## 2 Model
The \(c_{in}\) is the optical vacuum noise operator that is incident upon the beam splitter, it has zero mean value and is delta-correlated as  
\[\langle c_{in}(\tau)c_{in}^{\dagger}(\tau^{\prime})\rangle = \delta(\tau-\tau^{\prime}). \tag{5}\]  
Here, \(tc_{in}\) represents a part of the input vacuum noise \(c_{in}\) which is transmitted by the beam splitter to enter the cavity. Moreover, \(rc_{out}\) represents a part of the output field \(c_{out}\) which is reflected by the beam splitter to enter the cavity. The output field \(c_{out}\) is related to the cavity field \(c\) via the input-output relation \(c_{out}=\sqrt{2\kappa}c-tc_{in}\), which differs from the standard input-output relation [32] due to the presence of the beam splitter. Then the Equation (3) can be written as  
\[ \begin{array}{ll}\dot{Q}=&\omega_{m}P,\\ \dot{P}=&-\omega_{m}Q+g_{0}c^{\dagger}c-\gamma_{m}P+\xi,\\ \dot{c}=&-[\kappa_{eff}+i(\omega_{c}-\omega_{l})]c+ig_{0}cQ+t\varepsilon_{l}+ \sqrt{2\kappa}(1-r)tc_{in}.\end{array}  \tag{6}\]  
For the time evolution of the cavity field in Equation (6), \(\kappa_{eff}=\kappa(1-2r)\) is the effective cavity decay rate, we note that the input vacuum noise term always accompanies the cavity damping term, thus \(r\) must be less than \(\frac{1}{2}\) so that \(\kappa_{eff}>0\). In the steady state, the solution to Equation (6) is given by  
\[ \begin{array}{ll}P_{s}=&0,\\ Q_{s}=&\frac{g_{0}}{\omega_{m}}|c_{s}|^{2},\\ c_{s}=&\frac{\kappa_{eff}}{\kappa_{eff}+i\Delta},\end{array}  \tag{7}\]  
where \(\Delta=\omega_{c}-\omega_{l}-g_{0}Q_{s}\) is the effective frequency detuning of the cavity field with respect to the input laser, including the frequency shift \(g_{0}Q_{s}\) due to the radiation pressure. It is noted that the steady-state amplitude \(c_{s}\) of the cavity mode is related to the reflection and transmission coefficients \((r,t)\) of the beam splitter, the steady-state displacement \(Q_{s}\) of the movable mirror depends on the photon number \(|c_{s}|^{2}\) in the cavity, and the steady-state momentum \(P_{s}\) of the movable mirror is zero. We deal here with the case that the cavity field is driven by a strong laser so that the photon number in the cavity is large (\(|c_{s}|^{2}\gg 1\)). In this case, the nonlinear quantum Langevin Equation (6) can be linearized by writing each operator of the system in Equation (6) as \(Q=Q_{s}+\delta Q\), \(P=P_{s}+\delta P\), \(c=c_{s}+\delta c\), where \(\delta Q\), \(\delta P\), and \(\delta c\) represent the small fluctuations with zero mean value around the steady-state mean values \(Q_{s}\), \(P_{s}\), and \(c_{s}\), respectively. The linearized quantum Langevin equations for the fluctuation operators are found to be  
\[ \begin{array}{ll}\delta\dot{Q}=&\omega_{m}\delta P,\\ \delta\dot{P}=&-\omega_{m}\delta Q+\frac{1}{\sqrt{2}}(G^{*}\delta c+G\delta c^ {\dagger})-\gamma_{m}\delta P+\xi,\\ \delta\dot{c}=&-(\kappa_{eff}+i\Delta)\delta c+\frac{i}{\sqrt{2}}G\delta Q+ \sqrt{2\kappa}(1-r)t\epsilon_{in},\end{array}  \tag{8}\]  
where \(G=\sqrt{2}g_{0}c_{s}\) is the light-enhanced optomechanical coupling strength, being related to the input laser power. By introducing the quadrature operators for the optical mode \(\delta x=\frac{1}{\sqrt{2}}(\delta c+\delta c^{\dagger})\), \(\delta y=\frac{1}{\sqrt{2}}(\delta c-\delta c^{\dagger})\) and the quadrature operators for the input vacuum noise \(x_{in}=\frac{1}{\sqrt{2}}(c_{in}+c_{in}^{\dagger})\), \(y_{in}=\frac{1}{\sqrt{2}i}(c_{in}-c_{in}^{\dagger})\), we can write Equation (8) as  
\[\dot{f}(\tau)=Mf(\tau)+n(\tau), \tag{9}\]  
where \(f(\tau)=(\delta Q,\delta P,\delta x,\delta y)^{T}\) is the vector of the fluctuation operators, \(n(\tau)=(0,\xi,\sqrt{2\kappa}(1-r)tx_{in},\sqrt{2\kappa}(1-r)ty_{in})^{T}\) is the vector of the noise operators, and \(M\) is the \(4\times 4\) matrix given by  
\[M=\left( \begin{array}{cccc}0&\omega_{m}&0&0\\ -\omega_{m}&-\gamma_{m}&u&v\\ -v&0&-\kappa_{eff}&\Delta\\ u&0&-\Delta&-\kappa_{eff}\end{array} \right) \tag{10}\]  
with \(u=\frac{1}{2}(G+G^{*})\), \(v=\frac{1}{2}i(G^{*}-G)\).