Improving Mechanical Oscillator Cooling in a Double-Coupled Cavity Optomechanical System with an Optical Parametric Amplifier

Abstract

We investigate the cooling phenomenon of a mechanical oscillator in a double-coupled cavity optomechanical system. Our model includes two single-mode optical cavities. The left cavity is an optomechanical system with an optical parametric amplifier, and the right cavity is a standard optical cavity. The two optical cavities couple with each other by exchanging photons. The optomechanical system is effectively driven by an input laser field. By solving the linear quantum Langevin equation of the system under a steady-state condition, we can obtain the position fluctuation spectrum and momentum fluctuation spectrum of the mechanical oscillator, and then, the expression of its effective temperature is obtained. Through numerical analysis, we find the change in the effective temperature of the mechanical oscillator under different physical parameters. The results show that the cooling of the mechanical oscillator can be significantly improved in the presence of optical parameter amplification and adjustment of optical cavity parameters. Our cooling solutions have potential applications for the preparation of nonclassical states of mechanical oscillators, high-precision measurements, and quantum information processing.

1. Introduction

In theoretical and experimental fields, the cavity optomechanical system [1,2] has attracted extensive attention for its wide applications, such as the high-precision detection of gravitational waves and mass [3,4], mechanical entanglement [5], biological sensing [6], mechanical squeezing [7,8], mechanical displacement [9], quantum information processing [10], and superpositions of macroscopic quantum states [11]. The research on optomechanical systems shows that people can use photons, nonclassical states, and mechanical motion to manipulate and measure quantum effects in quantum regions. The optomechanical system can constitute a hybrid quantum system [12,13] with other different physical systems, providing a new approach for us to explore the quantum world. A specific optomechanical system is mainly composed of a mechanical oscillator and cavity. In order to observe and measure the quantum effect of macroscopic mechanical systems experimentally, it is necessary to overcome the thermal noise of the mechanical oscillator and prepare it in the quantum ground state [14]. Therefore, exploring mechanical oscillator cooling becomes a crucial adjective, which could be achieved in the optomechanical system, enhancing the interaction between the optical field and the mechanical motion to reduce the thermal noise [15].

Currently, considerable progress has been made in the cooling of mechanical oscillators. Generally, the most frequently used methods are self-cooling [16,17,18,19,20,21,22] and active feedback cooling [23,24,25,26,27]. In the active feedback cooling schemes, a viscous force is applied to a mechanical oscillator by measuring its position without generating thermal noise. However, the measurement accuracy of this method is affected by noise in the feedback loop. The self-cooling scheme is to achieve cooling by interacting with the mechanical oscillator and auxiliary system and absorbing energy from the mechanical oscillator. Resolved sideband cooling [28,29,30] is a universal study in self-cooling scheme. However, the precondition of this scheme is that the dissipation coefficient of the optical cavity is much smaller than the frequency of the mechanical oscillator, which has high requirements for the quality factor of the optical cavity. Later, people began to explore mechanical oscillator cooling under unresolved sideband conditions. For example, Guo et al. used electromagnetically induced transparency to break through the limitation of the unresolved conditions of the sideband on the fluctuation spectrum of light pressure [31]. Furthermore, Liu et al. demonstrated that the motional ground state can be achieved in the highly unresolved sideband regime through coherent auxiliary cavity interferences [32].

Cooling mechanical oscillators to lower temperatures has become a greater focus. A degenerate optical parametric amplifier [33] has been shown to improve the cooling of mechanical oscillators. For instance, Huang et al. used parametric to enhance the cavity cooling of a micromechanical mirror in the dispersive coupling [34]. Further, they investigated the cooling of the mechanical membrane in a dissipative optomechanical system with a degenerate optical parametric amplifier [35]. Researchers have demonstrated that coupling a degenerate optical parametric with a Fabry–Perot optical cavity can tremendously improve the cooling of the mechanical oscillator [22,34,36].

This paper analyzes the scheme for improving mechanical oscillator cooling in double-coupled cavity optomechanical systems with an optical parametric amplifier. The paper is organized as follows: Section 2 introduces the model of the system and provides the quantum Langevin equations. In Section 3, we calculate the position fluctuation spectrum and momentum fluctuation spectrum of the mechanical oscillator and derive the effective temperature of the mechanical oscillator. In Section 4, we show the effective resonance frequency and effective damping rate of the mechanical oscillator, and we also represent how the cooling of the mechanical oscillator can be effectively improved by using an optical parametric amplifier inside the cavity alongside adjusting the optical cavity parameters. Some conclusions are presented in Section 5.

2. Model

The model that we consider is a coupled single-mode double-cavity, as shown in Figure 1. The left and right optical cavities are cavity $a_1$ and cavity $a_2$, respectively. The left cavity is an optomechanical system and the right cavity is a standard optical cavity. The two cavities are coupled to each other through photon exchange coupling with the coupling coefficient $J$. A degenerate optical parametric amplifier is placed in the left cavity. The resonance frequency and decay rate of the left cavity are ${\omega }_1$ and ${\kappa }_1$, respectively, and the resonance frequency and decay rate of the right cavity are ${\omega }_2$ and ${\kappa }_2$, respectively. The movable cavity mirror in the left cavity can be regarded as a mechanical oscillator with a natural frequency of ${\omega }_m$, effective mass $m$, and damping rate ${\gamma }_m$. The system is in a state of thermal equilibrium at temperature $T$. In addition, the left cavity is driven with an input laser field with frequency ${\omega }_L$, power $P$ and amplitude $\epsilon$, which does not interact directly with the right cavity.

Therefore, the Hamiltonian of the system under the rotational frame of the input laser frequency ${\omega }_L$ can be expressed as:

$$\begin{array}{c}H=\hslash {\mathit{\Delta }}_1{a}_1^{†}{a}_1+\hslash {\mathit{\Delta }}_2{a}_2^{†}{a}_2+\frac{1}{2}\left(\frac{p^2}{m}+m{\omega }_m^2{q}^2\right)-\hslash g{a}_1^{†}{a}_{1}q+\hslash J\left(a_1^{†}{a}_2+a_1{a}_2^{†}\right)\\ +i\hslash \epsilon \left(a_1^{†}-a_1\right)+i\hslash G\left(e^{i\theta }{a}_1^{†}{}^2-e^{-i\theta }{a}_1^2\right).\end{array}$$

(1)

Here $a_1$ and $a_1{}^{†}$ are the annihilation and creation operators for the field inside the left cavity, respectively, and $a_2$ and $a_2{}^{†}$ are the annihilation and creation operators for the field inside the right cavity, respectively, which satisfy the commutative relationship $\left[a_j,a_j{}^{†}\right]=1\left(j=\mathrm{1,2}\right)$. ${\mathsf{\Delta }}_1={\omega }_1-{\omega }_L$, where ${\mathsf{\Delta }}_2={\omega }_2-{\omega }_L$ are the detuning of the left cavity and the right cavity to the input laser, respectively. Further, $p$ and $q$ are the momentum and position operators of the movable cavity mirror, respectively, satisfying $\left[q,p\right]=i\hslash$. While $g={\omega }_1/L$ is the single-photon optomechanical coupling coefficient, and $L$ is the length of the two cavities. Then, $\mathcal{E}={\left(2{\kappa }_{1}P/\hslash {\omega }_L\right)}^{1/2}$ is the amplitude of the input laser, $G$ is the nonlinear gain of the OPA, and $\theta$ is the drive field phase of the OPA. In Equation (1), the first two terms represent the energy of the left cavity and the right cavity, respectively, the third term represents the energy of the mechanical oscillator, the fourth term represents the interaction between the mechanical oscillator and the left cavity, the fifth term represents the interaction between the left cavity and the right cavity, the sixth term represents the interaction between the left cavity and the input driving light, and the last term represents the interaction between the left cavity and OPA. The quantum Langevin equation of the system can be provided by the Heisenberg equation:

$$\begin{array}{l}\dot{q}=\frac{p}{m},\\ \dot{p}=-m{\omega }_m^{2}q+\hslash g{a}_1^{†}{a}_1-{\gamma }_{m}p+\xi ,\\ {\dot{a}}_1=-\left(i{\mathit{\Delta }}_1+{\kappa }_1\right)a_1+ig{a}_{1}q-iJ{a}_2+2G{e}^{i\theta }{a}_1^{†}+\epsilon +\sqrt{2{\kappa }_1}{a}_{1,in},\\ {\dot{a}}_2=-\left(i{\mathit{\Delta }}_2+{\kappa }_2\right)a_2-iJ{a}_1+\sqrt{2{\kappa }_2}{a}_{2,in}.\end{array}$$

(2)

Here, we introduce the input vacuum noise operators $a_{1,in}$ and $a_{2,in}$ with zero mean values, and the relevant equations in the time domain are [37]:

$$\begin{array}{l}\langle \delta a_{1,in}\left(t\right)\delta a_{1,in}^{†}\left(t^{\prime }\right)\rangle =\delta \left(t-t^{\prime }\right),\\ \langle \delta a_{1,in}\left(t\right)\delta a_{1,in}\left(t^{\prime }\right)\rangle =\langle \delta a_{1,in}^{†}\left(t\right)\delta a_{1,in}\left(t^{\prime }\right)\rangle =0,\\ \langle \delta a_{2,in}\left(t\right)\delta a_{2,in}^{†}\left(t^{\prime }\right)\rangle =\delta \left(t-t^{\prime }\right),\\ \langle \delta a_{2,in}\left(t\right)\delta a_{2,in}\left(t^{\prime }\right)\rangle =\langle \delta a_{2,in}^{†}\left(t\right)\delta a_{2,in}\left(t^{\prime }\right)\rangle =0.\end{array}$$

(3)

Due to the interaction between the mechanical oscillator and the thermal bath, a force $\xi$ will be generated, which is a Brownian noise operator, and its mean value is also zero, and the relevant equation satisfied is [38]:

$$\langle \xi \left(t\right)\xi \left(t^{\prime }\right)\rangle =\frac{\hslash {\gamma }_{m}m}{2\pi }\int \omega e^{-i\omega \left(t-t^{\prime }\right)}\left[\coth \left(\frac{\hslash \omega }{2k_{B}t}\right)+1\right]d\omega ,$$

(4)

where $k_B$ is the Boltzmann constant and $\mathrm{T}$ is the bath temperature. By setting all-time derivatives of the quantum Langevin equation to zero, we can obtain the steady state average of the system:

$$\begin{array}{l}{p}_s=0,\\ q_s=\frac{\hslash g{\left|a_{1s}\right|}^2}{m{\omega }_m^2},\\ a_{2s}=\frac{-iJ{a}_{1s}}{i{\mathit{\Delta }}_2+{\kappa }_2},\\ a_{1s}=\frac{2G{e}^{i\theta }\epsilon \left({\mathit{\Delta }}_2^2+{\kappa }_2^2\right)+\epsilon \left(i{\mathit{\Delta }}_2+{\kappa }_2\right)\left[\left({\kappa }_1-i\mathit{\Delta }\right)\left({\kappa }_2-i{\mathit{\Delta }}_2\right)\right]}{\left({\kappa }_1^2+{\mathit{\Delta }}^2\right)\left({\kappa }_2^2+{\mathit{\Delta }}_2^2\right)+2J^2\left({\kappa }_1{\kappa }_2-\mathit{\Delta }{\mathit{\Delta }}_2\right)+J^4-4G^2\left({\kappa }_2^2+{\mathit{\Delta }}_2^2\right)}.\end{array}$$

(5)

Here $a_{1s}$ and $a_{2s}$ represent the steady-state amplitude of the two cavity fields, respectively. Moreover, $\mathit{\Delta }={\mathit{\Delta }}_1-g{q}_s$ is the detuning of the effective cavity, which is generated under the influence of the radiation pressure and is related to the average displacement of the mechanical oscillator. We can see that $q_s\ne 0$, indicating that after adding a series of coupled systems, the mechanical oscillator is moved around the new equilibrium position.

3. Radiation Pressure and Quantum Fluctuations

Since the fluctuation $\delta q=q-q_s$ is the real operator of the mechanical oscillator, and it is also the object that is processed by cooling the mechanical oscillator, we first define all the fluctuation operators of the system:

$$\delta q=q-q_s,\delta p=p-p_s,\delta a_1=a_1-a_{1s},\delta a_2=a_2-a_{2s}.$$

(6)

Substituting the fluctuation operator into the equation, we can get the linear quantum Langevin equation:

$$\begin{array}{l}\delta \dot{q}=\frac{\delta p}{m},\\ \delta \dot{p}=-m{\omega }_m^2\delta q+\hslash g\left(a_{1s}\delta a_1^{†}+a_{1s}^{\ast }\delta a_1\right)-{\gamma }_m\delta p+\xi ,\\ \delta {\dot{a}}_1=-\left(i\mathit{\Delta }+{\kappa }_1\right)\delta a_1+ig{a}_{1s}\delta q-iJ\delta a_2+2G{e}^{i\theta }\delta a_1^{†}+\sqrt{2{\kappa }_1}{a}_{1,in},\\ \delta {\dot{a}}_2=-\left(i{\mathit{\Delta }}_2+{\kappa }_2\right)\delta a_2-iJ\delta a_1+\sqrt{2{\kappa }_2}{a}_{2,in}. \end{array}$$

(7)

Here we introduce the quadrature for the two cavity fields $\delta x_1=\delta a_1+\delta a_1^{†}$, $\delta y_1=i\left(\delta a_1^{†}-\delta a_1\right)$ and $\delta x_2=\delta a_2+\delta a_2^{†}, \delta y_2=i\left(\delta a_2^{†}-\delta a_2\right)$, alongside the input noise quadrature operator: $x_{1,in}=a_{1,in}+a_{1,in}^{†}, y_{1,in}=i\left(a_{1,in}^{†}-a_{1,in}\right)$ and $x_{2,in}=a_{2,in}+a_{2,in}^{†}, y_{2,in}=i\left(a_{2,in}^{†}-a_{2,in}\right)$. Equation (7) can be expressed in matrix form:

$$\dot{A}\left(t\right)=MA\left(t\right)+\eta \left(t\right),$$

(8)

here $A\left(t\right)$ is a column vector of fluctuations, $\eta \left(t\right)$ is a column vector of noise terms, and their transpose can be written as:

$$\begin{array}{l}A{\left(t\right)}^T=\left(\delta q,\delta p,\delta x_1,\delta y_1,\delta x_2,\delta y_2\right),\\ \eta {\left(t\right)}^T=\left(0,\xi ,\sqrt{2{\kappa }_1}{x}_{1,in},\sqrt{2{\kappa }_1}{y}_{1,in},\sqrt{2{\kappa }_2}{x}_{2,in},\sqrt{2{\kappa }_2}{y}_{2,in}\right).\end{array}$$

(9)

The matrix $M$ is represented as:

$$\left(\begin{array}{cccccc}0& \frac{1}{m}& 0& 0& 0& 0\\ -m{\omega }_m^2& -{\gamma }_m& \hslash \varphi & \hslash \psi & 0& 0\\ -2\psi & 0& -\left({\kappa }_1-2G\cos \theta \right)& \mathit{\Delta }+2G\sin \theta & 0& J\\ 2\varphi & 0& -\mathit{\Delta }+2G\sin \theta & -\left({\kappa }_1+2G\cos \theta \right)& -J& 0\\ 0& 0& 0& J& -{\kappa }_2& {\mathit{\Delta }}_2\\ 0& 0& -J& 0& -{\mathit{\Delta }}_2& -{\kappa }_2\end{array}\right),$$

(10)

here $\varphi =g\left(a_{1s}+a_{1s}^{\ast }\right)/2$ and $\psi =g\left(a_{1s}-a_{1s}^{\ast }\right)/2i$. The solution of Equation (7) is stable when all the eigenvalues of the matrix have negative real parts. According to the Rouse-Hurwitz stability criterion [39], we can obtain the stability condition. Since it is too complicated, it will not be shown here, and we will perform numerical simulations on the parameters we selected in order to satisfy the stability conditions. By performing the Fourier transform on the formula, we obtain the position fluctuation of the mechanical oscillator in the frequency domain:

$$\begin{array}{c}\delta q\left(\omega \right)=\chi \left(\omega \right)[A_1\left(\omega \right)a_{1,in}\left(\omega \right)+A_2\left(\omega \right)a_{1,in}^{†}\left(\omega \right)+A_3\left(\omega \right)a_{2,in}\left(\omega \right)\\ +A_4\left(\omega \right)a_{2,in}^{†}\left(\omega \right)+Z\left(\omega \right)\xi \left(\omega \right)],\end{array}$$

(11)

where

$$\begin{gathered} \begin{array}{l} {\chi }^{-1}\left(\omega \right)=m\left({\omega }_m^2-i\omega {\gamma }_m-{\omega }^2\right)Z\left(\omega \right)-i\hslash {\left|g_0\right|}^2\left[v_{+}\left(u_{-}{v}_{-}+J^2\right)-v_{-}\left(u_{+}{v}_{+}+J^2\right)\right] \\ -2i\hslash G{v}_{+}{v}_{-}\left(g_0^2{e}^{-i\theta }-g_0^{\ast }{}^2{e}^{i\theta }\right),\\ Z \left(\omega \right)=\left(u_{+}{v}_{+}+J^2\right)\left(u_{-}{v}_{-}+J^2\right)-4G^2{v}_{+}{v}_{-}, g_0=g{a}_{1s}, g_0^{\ast }=g{a}_{1s}^{\ast },\\ u_{+}={\kappa }_1-i\omega +i\mathit{\Delta },u_{-}={\kappa }_1-i\omega -i\mathit{\Delta }, v_{+}={\kappa }_2-i\omega +i{\mathit{\Delta }}_2, v_{-}={\kappa }_2-i\omega -i{\mathit{\Delta }}_{2,}\\ A_1\left(\omega \right)=2\hslash g_{0}G{e}^{-i\theta }{v}_{+}{v}_{-}\sqrt{2{\kappa }_1}+\hslash g_0^{\ast }{v}_{+}\sqrt{2{\kappa }_1}\left(u_{-}{v}_{-}+J^2\right),\\ A_2\left(\omega \right)=2\hslash g_0^{\ast }G{e}^{i\theta }{v}_{+}{v}_{-}\sqrt{2{\kappa }_1}+\hslash g_0{v}_{-}\sqrt{2{\kappa }_1}\left(u_{+}{v}_{+}+J^2\right),\\ A_3\left(\omega \right)=-2i\hslash g_{0}JG{e}^{-i\theta }{v}_{-}\sqrt{2{\kappa }_2}-iJ\hslash g_0^{\ast }\left(u_{-}{v}_{-}+J^2\right)\sqrt{2{\kappa }_2},\\ A_4\left(\omega \right)=2i\hslash g_0^{\ast }JG{e}^{i\theta }{v}_{+}\sqrt{2{\kappa }_2}+iJ\hslash g_0\left(u_{+}{v}_{+}+J^2\right)\sqrt{2{\kappa }_2}.\end{array} \end{gathered}$$

(12)

In Equation (11), the first four terms are caused by the input vacuum noise caused by mutual coupling in the system, and the last term is derived from the thermal noise in the environment. Here $\chi \left(\omega \right)$ is the susceptibility of the mechanical oscillator in the presence of the coupled system, which can be written as:

$$\chi \left(\omega \right)=\frac{1}{m\left[{\omega }_{eff}{\left(\omega \right)}^2-i\omega {\gamma }_{eff}\left(\omega \right)-{\omega }^2\right]},$$

(13)

where ${\omega }_{eff}\left(\omega \right)$ and ${\gamma }_{eff}\left(\omega \right)$ are the effective resonance frequency and effective damping rate of the mechanical oscillator, respectively, and they can be expressed as:

$$\begin{array}{c}{\omega }_{eff}\left(\omega \right)={\left\{{\omega }_m^2-Re\left[\frac{B_1\left(\omega \right)+B_2\left(\omega \right)}{mZ\left(\omega \right)}\right]\right\}}^{\frac{1}{2}}, \\ {\gamma }_{eff}\left(\omega \right)={\gamma }_m+Im\left[\frac{B_1\left(\omega \right)+B_2\left(\omega \right)}{m\omega Z\left(\omega \right)}\right], \end{array}$$

(14)

here,

$$\begin{array}{l}{B}_1\left(\omega \right)=i\hslash {\left|g_0\right|}^2{v}_{+}\left(u_{-}{v}_{-}+J^2\right)-i\hslash {\left|g_0\right|}^2{v}_{-}\left(u_{+}{v}_{+}+J^2\right),\\ B_2\left(\omega \right)=2i\hslash G{v}_{+}{v}_{-}\left(g_0^2{e}^{-i\theta }-g_0^{\ast }{}^2{e}^{i\theta }\right).\end{array}$$

(15)

We note that the effective resonance frequency ${\omega }_{eff}\left(\omega \right)$ and the effective damping rate ${\gamma }_{eff}\left(\omega \right)$ of the mechanical oscillator are related to the enhanced optomechanical coupling coefficient, the parametric gain coefficient $\mathrm{G}$ of the OPA, and the phase $\mathsf{\theta }$. In the case of an uncoupled system ($g=0$), the mechanical oscillator performs Brownian motion, and its effective susceptibility $\chi \left(\omega \right)=m^{-1}{\left({\omega }_m^2-i\omega {\gamma }_m-{\omega }^2\right)}^{-1}$ is a standard Lorentzian function [40]. The position fluctuation spectrum of the mechanical oscillator is defined as:

$$2\pi S_q\left(\omega \right)\delta \left(\omega +\Omega \right)=\frac{1}{2}\left[\langle \delta q\left(\omega \right)\delta q\left(\Omega \right)\rangle +\langle \delta q\left(\Omega \right)\delta q\left(\omega \right)\rangle \right],$$

(16)

whereby we introduce the non-zero correlation of the noise term in the frequency domain:

$$\begin{array}{l}\langle a_{1,in}\left(\omega \right)a_{1,in}^{†}\left(-\Omega \right)\rangle =2\pi \delta \left(\omega +\Omega \right), \\ \langle a_{2,in}\left(\omega \right)a_{2,in}^{†}\left(-\Omega \right)\rangle =2\pi \delta \left(\omega +\Omega \right),\\ \langle \xi \left(\omega \right)\xi \left(\Omega \right)\rangle =2\pi \hslash {\gamma }_{m}m\omega \left[1+coth\left(\frac{\hslash \omega }{2k_{B}T}\right)\right]\delta \left(\omega +\Omega \right),\end{array}$$

(17)

then, we obtain the position fluctuation spectrum of the mechanical oscillator:

$$S_q\left(\omega \right)=\chi \left(\omega \right)\chi \left(-\omega \right)\left[A_{12}\left(\omega \right)+A_{34}\left(\omega \right)+\hslash m\omega {\gamma }_{m}Z\left(\omega \right)Z\left(-\omega \right)coth\left(\frac{\hslash \omega }{2k_{B}T}\right)\right],$$

(18)

where

$$\begin{array}{c}{A}_{12}\left(\omega \right)=\frac{A_1\left(\omega \right)A_2\left(-\omega \right)+A_1\left(-\omega \right)A_2\left(\omega \right)}{2}, \\ A_{34}\left(\omega \right)=\frac{A_3\left(\omega \right)A_4\left(-\omega \right)+A_3\left(-\omega \right)A_4\left(\omega \right)}{2}.\end{array}$$

(19)

In Equation (18), the first two terms are the contribution of the input vacuum noise, and the last term is the contribution of the thermal noise. When we take the Fourier transform of $\stackrel{·}{\delta q}=\delta p/m$ in Equation (7), we can get $S_p\left(\omega \right)=m^2{\omega }^2{S}_q\left(\omega \right)$, which makes the momentum fluctuation spectrum of the mechanical oscillator $S_p\left(\omega \right)=m^2{\omega }^2{S}_q\left(\omega \right)$. For a system in thermal equilibrium, we can use the energy equipartition theorem to define the effective temperature $k_B{T}_{eff}=m{\omega }_m^2\langle q^2\rangle /2+\langle p^2\rangle /2m$, which is determined by the total energy of the mechanical oscillator. Note, we are dealing with a drive system here, and $m{\omega }_m^2\langle q^2\rangle /2\ne \langle p^2\rangle /2m$ [34], where $\langle q^2\rangle =\frac{1}{2\pi }{{\displaystyle \int }}_{-\infty }^{\infty }{S}_q\left(\omega \right)d\omega$, $\langle p^2\rangle =\frac{1}{2\pi }{{\displaystyle \int }}_{-\infty }^{\infty }{S}_p\left(\omega \right)d\omega$. Through analysis, we find that the cooling of the mechanical oscillator is related to the decay rate, detuning, coupling coefficient, and OPA gain coefficient of the two cavities.

4. Results and Discussions

4.1. Effective Resonance Frequency and Effective Damping Rate of Mechanical Oscillator

Firstly, we show the effect of coupled dual cavity and OPA on the effective resonance frequency and effective damping rate of the mechanical oscillator. In the numerical calculations we have chosen some similar parameters that have been used in past experiments: $m=145 \mathrm{ng},L=25 \mathrm{mm},{\omega }_m/(2\pi )=215 \mathrm{kHz},P=0.2 \mathrm{Mw},{\lambda }_L=2\pi c/{\omega }_L=1064 \mathrm{nm},$ and mechanical quality factor $Q={\omega }_m/{\gamma }_m=8000$. We take the same value for the decay rate of the two cavities ${\kappa }_1={\kappa }_2={\omega }_m$; therefore, our system is in an unresolved sideband condition. We plotted the normalized effective resonance frequency ${\omega }_{eff}\left(\omega \right)/{\omega }_m$ and normalized effective damping rate ${\gamma }_{eff}\left(\omega \right)/{\gamma }_m$ of the mechanical oscillator versus the normalized frequency $\omega /{\omega }_m$ for different coupling coefficients $\mathrm{J}$ and parametric gain $\mathrm{G}$. We take the cavity detuning of the two cavities as $\mathit{\Delta }=1.2{\omega }_m$, ${\mathit{\Delta }}_2=2{\omega }_m$, and the parametric phase $\theta =0$.

In Figure 2, we fixed $J=1.5{\kappa }_1$ and plotted the curves of the normalized effective resonance frequency ${\omega }_{eff}\left(\omega \right)/{\omega }_m$, which normalized the effective damping rate ${\gamma }_{eff}\left(\omega \right)/{\gamma }_m$ versus the normalized effective resonance frequency ${\omega }_{eff}\left(0\right)/{\omega }_m$, when $G=0$ (red solid line), $G=0.15{\kappa }_1$ (green solid line), and $G=0.19{\kappa }_1$ (black solid line), respectively. According to Figure 2a,b, it can be seen that both the normalized effective resonance frequency and the normalized effective damping rate are symmetrically distributed within the selected interval as the normalized effective frequency changes, and are symmetric about the longitudinal axis. When $G=0$, there is no optical parametric amplifier placed in the left cavity. In Figure 2a, while $\omega$ is from $-7{\omega }_m$ to $-5{\omega }_m$ and from $5{\omega }_m$ to $7{\omega }_m$, the normalized effective resonance frequency of the mechanical oscillator remained basically unchanged. Then, as the value of $\omega$ increases in the left half-axis and decreases in the right half-axis, the normalized effective resonance frequency will suddenly decrease. When $\omega =0$, the normalized effective resonance frequency has a minimum value. When an optical parametric amplifier is placed in the left cavity, the image of the normalized effective resonance frequency still shows a symmetrical distribution. However, the difference between this situation and the absence of an optical parametric amplifier in the cavity is that when the normalized effective resonance frequency reaches a subpeak, it rapidly decreases and reaches a minimum value. It is worth noting that when $G=0.19{\kappa }_1$, its normalized effective resonance frequency is lower than the value when $G=0.15{\kappa }_1$. In Figure 2b, when $G=0$, the normalized effective damping rate of the mechanical oscillator shows three peaks in the selected region, with two subpeaks appearing at approximately $\omega =-3{\omega }_m$ and $\omega =3{\omega }_m$, with the maximum value appearing at $\omega =0$. However, the difference between the maximum and sub-maximum values is not significant, and as it extends along the left and right half axes, the value of the normalized effective damping rate gradually approaches zero. After adding OPA, the overall trend of the normalized effective damping rate is similar to without OPA, yet the difference is that the peak value of the normalized effective damping rate suddenly increases several times. Moreover, the peak value at $G=0.19{\kappa }_1$ is greater than at $G=0.15{\kappa }_1$, and it is found that this pattern corresponds to the normalized effective resonance frequency. That is, at $\omega =0$, the larger the normalized effective frequency ${\omega }_{eff}\left(0\right)/{\omega }_m$ of the mechanical oscillator, the smaller its normalized effective damping rate ${\gamma }_{eff}\left(0\right)/{\gamma }_m$. Figure 2 also shows that with the increase of parameter gain G, the normalized effective resonance frequency ${\omega }_{eff}\left(0\right)/{\omega }_m$ decreases, while the normalized effective damping rate increases ${\gamma }_{eff}\left(0\right)/{\gamma }_m$. However, it is worth noting that a larger value of G is not necessarily better, which indicates that adding OPA to the dual cavity can effectively assist in the cooling of the mechanical oscillator. Similarly, we also selected $J=1.25{\kappa }_1$, $\mathit{\Delta }=0.9{\omega }_m$, and ${\mathit{\Delta }}_2=2{\omega }_m$, as well as the parameter phase $\theta =0$, to observe the changes in the normalized effective resonance frequency and normalized effective damping rate of the mechanical oscillator by increasing the parameter gain $G$. As shown in Figure 3, it can be observed that the change in curves presented in Figure 3 is similar to Figure 2. In addition, the influence of the values on the normalized effective resonance frequency and normalized effective damping rate is also discussed. It is found that when $\theta =2k\pi \left(k\in N\right)$, the normalized effective resonance frequency achieves the minimum value, while the normalized effective damping rate reaches the maximum value. Therefore, in the following discussion, we take $\theta =0$ for simplicity.

4.2. Cooling the Mechanical Oscillator

We assumed that the initial environment temperature is $T=300K$, which means that the thermal phonon number of the mechanical oscillator is $n_m^{th}=2.90743\times {10}^7$, where $n_m^{th}=1/\{\exp \left[\hslash {\omega }_m/(k_{B}T\right)]-1\}$. We plotted the relationship between the effective temperature of the mechanical oscillator and the normalized detuning $\mathit{\Delta }/{\omega }_m$ of the left cavity. We fixed $J=1.5{\kappa }_1$ and plotted the curves when $G=0$, $G=0.15{\kappa }_1$, and $G=0.19{\kappa }_1$, respectively, as shown in Figure 4.

It can be seen that when the parameter gain $G$ exists, the effective temperature of the mechanical oscillator is significantly improved. As the parameter gain increases, the cooling effect of the mechanical oscillator becomes more obvious, and the oscillator has a lower temperature. Furthermore, for the different parametric gain, the process of cooling corresponds to different ranges of the detuning of the effective cavity. For the parametric gain $G=0.15{\kappa }_1$, the temperature can drop to $0.212K$, when $\mathit{\Delta }=1.35{\omega }_m$. For the parametric gain $G=0.19{\kappa }_1$, the effective temperature can arrive at $0.135K$, when $\mathit{\Delta }=1.6{\omega }_m$. Similarly, we also plotted the effective temperature with or without the parametric gain $G$, when $J=1.25{\kappa }_1$ and $J={\kappa }_1$, as shown in Figure 5.

According to the normalized effective mechanical damping ${\gamma }_{eff}\left(\omega \right)/{\gamma }_m$ in Figure 2 and Figure 3, we can know that ${\gamma }_{eff}\left(\omega \right)/{\gamma }_m$ significantly increases when a parametric gain G is introduced, which means that the damping feedback force on the mechanical oscillator increases; thus, adding OPA can improve the cooling of the mechanical oscillator. In Figure 5a, when $J=1.25{\kappa }_1$ and $\mathit{\Delta }=0.8{\omega }_m$, then, $G=0$, $G=0.05{\kappa }_1$, and $G=0.12{\kappa }_1$, the effective temperature of the mechanical oscillator is $0.548K$, $0.437K$, and $0.298K$, respectively. When $J={\kappa }_1$ and $\mathit{\Delta }=0.5{\omega }_m$, then, $G=0$,$G=0.05{\kappa }_1 \mathrm{and} G=0.1{\kappa }_1$, the effective temperature of the mechanical oscillator is $0.594K, 0.469K, \mathrm{and} 0.349K$, respectively.

In addition, we also discussed the effect of the decay rate of the right cavity on the mechanical oscillator cooling. In the case of $J=1.25{\kappa }_1,G=0$, and $J=1.25{\kappa }_1,G=0.05{\kappa }_1$, we chose parameter ${\kappa }_2$ with different values ${\kappa }_1, 1.5{\kappa }_1, 2{\kappa }_1$, and explored the relationship between the temperature of the mechanical oscillator and normalized detuning $\mathit{\Delta }/{\omega }_m$ of the left cavity, as shown in Figure 6. With the increase in the decay rate of the right cavity, the cooling of the mechanical oscillator became more obvious. In addition, with or without the existence of the parameter gain, the appearance of the lowest temperature corresponds to different detuning.

We found that with the increase of ${\kappa }_2$, the cooling effect of the mechanical oscillator was enhanced. In the same way, we also found a similar rule in the case of $J=1.5{\kappa }_1,G=0$, and $J=1.5{\kappa }_1,G=0.15{\kappa }_1$, as shown in Figure 7. Under appropriate parameter choices, a lower temperature cooling process can be achieved when the decay rate of the right cavity gradually increases. From Figure 7a,b, in this case we can find that the presence of parametric gain has little effect on the cooling of the mechanical oscillator, and the minimum temperature for both processes is around $0.2K$. Comparing Figure 7 with Figure 6, the cooling process has a strong dependence on the coupling between the two cavities. In Figure 7, we chose a larger coupling between the two cavities, and the cooling effect was reduced, as shown in Figure 6, where the coupling between the two cavities demonstrates smaller values. Thus, the parameter choices during the cooling process are very strict.

5. Conclusions

In conclusion, we investigated mechanical oscillator cooling in two optical cavities, where one optical cavity is an optomechanical system and the other is a standard optical cavity. Photon exchange coupling existed between the two cavities. In addition, we placed an optical parametric amplifier in the left cavity. When the optomechanical system is driven by an input laser, we discussed the cooling of the mechanical oscillator under this condition using different parameters. Our study shows that we can improve the cooling effect of the mechanical oscillator under unresolved conditions. For appropriate physical parameters, we demonstrated that the cooling effect of the mechanical oscillator can also be improved with an increase in the gain factor of the optical parametric amplifier. In addition, we can also see that the mechanical oscillator can be cooled to a lower temperature by adjusting the decay rate of the right cavity. Our studies show that the photon exchange coupling between two cavities can also affect the cooling of the mechanical oscillator. Recently, with the development of micro/nano-processing technology, the experimental progress of cavity optomechanical systems has also developed rapidly [2,3,4,5], which provides an ideal platform for the study of quantum nonlinear phenomena in cavity optomechanical systems. A scheme for cooling the mechanical resonator close to its ground state, via an electromagnetically-induced-transparency-like mechanism in a double-cavity optomechanical system is proposed in ref. [31]. Our theoretical model is based on these research developments; therefore, our study has feasibility in these experiments.