1. Introduction

Recently, the cavity optomechanics has become an exciting research field because of its potential applications in quantum information processing. A typical optomechanical system couples the cavity field and a movable mirror by the radiation pressure [1, 2]. As the deepening of the theoretical and experimental research, many properties and applications of the optomechanical system, for example, the optomechanically induced transparency (OMIT) [3–6], the quantum ground state cooling of the nanomechanical resonators [7, 8], and quantum information processing [9,10] have been confirmed. Various kinds of nonclassical effects, which are also generated by optomechanical coupling, have attracted much attention, such as the squeezing effect and the photon anti-bunching effect. It has been demonstrated that nonclassical effects are strongly related to the study of quantum communication and quantum computation.

The quantum squeezing is a typical nonclassical effect and is a key resource for many applications, such as ultrahigh precision measurements [11–13] and gravitational wave detection [14–16]. Many schemes for creating quantum squeezing of the moving mirror in cavity optomechanics have been proposed, including injection of nonclassical light [17]. While such schemes can generate quantum squeezing of mechanical modes well, they are difficult to be implemented. To date, experimental squeezing of a nanomechanical object has been achieved through a nonlinear Duffing resonator [18]. A mechanical resonator can also be squeezed by coupling it to an auxiliary nonlinear system, such as a superconducting quantum interference device loop [19], a Cooper-pair box circuit [20], or an optical cavity containing an atomic medium [21]. The photon anti-bunching is another kind of nonclassical effects, which reveals the quantum nature of light and plays a key role in the fields of optical communication and signal detection [22,23].

In this paper, we discuss a coupled two-cavity optomechanical system for generating the mechanical squeezing and the photon anti-bunching. The power of the driving lasers and the detuning between them play a key role in generating squeezing. We find that the steady-state squeezing of the quadrature operators for the mechanical mode has analytical solutions when we choose the suitable detuning. This is different from the systems proposed in [24, 25], in which the mechanical mode is coupled to an auxiliary system to increase nonlinearity. Comparing with [26], we do not limit the mechanical mode in vacuum state. Moreover, strong steady-state squeezing can be achieved even at a high temperature. We also find that the cavity-modes can show photon anti-bunching under suitable conditions.

The organization of this paper is as follows. In section 2 we introduce the model and discuss the Hamiltonian. In sections 3 and 4 we study the squeezing of the mechanical mode and the anti-bunching of the photonic modes, respectively. Section 5 presents a brief summary.

2. The model and the Hamiltonian

As shown in Fig. 1, the system consists of a high-finesse Fabry-Pérot (FP) cavity, and there is an oscillating dielectric mirror in the middle of the FP cavity. With the presence of the middle mirror, the FP cavity is divided into two subcavities, denoted by the left cavity and the right cavity, respectively. Here we assume that the middle mirror has a nonzero transmission, which allows the exchange of photons between the left and right cavity modes and thus leads to an effective coupling between them [27]. The two cavity modes are driven simultaneously by two driving lasers, respectively.

 

Fig. 1 Sketch of the system. Two cavity-modes couple with a common oscillating mirror, and each cavity-mode is driven by two external lasers.

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The Hamiltonian of the system is

In the rotating frame with respect to U exp[−i(ω₁a^†a + ω₂b^†b)t/ħ], the Hamiltonian of the system is given by

By using this effective Hamiltonian we can obtain following quantum Langevin equations

In order to investigate the fluctuation properties of the system, we divide each operator into two parts, a mean-value and a small fluctuation, a α + δa, b β + δb, c η + δc. Substituting these equations into Eq. (5) and neglecting some nonlinear terms such as $g_0^2\left(2\eta *+2\eta \right)\delta a\left(\delta c^{†}+\delta c\right)/\mathrm{\Delta }$, $g_0^2\delta b{\left(\delta c^{†}+\delta c\right)}^2/\mathrm{\Delta }$ and $g_0^2\left(\delta a^{†}\delta a-\delta b^{†}\delta b\right)\left(2\delta c^{†}+2\delta c\right)/\mathrm{\Delta }$, we obtain following two sets of equations

From Eq. (6), we can obtain the linearized Hamiltonian

The presence of the term (δc^†2 + δc²) suggests the possibility to generate squeezing in the mechanical mode, and the parametric interaction strength Λ plays a key role in generating this kind of squeezing. We can see that the parametric interaction strength is determined by two factors, one is the coupling coefficient $g_0^2/\mathrm{\Delta }$, and the other is the photon-number difference |β|² − |α|². We can obtain the steady-state mean values by setting all the time derivatives of Eq. (7) to be zero, from which we can obtain the photon-number difference, and in turn, the parametric interaction strength

As shown in Fig. 2, the parametric interaction strength Λ decreases with the increase of the detuning and it increases linearly with the laser power P₂. In fact, this can also be seen from Eq. (10) in which |ε₂|² 2P₂κ/ħω₂. Here we choose Δ/ω_m starting from unity to satisfy the condition of large detuning.

 

Fig. 2 (a) The parametric interaction strength Λ versus the detuning Δ with P₂ 1 mW. (b)The parametric interaction strength Λ versus the driving power P₂ with Δ ω_m. Here we choose the parameters from [30]: the frequency of the mechanical resonator ω_m/2π 1 MHz, the mechanical decay rate γ 10⁻⁶ω_m, the parameters of cavity are g₀ 10⁻⁴ω_m, κ 0.1ω_m, the frequencies of the external optical fields are ω₁/2π 320 THz and ω₂ ω₁ +Δ, the driving power P₁ 0.65 μW. The coupling coefficient J κ which is based on [31].

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For simplifying the calculation, we apply a squeezing transformation to the Hamiltonian, H S⁺rH_{eff, L}Sr, here S(r) exp[r(δc² − δc^†2)] is the squeezing operator, and r (1/4) ln (1 + 4Λ/ω_m) is the squeezing amplitude. By using the relation

In the new rotating frame, the quantum Langevin equations for the fluctuations are given by

For the situation in which the two cavity modes are coupled to a vacuum reservoir and the mechanical mode is coupled to a thermal reservoir, the only nonzero correlation functions are

3. Squeezing of the mechanical mode

In the same picture as the Hamiltonian H S⁺rH_eff,LSr, the transformed quadrature operator of the mechanical mode is X S^† (r)(δc^† + δc) S (r), its variance can be derived as

When we choose an appropriate detuning Δ ω_m, we can obtain the analytical results of the variance of the quadrature operator using the Laplace transform (see Appendix A),

From Fig. 3 we can see that the variance 〈δX²〉_ss of the quadrature operator falls below the standard quantum limit 〈δX²〉_ss 1, so the steady-state squeezing takes place. The figure shows the dependence of the variance on the driving power, and the mechanical squeezing becomes stronger as the driving power increases. This can be explained as follows. The linearized Hamiltonian (8) contains the term (δc^†2 + δc²) which can leads to squeezing of the mechanical mode, and the coefficient of this term Λ corresponds to the squeezing amplitude r (r (1/4) ln (1 + 4Λ/ω_m)). Form Eq. (10) and Fig. 2, we can find that with the increase of the laser power P₂, the coefficient Λ increases, and in turn, the squeezing amplitude r increases. So the squeezing becomes stronger as the laser power P₂ increases. We can also see that the squeezing weakens with the increase of the average thermal phonon number ${\overline{n}}_{th}$

 

Fig. 3 The steady state variance 〈δX²〉_ss with the thermal phonons ${\overline{n}}_{th}$ as a parameter. Here we choose Δ ω_m, and other parameters are the same as in Fig. 2.

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Above we discussed the steady-state squeezing under the conditions Δ ω_m. Now we discuss the general situation. For getting the variance of the quadrature operator, we apply Fourier transform to Eq. (13) by F (t) $\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{+\infty }}$ F (ω) exp (−iωt) dω and solve it in the frequency domain, the variance is given by (see Appendix B)

In Fig. 4 we plot the variance of the quadrature operator as a function of the detuning through the numerical results. We see that the squeezing weakens with the increase of the detuning. This result can be understood as follows. Form Eq. (17) and Eq. (18) we find that the variance of the quadrature operator is determined by the transformed oscillator frequency ${{\omega }^{\prime }}_m$ ${\omega }_m\sqrt{1+4\mathrm{\Lambda }/{\omega }_m}$, and the transformed optomechanical coupling coefficients ${G^{\prime }}_a$ $G_a{\left(1+4\mathrm{\Lambda }/{\omega }_m\right)}^{-1/4}$ and ${G^{\prime }}_b$ $G_b{\left(1+4\mathrm{\Lambda }/{\omega }_m\right)}^{-1/4}$, and in turn, is determined by the parametric interaction strength Λ. However, we know that Λ corresponds to the squeezing amplitude r (r(1/4) ln (1 + 4Λ/ω_m)), and we find from Eq. (10) and Fig. 2 that Λ decreases with the increase of the detuning, which means that the squeezing amplitude r decreases with the increase of the detuning. So the squeezing will weaken until disappear with the increase of the detuning. Besides, the driving power P₂ can also influence the degree of squeezing. Form Eq. (10) and Fig. 2, we can find that Λ increases with the increase of the laser power P₂, and in turn, the squeezing becomes stronger as the laser power P₂ increases.

 

Fig. 4 Plot of the variance 〈δX²〉 as a function of Δ/ω_m with different P₂. Here we choose the thermal phonons ${\overline{n}}_{th}$ 1000, and other parameters are the same as in Fig. 2.

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4. Second-order correlation of the photons

In this section, we analyze the second-order correlation of the photons. Second-order correlation is a measurement of the photons correlations between some time t and a later time t+τ. It is also used to judge bunching and anti-bunching. When the condition g²0 < 1 or g⁽²⁾ (τ) > g⁽²⁾ (0) are met, the photons show an anti-bunching property.

When we choose an appropriate detuning Δ, the coupling between mode b and the mechanical mode can be zero, i.e., ${G^{\prime }}_b$ 0, this means that mode b is decoupled and the two-cavity-mode system is reduced to an effective single-cavity-mode system. So we just use the second-order correlation function for mode a to analyze the second-order correlation of the photons, and the function is

Solving the Heisenberg-Langevin equations, we obtain (see Appendix C)

The first term in Eq. (20) represents the second-order correlation function for a free single-mode field, while the second term comes from the interaction between the single-mode field and the mechanical mode. We plot g⁽²⁾ (τ) in Fig. 5. It can be seen that g²0 < 1 and g⁽²⁾ (τ) > g⁽²⁾ 0, this means that the photons show an anti-bunching property.

 

Fig. 5 The second-order correlation function g⁽²⁾ (τ) of photons.

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5. Conclusion

In this work, we propose a scheme for generating the squeezing of the mechanical mode and the anti-bunching of the photonic mode in a two-cavity optomechanical system. Our results show that the power of the driving lasers and the detuning between them play a key role in generating squeezing. We find that the squeezing of the mechanical mode can be achieved even at a high temperature by increasing the power of the driving lasers. We also find that the photonic modes can show anti-bunching under some conditions.

Funding

National Natural Science Foundation of China (NSFC) (11574092, 61378012, 91121023, 60978009); National Basic Research Program of China(2013CB921804); Innovative Research Team in University (IRT1243).

Appendix A Laplace transform

Applying the Laplace transform to Eq. (13) of the main text we obtain

(22)$$\begin{array}{l}sA\left(s\right)-\delta a\left(0\right)-\left(i{\mathrm{\Delta }}_a+\frac{\kappa }{2}\right)A\left(s\right)+i{G^{\prime }}_{a}C\left(s\right)+\sqrt{\kappa }{A}_{in}\left(s\right),\\ sB\left(s\right)-\delta b\left(0\right)-\left(i{\mathrm{\Delta }}_b+\frac{\kappa }{2}\right)B\left(s\right)+i{G^{\prime }}_{b}C\left(s\right)+\sqrt{\kappa }{B}_{in}\left(s\right),\\ sC\left(s\right)-\delta c\left(0\right)-\left(i{{\mathit{\omega }}^{\prime }}_m+\frac{\gamma }{2}\right)C\left(s\right)+i{G^{\prime }}_{a}A\left(s\right)+i{G^{\prime }}_{b}B\left(s\right)+\sqrt{\gamma }{C}_{in}\left(s\right),\end{array}$$

where A(s) ℒ(δa), B(s) ℒ(δb) and C(s) ℒ (δc) with ℒ denoting Laplace transform.

In the condition of Δ ω_m, we can obtain the solution of Eq. (22) as

(23)$$\begin{array}{l}C\left(s\right)\frac{1}{\chi \left(s\right)}i{G^{\prime }}_a\left[\sqrt{\kappa }{A}_{in}\left(s\right)+\delta a\left(0\right)\right]+\frac{1}{\chi \left(s\right)}i{G^{\prime }}_b\left[\sqrt{\kappa }{B}_{in}\left(s\right)+\delta b\left(0\right)\right]\\ +\frac{1}{\chi \left(s\right)}\left(s+i{\mathrm{\Delta }}_a+\frac{\kappa }{2}\right)\left[\sqrt{\gamma }{C}_{in}\left(s\right)+\delta c\left(0\right)\right],\end{array}$$

where

(24)$$\chi \left(s\right)\left(s+i{{\omega }^{\prime }}_m+\frac{\gamma }{2}\right)\left(s+i{\mathrm{\Delta }}_a+\frac{\kappa }{2}\right)+{G^{\prime }}_a^2+{G^{\prime }}_b^2.$$

Making inverse Laplace transform we have

(25)$$\begin{array}{l}\delta c\left(t\right)f_1\left(t\right)\delta c\left(0\right)-f_2\left(t\right)\delta a\left(0\right)-f_3\left(t\right)\delta b\left(0\right)+\sqrt{\gamma }{\displaystyle {\int }_0^{t}d{t}^{\prime }{f}_1\left(t-t^{\prime }\right)\delta c_{in}}\left(t^{\prime }\right)\\ -\sqrt{\kappa }{\displaystyle {\int }_0^{t}d{t}^{\prime }}{f}_2\left(t-t^{\prime }\right)\delta a_{in}\left(t^{\prime }\right)-\sqrt{\kappa }{\displaystyle {\int }_0^{t}d{t}^{\prime }{f}_3}\left(t-t^{\prime }\right)\delta b_{in}\left(t^{\prime }\right),\end{array}$$

where

(26)$$\begin{array}{l}{f}_1\left(t\right)\frac{1}{2\mu }\exp \left[\frac{-\gamma -\kappa -2i\left({\mathrm{\Delta }}_a+{{\omega }^{\prime }}_m\right)}{4}t\right]\{\mu [\exp \left(\mu t/4\right)+\exp \left(-\mu t/4\right)]\\ +\left(\gamma -\kappa -2i{\mathrm{\Delta }}_a+2i{{\omega }^{\prime }}_m\right)\left[\exp \left(\mu t/4\right)-\exp \left(-\mu t/4\right)\right]\},\\ f_2\left(t\right)2i{G^{\prime }}_a\exp \left[\frac{-\gamma -\kappa -2i\left({\mathrm{\Delta }}_a+{{\omega }^{\prime }}_m\right)}{4}t\right]\frac{\exp \left(\mu t/4\right)-\exp \left(-\mu t/4\right)}{\mu },\\ f_3\left(t\right)2i{G^{\prime }}_b\exp \left[\frac{-\gamma -\kappa -2i\left({\mathrm{\Delta }}_a+{{\omega }^{\prime }}_m\right)}{4}t\right]\frac{\exp \left(\mu t/4\right)-\exp \left(-\mu t/4\right)}{\mu },\\ \mu \sqrt{-16{G^{\prime }}_a^2-16{G^{\prime }}_b^2+{\left(\gamma -\kappa -2i{\mathrm{\Delta }}_a+2i{{\omega }^{\prime }}_m\right)}^2}.\end{array}$$

We assume that the cavity modes and the mechanical mode are initially in the vacuum state [〈δa^† (0) δa(0)〉 0, 〈δb^†(0) δb(0)〉 0, 〈δc^† (0) δc (0)〉 0]. Using Eq. (25) and taking into account Eq. (14), the mean phonon number of the transformed system can be derived as

(27)$$\begin{array}{l}〈\delta c^{†}\left(t\right)\delta c\left(t\right)〉{\lambda }_1\frac{\gamma {\overline{n}}_{th}}{\mu \nu }\frac{\exp \left[-t\left(2\gamma +2\kappa +\mu +\nu \right)/4\right]}{\left(2\gamma +2\kappa +\mu +\nu \right)}-{\lambda }_2\frac{\gamma {\overline{n}}_{th}}{\mu \nu }\frac{\exp \left[-t\left(2\gamma +2\kappa +\mu -\nu \right)/4\right]}{\left(2\gamma +2\kappa +\mu -\nu \right)}\\ -{\lambda }_3\frac{\gamma {\overline{n}}_{th}}{\mu \nu }\frac{\exp \left[-t\left(2\gamma +2\kappa -\mu +\nu \right)/4\right]}{\left(2\gamma +2\kappa +\mu +\nu \right)}+{\lambda }_4\frac{\gamma {\overline{n}}_{th}}{\mu \nu }\frac{\exp \left[-t\left(2\gamma +2\kappa -\mu -\nu \right)/4\right]}{\left(2\gamma +2\kappa -\mu -\nu \right)}\\ +\frac{4{G^{\prime }}_a^2\left(\gamma +\kappa \right)+4{G^{\prime }}_b^2\left(\gamma +\kappa \right)+4\kappa {\left({\mathrm{\Delta }}_a-{{\omega }^{\prime }}_m\right)}^2+\kappa {\left(\gamma +\kappa \right)}^2}{4{G^{\prime }}_a^2{\left(\gamma +\kappa \right)}^2+4{G^{\prime }}_b^2{\left(\gamma +\kappa \right)}^2+4\gamma \kappa {\left({\mathrm{\Delta }}_a-{{\omega }^{\prime }}_m\right)}^2+\gamma \kappa {\left(\gamma +\kappa \right)}^2}\gamma {\overline{n}}_{th}.\end{array}$$

where

(28)$$\begin{array}{l}{\lambda }_1(2i{\mathrm{\Delta }}_a-2i{{\omega }^{\prime }}_m-\gamma +\kappa -\mu )(2i{\mathrm{\Delta }}_a-2i{{\omega }^{\prime }}_m+\gamma -\kappa +v),\\ {\lambda }_2(2i{\mathrm{\Delta }}_a-2i{{\omega }^{\prime }}_m-\gamma +\kappa -\mu )(2i{\mathrm{\Delta }}_a-2i{{\omega }^{\prime }}_m+\gamma -\kappa -v),\\ {\lambda }_3(2i{\mathrm{\Delta }}_a-2i{{\omega }^{\prime }}_m-\gamma +\kappa +\mu )(2i{\mathrm{\Delta }}_a-2i{{\omega }^{\prime }}_m+\gamma -\kappa +v),\\ {\lambda }_4(2i{\mathrm{\Delta }}_a-2i{{\omega }^{\prime }}_m-\gamma +\kappa +\mu )(2i{\mathrm{\Delta }}_a-2i{{\omega }^{\prime }}_m+\gamma -\kappa -v),\\ \mu \sqrt{-16{G^{\prime }}_a^2-16{G^{\prime }}_b^2+{\left(\gamma -\kappa -2i{\mathrm{\Delta }}_a+2i{{\omega }^{\prime }}_m\right)}^2},\\ v\sqrt{-16{G^{\prime }}_a^2-16{G^{\prime }}_b^2+{\left(-\gamma +\kappa -2i{\mathrm{\Delta }}_a+2i{{\omega }^{\prime }}_m\right)}^2}.\end{array}$$

In Fig. 6, we plot the mean phonon number [Eq. (27)] as a function of time. It is seen that the mean phonon number approaches the steady-state value when the time is large enough. In fact, when the time is large enough, Eq. (27) reduces to

(29)$${〈\delta c^{†}(t)\delta c(t)〉}_{ss}\gamma {\overline{n}}_{th}\frac{4{G^{\prime }}_a^2(\gamma +\kappa )+4{G^{\prime }}_b^2(\gamma +\kappa )+4\kappa {\left({\mathrm{\Delta }}_a-{{\omega }^{\prime }}_m\right)}^2+\kappa {\left(\gamma +\kappa \right)}^2}{4{G^{\prime }}_a^2{\left(\gamma +\kappa \right)}^2+4{G^{\prime }}_b^2{\left(\gamma +\kappa \right)}^2+4\gamma \kappa {\left({\mathrm{\Delta }}_a-{{\omega }^{\prime }}_m\right)}^2+\gamma \kappa {\left(\gamma +\kappa \right)}^2}.$$

 

Fig. 6 Plot of the mean phonon number 〈δc^† (t)δc(t)〉 as a function of t. Here we choose the thermal phonons ${\overline{n}}_{th}$ 10000, P₂ 2mW, Δ ω_m, and other parameters are the same as in Fig. 2.

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By substituting Eq. (29) into Eq. (15), we can obtain Eq. (16) for the steady-state variance 〈δX²〉_ss.

Appendix B Fourier transform

In the frequency domain, Eq. (13) becomes to

(30)$$\begin{array}{l}i\omega \delta a(\omega )-\left(i{\mathrm{\Delta }}_a+\frac{\kappa }{2}\right)\delta a(\omega )+i{G^{\prime }}_a\delta c(\omega )+\sqrt{\kappa }\delta a_{in}(\omega ),\\ i\omega \delta b(\omega )-\left(i{\mathrm{\Delta }}_b+\frac{\kappa }{2}\right)\delta b(\omega )+i{G^{\prime }}_b\delta c(\omega )+\sqrt{\kappa }\delta b_{in}(\omega ),\\ i\omega \delta c(\omega )-\left(i{{\omega }^{\prime }}_m+\frac{\gamma }{2}\right)\delta c(\omega )+i{G^{\prime }}_a\delta a(\omega )+i{G^{\prime }}_b\delta b(\omega )+\sqrt{\gamma }\delta c_{in}(\omega ).\end{array}$$

Solving the above equations, we obtain

(31)$$\begin{array}{l}\delta c(\omega )\frac{i{G^{\prime }}_a^2(i\omega +i{\mathrm{\Delta }}_b+\frac{\kappa }{2})}{J(\omega )}\sqrt{\kappa }\delta a_{in}(\omega )+\frac{i{G^{\prime }}_b^2(i\omega +i{\mathrm{\Delta }}_a+\frac{\kappa }{2})}{J(\omega )}\sqrt{\kappa }\delta b_{in}(\omega )\\ +\frac{(i\omega +i{\mathrm{\Delta }}_a+\frac{\kappa }{2})(i\omega +i{\mathrm{\Delta }}_b+\frac{\kappa }{2})}{J(\omega )}\sqrt{\gamma }\delta c_{in}(\omega ),\end{array}$$

where

(32)$$\begin{array}{l}J(\omega )\left(i\omega +i{{\omega }^{\prime }}_m+\frac{\gamma }{2}\right)\left(i\omega +i{\mathrm{\Delta }}_a+\frac{\kappa }{2}\right)\left(i\omega +i{\mathrm{\Delta }}_b+\frac{\kappa }{2}\right)\\ +{G^{\prime }}_a^2\left(i\omega +i{\mathrm{\Delta }}_b+\frac{\kappa }{2}\right)+{G^{\prime }}_b^2\left(i\omega +i{\mathrm{\Delta }}_a+\frac{\kappa }{2}\right).\end{array}$$

The mean phonon number is determined by

(33)$$〈\delta c^{†}(t)\delta c(t)〉\frac{1}{4{\pi }^2}{\displaystyle \int {\displaystyle {\int }_{-\infty }^{+\infty }d\omega d\mathrm{\Omega }{e}^{-i(\omega +\mathrm{\Omega })t}〈\delta c^{†}(\omega )\delta c(\mathrm{\Omega })〉.}}$$

To calculate the mean phonon number, we require the correlation functions of the noise sources in the frequency domain. Fourier transforming Eq. (14) gives the correlation functions in the frequency domain

(34)$$\begin{array}{l}〈\delta a_{in}(\omega )\delta a_{in}^{†}(-\mathrm{\Omega })〉〈\delta b_{in}(\omega )\delta b_{in}^{†}(-\mathrm{\Omega })〉2\pi \delta (\omega +\mathrm{\Omega }),\\ 〈\delta c_{in}(\omega )\delta c_{in}^{†}(-\mathrm{\Omega })〉2\pi ({\overline{n}}_{th}+1)\delta (\omega +\mathrm{\Omega }),\\ 〈\delta c_{in}^{†}(-\mathrm{\Omega })\delta c_{in}(\omega )〉2\pi {\overline{n}}_{th}\delta (\omega +\mathrm{\Omega }).\end{array}$$

Upon substituting Eq. (31) into Eq. (33) and taking into account Eq. (34), we obtain

(35)$$〈\delta c^{†}(t)\delta c(t)〉\frac{\gamma {\overline{n}}_{th}}{2\pi }{\displaystyle {\int }_{-\infty }^{+\infty }\frac{(-i\omega -i{\mathrm{\Delta }}_a+\frac{\kappa }{2})(-i\omega -i{\mathrm{\Delta }}_b+\frac{\kappa }{2})}{D(\omega )}\frac{(-i\omega -i{\mathrm{\Delta }}_a+\frac{\kappa }{2})(-i\omega -i{\mathrm{\Delta }}_b+\frac{\kappa }{2})}{H(\omega )}}d\omega ,$$

where

(36)$$\begin{array}{l}D(\omega )\left(-i\omega -i{{\omega }^{\prime }}_m+\frac{\gamma }{2}\right)\left(-i\omega -i{\mathrm{\Delta }}_a+\frac{\kappa }{2}\right)\left(-i\omega -i{\mathrm{\Delta }}_b+\frac{\gamma }{2}\right)\\ +{G^{\prime }}_a^2\left(-i\omega -i{\mathrm{\Delta }}_b+\frac{\kappa }{2}\right)+{G^{\prime }}_a^2\left(-i\omega -i{\mathrm{\Delta }}_a+\frac{\kappa }{2}\right),\\ H(\omega )\left(-i\omega +i{{\omega }^{\prime }}_m+\frac{\gamma }{2}\right)\left(-i\omega +i{\mathrm{\Delta }}_a+\frac{\kappa }{2}\right)\left(-i\omega -i{\mathrm{\Delta }}_b+\frac{\kappa }{2}\right)\\ +{G^{\prime }}_a^2\left(-i\omega +i{\mathrm{\Delta }}_b+\frac{\kappa }{2}\right)+{G^{\prime }}_b^2\left(-i\omega +i{\mathrm{\Delta }}_a+\frac{\kappa }{2}\right).\end{array}$$

By substituting Eq. (35) into Eq. (15) and making numerical calculation we obtain Fig. 4.

Appendix C Calculation of the mean photon number 〈δa^† (t)δa(t)〉_ss

In the special situation of ${G^{\prime }}_b$ 0, following the method outlined in Appendix A, we obtain the solution of Eq. (13) to be

(37)$$\begin{array}{l}\delta a(t)g_1(t)\delta a(0)+\sqrt{\kappa }{\displaystyle {\int }_0^{t}d{t}^{\prime }{g}_1(t-t^{\prime })\delta a_{in}(t^{\prime })}\\ +g_2(t)\delta c(0)+\sqrt{\gamma }{\displaystyle {\int }_0^{t}d{t}^{\prime }{g}_2(t-t^{\prime })\delta c_{in}(t^{\prime })},\end{array}$$

where

(38)$$\begin{array}{l}{g}_1(t)\frac{1}{2\sigma }\exp \left[\frac{-\gamma -\kappa -2i({\mathrm{\Delta }}_a+{{\omega }^{\prime }}_m)}{4}t\right]\{\sigma [\exp (\sigma t/4)+\exp (\sigma t/4)]\\ +(\gamma -\kappa -2i{\mathrm{\Delta }}_a+2i{{\omega }^{\prime }}_m)[\exp (\sigma t/4)-\exp (-\sigma t/4)]\},\\ g_2(t)2i{G^{\prime }}_a\exp \left[\frac{-\gamma -\kappa -2i({\mathrm{\Delta }}_a+{{\omega }^{\prime }}_m)}{4}t\right]\frac{\exp (\sigma t/4)-\exp (-\sigma t/4)}{\sigma },\\ \sigma \sqrt{-16{G^{\prime }}_a^2+{\left(\gamma -\kappa -2i{\mathrm{\Delta }}_a+2i{{\omega }^{\prime }}_m\right)}^2.}\end{array}$$

Using these solutions, the mean photon number can be derived as

(39)$$\begin{array}{l}〈\delta a^{†}(t)\delta a(t)〉\frac{\gamma {\overline{n}}_{th}}{\sigma \rho }\frac{4\exp [-t(2\gamma +2\kappa +\rho -\sigma )/4]}{(2\gamma +2\kappa +\rho -\sigma )}+\frac{\gamma {\overline{n}}_{th}}{\sigma \rho }\frac{4\exp [-t(2\gamma +2\kappa -\rho +\sigma )/4]}{(2\gamma +2\kappa -\rho +\sigma )}\\ -\frac{\gamma {\overline{n}}_{th}}{\sigma \rho }\frac{4\exp [-t(2\gamma +2\kappa +\rho +\sigma )/4]}{(2\gamma +2\kappa +\rho +\sigma )}-\frac{\gamma {\overline{n}}_{th}}{\sigma \rho }\frac{4\exp [-t(2\gamma +2\kappa -\rho -\sigma )/4]}{(2\gamma +2\kappa -\rho -\sigma )}\\ \gamma {\overline{n}}_{th}\frac{4{G^{\prime }}_a^2(\gamma +\kappa )}{4{G^{\prime }}_a^2{\left(\gamma +\kappa \right)}^2+4\gamma \kappa {\left({\mathrm{\Delta }}_a-{{\omega }^{\prime }}_m\right)}^2+\gamma \kappa {\left(\gamma +\kappa \right)}^2},\end{array}$$

where

(40)$$\begin{array}{l}\rho \sqrt{-16{G^{\prime }}_a^2+{\left(\gamma -\kappa +2i{\mathrm{\Delta }}_a-2i{{\omega }^{\prime }}_m\right)}^2},\\ \sigma \sqrt{-16{G^{\prime }}_a^2+{\left(\gamma -\kappa -2i{\mathrm{\Delta }}_a+2i{{\omega }^{\prime }}_m\right)}^2}.\end{array}$$

In Fig. 7, we plot the mean photon number [Eq. (39)] as a function of time. It is seen that the mean photon number approaches the steady-state value at large enough time. In fact, when the time is large enough, Eq. (39) reduces to

(41)$${〈\delta a^{†}(t){\delta }_a(t)〉}_{ss}\gamma {\overline{n}}_{th}\frac{4{G^{\prime }}_a^2(\gamma +\kappa )}{4{G^{\prime }}_a^2{\left(\gamma +\kappa \right)}^2+4\gamma \kappa {\left({\mathrm{\Delta }}_a-{{\omega }^{\prime }}_m\right)}^2+\gamma \kappa {\left(\gamma +\kappa \right)}^2}.$$

 

Fig. 7 Plot of the mean photon number 〈δa^† (t) δa (t) as a function of t. Here we choose the thermal phonons ${\overline{n}}_{th}$ 10000, P₂ 1μW, and other parameters are the same as in Fig. 2.

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The form of the operator 〈δa^† (t)δa(t)〉_ss can be used to calculate the second-order correlation function of the cavity field.

References and links

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