setup. In a frame with the system modes rotating at the cavity frequency $\omega_c$ and the mechanical frequency $\omega_m$, respectively, the process is governed by the following Hamiltonians [@supp]: (1) $H_e(t)=iE(\hat{a}^{\dagger}e^{i\Delta t}-\hat{a}e^{-i\Delta t})$ for an external drive with the intensity $E$ and the detuning $\Delta=\omega_c-\omega_l$ of its frequency $\omega_l$; (2) $H_{om}(t)=-g\hat{a}^\dagger\hat{a}(\hat{b}e^{-i\omega_m t}+\hat{b}^\dagger e^{i\omega_m t})$ for the optomechanical coupling; (3) the stochastic Hamiltonian $H_{sr}(t)=i\sqrt{2\kappa }\{\hat{a}^{\dagger }\hat{\xi}
_{c}(t)-\hat{a}\hat{\xi}^{\dagger }
_{c}(t)\}+i\sqrt{2\gamma_m }\{\hat{b}^{\dagger }\hat{\xi}
_{m}(t)-\hat{b}\hat{\xi}^{\dagger }
_{m}(t)\}$ accounting for the cavity (mechanical) damping at the rate $\kappa$ ($\gamma_m$), with the correlations of the quantum noise operators satisfying [@rev] $$\begin{aligned}
&&\langle \hat{\xi}^\dagger_{c}(t)\hat{\xi}_{c}(t')\rangle=0, ~~\langle \hat{\xi}^\dagger_{m}(t)\hat{\xi}_{m}(t')\rangle=n_{th}\delta(t-t'),
\label{noise}\end{aligned}$$ where a zero thermal occupation for the cavity reservoir is assumed and $n_{th}$ is the thermal occupation of the mechanical reservoir. Then we take an interaction picture with respect to $H_e(t)$. The transformed Hamiltonian $H^{in}(t)=U_0^\dagger(t)(H_{om}+H_{sr})U_0(t)$, where $U_0(t)={\cal T}e^{-i\int_0^t d\tau H_e(\tau)}$, gives the dynamical equations $$\begin{aligned}
\dot{\hat{a}}&=&-\kappa\hat{a}+g E f(t)e^{-i\omega_mt} \hat{b}+g Ef(t)e^{i\omega_mt} \hat{b}^\dagger\nonumber\\
&+&i\kappa f(t) E+\sqrt{2\kappa}\hat{\xi}_c(t),\\
\dot{\hat{b}}&=&-\gamma_m\hat{b}-g Ef^{\ast}(t)e^{i\omega_mt}\hat{a}+g E f(t)e^{i\omega_mt} \hat{a}^\dagger\nonumber\\
&+&ig|f(t)E|^2e^{i\omega_m t}+\sqrt{2\gamma_m}\hat{\xi}_m(t),\end{aligned}$$ with $f(t)=(e^{i\Delta t}-1)/\Delta$. A cubic term in $H^{in}(t)$ is neglected in deriving these linear dynamical equations, since we are dealing with a weakly coupled OMS [@supp].

We first look at the mechanical oscillator’s thermal equilibrium with the environment. In 