1)}_+\hat{b}^{{\dagger} 2}_1}e^{\frac{\ln(\zeta^{(1)}_3)}{2}\hat{b}^{{\dagger}}_1\hat{b}_1}e^{\zeta^{(1)}_-\hat{b}^2_1}, \end{aligned}$$

where the coefficients are

(37)$$\begin{aligned} &\eta^{(1)}_+{=}-i\mathcal{G}_{11}\beta_1\tau, \ \ \ \ \ \ \ \ \ \ \ \eta^{(1)}_-{=}-i\mathcal{G}_{11}\beta^*_1\tau, \ \ \ \ \ \ \ \ \ \ \ \eta^{(1)}_3={-}2\gamma_1\tau, \end{aligned}$$

and

(38)$$\begin{aligned} &\zeta^{(1)}_0=\frac{2\gamma_1\tau+\ln(\zeta^{(1)}_3)}{4}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \zeta^{(1)}_1=({-}i\omega^{\prime\prime}_2-\gamma_2)\tau, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \zeta^{(1)}_3=\left(\cosh \epsilon_1-\frac{\eta^{(1)}_3}{2\epsilon_1} \sinh\epsilon_1\right)^{{-}2},\\ & \zeta^{(1)}_{{\pm}}=\frac{\eta^{(1)}_{{\pm}} \sinh\epsilon_1}{2\epsilon_1 \cosh\epsilon_1-\eta^{(1)}_3 \sinh\epsilon_1}, \ \ \ \ \ \ \ \ \ \epsilon^{2}_1=\frac{1}{4} (\eta_{3}^{(1)})^{2}-\eta^{(1)}_{+}\eta^{(1)}_{-}. \end{aligned}$$

The dynamics governed by the Hamiltonian $\hat {H}_2$ is described by the operator

(39)$$\begin{aligned} e^{{-}i\hat{H}_2t}=&e^{\zeta^{(2)}_0(\hat{b}^{{\dagger}}_1\hat{b}_1+\hat{b}^{{\dagger}}_2\hat{b}_2)}e^{\zeta^{(2)}_+\hat{b}^{{\dagger} }_1\hat{b}_2}e^{\frac{\ln(\zeta^{(2)}_3)}{2} (\hat{b}^{{\dagger}}_1\hat{b}_1-\hat{b}^{{\dagger}}_2\hat{b}_2)}e^{\zeta^{(2)}_-\hat{b}_1\hat{b}^{{\dagger}}_2}, \end{aligned}$$

where the corresponding parameters are

(40)$$\begin{aligned} &\eta^{(2)}_+{=}-i\mathcal{G}_{12}\beta_1\beta^*_2t, \ \ \ \ \ \ \ \ \ \ \eta^{(2)}_-{=}-i\mathcal{G}_{12}\beta^*_1\beta_2t, \ \ \ \ \ \ \ \ \ \ \ \zeta^{(2)}_0={-}(i\omega^{\prime\prime}_1+\gamma_1)t,\\ & \zeta^{(2)}_{3}=\left(\cosh \epsilon_2\right)^{{-}2}, \ \ \ \ \ \ \ \ \ \ \ \zeta^{(2)}_{{\pm}}=\frac{ \eta^{(2)}_{{\pm}} \sinh \epsilon_2}{\epsilon_2 \cosh \epsilon_2}, \ \ \ \ \ \ \ \ \ \ \ \epsilon^{2}_2=\eta^{(2)}_{+}\eta^{(2)}_{-}. \end{aligned}$$

The dynamics depending on the Hamiltonian $\hat {H}_3$ has the following form, i.e.,

(41)$$\begin{aligned} e^{{-}i\hat{H}_3\tau}&=e^{\zeta^{(3)}_0}e^{\zeta^{(3)}_1\hat{b}^{{\dagger}}_1\hat{b}_1}e^{\zeta^{(3)}_+\hat{b}^{{\dagger} 2}_2}e^{\frac{\ln(\zeta^{(3)}_3)}{2}\hat{b}^{{\dagger}}_2\hat{b}_2}e^{\zeta^{(3)}_-\hat{b}^2_2}, \end{aligned}$$

where the parameters are

(42)$$\begin{aligned} &\eta^{(3)}_+{=}-i\mathcal{G}_{22}\beta_2\tau, \ \ \ \ \ \ \ \ \ \ \ \eta^{(3)}_-{=}-i\mathcal{G}_{22}\beta^*_2\tau, \ \ \ \ \ \ \ \ \ \ \ \eta^{(3)}_3={-}2\gamma_2\tau, \end{aligned}$$

and

(43)$$\begin{aligned} &\zeta^{(3)}_0=\frac{2\gamma_2\tau+\ln(\zeta^{(3)}_3)}{4}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \zeta^{(3)}_1=({-}i\omega^{\prime\prime}_1-\gamma_1)\tau, \ \ \ \ \ \ \ \ \ \ \ \zeta^{(3)}_3=\left(\cosh \epsilon_3-\frac{\eta^{(3)}_{3}}{2\epsilon_3} \sinh\epsilon_3\right)^{{-}2},\\ &\zeta^{(3)}_{{\pm}}=\frac{\eta^{(3)}_{{\pm}} \sinh\epsilon_3}{2\epsilon_3 \cosh\epsilon_3-\eta^{(3)}_{3}\sinh\epsilon_3}, \ \ \ \ \ \ \ \ \ \ \ \epsilon^{2}_3=\frac{1}{4} (\eta_{3}^{(3)})^{2}-\eta^{(3)}_{+}\eta^{(3)}_{-}. \end{aligned}$$

Finally, according to the operators in Eqs. (36), (39), and (41), we analytically simplify the fidelity $\mathcal {F}$ defined in Eq. (24). The corresponding analytical expressions of the four terms are simplified as the following four forms, that is,

(44)$$\begin{aligned} \langle\chi|_2\langle 0 |_1e^{{-}i\hat{H}_3\tau}e^{{-}i\hat{H}_2t}e^{{-}i\hat{H}_1\tau}|\chi\rangle_1| 0\rangle_2 =& e^{(\zeta_{0}^{(1)}+\zeta_{0}^{(3)}+\zeta^{(1)}_-\chi^2+\frac{|\chi|^2}{2}(|\sqrt{\zeta^{(1)}_3}|^2-1)+\zeta^{(3)}_+\chi^{*2})} e^{\frac{|\chi|^2}{2}(|\sqrt{\zeta^{(3)}_3}|^2-1))}\\ & \times e^{(\chi^*\zeta^{(2)}_-\sqrt{\frac{\zeta^{(3)}_3}{\zeta^{(2)}_3}}e^{\zeta^{(2)}_0}\chi\sqrt{\zeta^{(1)}_3})} e^{\frac{\zeta^{(1)}_+(\zeta^{(2)}_-)^2\chi^{*2}\zeta^{(1)}_3}{\zeta^{(3)}_3}e^{2\zeta^{(2)}_0}}, \end{aligned}$$

(45)$$\begin{aligned} \langle\chi^*|_2\langle 0 |_1e^{{-}i\hat{H}_3\tau}e^{{-}i\hat{H}_2t}e^{{-}i\hat{H}_1\tau}|\chi\rangle_1| 0\rangle_2 =& e^{(\zeta_{0}^{(1)}+\zeta_{0}^{(3)}+\zeta^{(1)}_-\chi^2+\frac{|\chi|^2}{2}(|\sqrt{\zeta^{(1)}_3}|^2-1)+\zeta^{(3)}_+\chi^{2})} e^{\frac{|\chi|^2}{2}(|\sqrt{\zeta^{(3)}_3}|^2-1))}\\ & \times e^{(\chi\zeta^{(2)}_-\sqrt{\frac{\zeta^{(3)}_3}{\zeta^{(2)}_3}}e^{\zeta^{(2)}_0}\chi\sqrt{\zeta^{(1)}_3})}e^{\frac{\zeta^{(1)}_+(\zeta^{(2)}_-)^2\chi^{2}\zeta^{(1)}_3}{\zeta^{(3)}_3}e^{2\zeta^{(2)}_0}} , \end{aligned}$$

(46)$$\begin{aligned} \langle\chi|_2\langle 0 |_1e^{{-}i\hat{H}_3\tau}e^{{-}i\hat{H}_2t}e^{{-}i\hat{H}_1\tau}|\chi^*\rangle_1| 0\rangle_2 =& e^{(\zeta_{0}^{(1)}+\zeta_{0}^{(3)}+\zeta^{(1)}_-\chi^{*2}+\frac{|\chi|^2}{2}(|\sqrt{\zeta^{(1)}_3}|^2-1)+\zeta^{(3)}_+\chi^{2})} e^{\frac{|\chi|^2}{2}(|\sqrt{\zeta^{(3)}_3}|^2-1))}\\ & \times e^{(\chi^*\zeta^{(2)}_-\sqrt{\frac{\zeta^{(3)}_3}{\zeta^{(2)}_3}}e^{\zeta^{(2)}_0}\chi^*\sqrt{\zeta^{(1)}_3})}e^{\frac{\zeta^{(1)}_+(\zeta^{(2)}_-)^2\chi^{*2}\zeta^{(1)}_3}{\zeta^{(3)}_3}e^{2\zeta^{(2)}_0}} , \end{aligned}$$

(47)$$\begin{aligned} \langle\chi^*|_2\langle 0 |_1e^{{-}i\hat{H}_3\tau}e^{{-}i\hat{H}_2t}e^{{-}i\hat{H}_1\tau}|\chi^*\rangle_1| 0\rangle_2 =& e^{(\zeta_{0}^{(1)}+\zeta_{0}^{(3)}+\zeta^{(1)}_-\chi^{*2}+\frac{|\chi|^2}{2}(|\sqrt{\zeta^{(1)}_3}|^2-1)+\zeta^{(3)}_+\chi^{2})}e^{\frac{|\chi|^2}{2}(|\sqrt{\zeta^{(3)}_3}|^2-1))}\\ &\times e^{(\chi\zeta^{(2)}_-\sqrt{\frac{\zeta^{(3)}_3}{\zeta^{(2)}_3}}e^{\zeta^{(2)}_0}\chi^*\sqrt{\zeta^{(1)}_3})}e^{\frac{\zeta^{(1)}_+(\zeta^{(2)}_-)^2\chi^{2}\zeta^{(1)}_3}{\zeta^{(3)}_3}e^{2\zeta^{(2)}_0}} . \end{aligned}$$

Funding

National Natural Science Foundation of China (11574041, 11375036); Natural Science Foundation of Liaoning Province (201801156).

Acknowledgments

The authors thank Feng-Yang Zhang, Wen-Lin Li, and Biao Xiong for the useful discussion.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Macroscopic Schrödinger cat state swapping in optomechanical system

Abstract

1. Introduction

2. Physical model

3. Specific scheme of QSS

4. Analyse of squeezed parameter

5. Conclusion

A. Equations for the coefficients

B. Dynamical of the system

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (9)

Equations (47)

Optics Express