## 2 Model  
We consider an optical cavity with two mirrors separated by a distance \(L\), as depicted in Figure 1. An input laser at frequency \(\omega_{l}\) and amplitude \(\varepsilon_{l}\) is split into two parts by an asymmetric beam splitter which has a \(2\times 2\) scattering matrix [31]  
\[\left( \begin{array}{cc}t&r\\ -r&t\end{array} \right) \tag{1}\]  
where \(r\) and \(t\) are the amplitude reflection and transmission coefficients of the beam splitter, respectively, they are real, and satisfy the relation \(r^{2}+t^{2}=1\), thus there is no absorption within the beam splitter itself. We assume that \(t\) is positive. Hence the transmitted part of the input laser has the amplitude \(t\varepsilon_{l}\), the reflected part of the input laser has the amplitude \(-r\varepsilon_{l}\). A cavity field \(c\) at frequency \(\omega_{c}\) is pumped by the transmitted part of the input laser. The movable mirror with 100% reflectivity is coupled to the cavity field via radiation pressure. Moreover, a part of the output field from the cavity is fed back into the cavity via a totally reflecting mirror and the asymmetric beam splitter. If \(r=0\) and \(t=1\), the input laser is totally sent into the cavity, and the output field is not fed back into the cavity. The Hamiltonian of the system in a frame rotating at the laser frequency \(\omega_{l}\) has a form  
\[H = \hbar(\omega_{c}-\omega_{l})c^{\dagger}c+\frac{1}{2}\hbar\omega_{ m}(Q^{2}+P^{2})-\hbar g_{0}c^{\dagger}cQ+i\hbar t\varepsilon_{l}(c^{\dagger}-c), \tag{2}\]  
Figure 1: A schematic diagram of the optomechanical system with a coherent feedback. An input laser is sent into the cavity through an asymmetric beam splitter (BS). A portion of the output field is fed back into the cavity via a completely reflecting mirror and the BS. The \(c_{in}\) is the optical vacuum noise that is incident on the BS. where \(Q\) and \(P\) are the dimensionless position and momentum operators with \(Q=\sqrt{\frac{m\omega_{m}}{\hbar}}q\), \(P=\frac{p}{\sqrt{m\omega_{m}}}\), and \([Q,P]=i\). In Equation (2), the first term is the energy of the cavity field, the second term is the energy of the movable mirror with the effective mass \(m\) and resonance frequency \(\omega_{m}\), the third term describes the optomechanical interaction between the cavity field and the movable mirror, \(g_{0}=\frac{\omega_{c}}{L}\sqrt{\frac{\hbar}{m\omega_{m}}}\) is the single-photon optomechanical coupling strength, the last term represents the cavity field driven by a part of the input laser which is transmitted by the beam splitter, and the laser amplitude \(\varepsilon_{l}\) depends on the laser power \(\wp\) by \(\varepsilon_{l}=\sqrt{\frac{2\kappa_{l}}{\hbar\omega_{l}}}\) where \(\kappa\) is the decay rate of the cavity mode due to the light escaping the cavity from the fixed mirror with partial reflectivity. The time evolution of the system operators can be obtained by using the Heisenberg equation of motion, adding the corresponding damping and noise terms, and taking into account the feedback term. Thus the quantum [PERSON] equations of the system are given by  
\[ \begin{array}{ll}\dot{Q}&=\omega_{m}P,\\ \dot{P}&=-\omega_{m}Q+g_{0}c^{\dagger}c-\gamma_{m}P+\xi,\\ \dot{c}&=-[\kappa+i(\omega_{c}-\omega_{l})]c+ig_{0}cQ+t\varepsilon_{l}+\sqrt {2\kappa}(tc_{in}+rc_{out}).\end{array}  \tag{3}\]  
Here \(\gamma_{m}\) is the damping rate of the movable mirror, \(\xi\) is the Brownian stochastic force due to the interaction of the movable mirror with the thermal bath at temperature \(T\). The \(\xi\) has zero mean value and obeys the two-time non-Markovian correlation function  
\[\langle\xi(\tau)\xi(\tau^{\prime})\rangle=\frac{1}{2\pi}\frac{\gamma_{m}}{ \omega_{m}}\int\omega e^{-i\omega(\tau-\tau^{\prime})}\left[1+\coth(\frac{\hbar \omega}{2k_{B}T})\right]d\omega, \tag{4}\]  
where \(k_{B}\) is the Boltzmann constant.