$\hat{O}$: $$\begin{aligned}
\langle \hat{O}(t)\rangle&=&\mbox{Tr}_s\{\hat{O}\rho(t)\}=\mbox{Tr}_s\{\hat{O}\mbox{Tr}_r(U(t)\rho(0)\rho_r U^\dagger(t))\}\nonumber\\
&=&\mbox{Tr}_{s,r}\{{\cal T}e^{i\int_0^t d\tau U_{nl}(t,\tau)\{H^{in}(\tau)+H_{nl}(\tau)\}U^\dagger_{nl}(t,\tau)}U^\dagger_0(t)\hat{O}U_0(t){\cal T}e^{-i\int_0^t d\tau U_{nl}(t,\tau)\{H^{in}(\tau)+H_{nl}(\tau)\}U^\dagger_{nl}(t,\tau)}\nonumber\\
&\times & (U_{nl}(t,0)\rho(0)\rho_r U^\dagger_{nl}(t,0))\}\nonumber\\
&\approx &\mbox{Tr}_{s,r}\{{\cal T}e^{i\int_0^t d\tau \{H^{in}(\tau)+H_{nl}(\tau)\}}U^\dagger_0(t)\hat{O}U_0(t){\cal T}e^{-i\int_0^t d\tau \{H^{in}(\tau)+H_{nl}(\tau)\}}\rho(0)\rho_r\},
\label{expect}\end{aligned}$$ where $U(t)={\cal T}\exp\{-i\int_0^t d\tau H(\tau)\}$ is the action including that of the stochastic part $H_{sr}(t)$ [@supp1], and $\rho_r$ represents the total quantum state of the reservoirs. Because a concerned dynamical evolution starts from a vacuum state $|0\rangle_c$ of the cavity and a thermal state $\rho_m(0)=\sum_{n=0}^{\infty} \frac{n_{th}^n}{(1+n_{th})^{n+1}}\left\vert n\right\rangle_m\left\langle n\right\vert$ of the mechanical oscillator ($\rho(0)$ is their product and $n_{th}$ is the initial thermal occupation), the action $U_{nl}(t,0)\rho(0)U^\dagger_{nl}(t,0)$ keeps this initial state invariant on the third line of Eq. (\[expect\]), as a consequence of the relation $U_{nl}(t,0)|0\rangle_c=|0\rangle_c$. Note that, in Eq. (\[expect\]), the operators $\hat{c}=\hat{a},\hat{b}$ in $H^{in}(\tau)+H_{nl}(\tau)$ are transformed by the action in Eq. (\[NL\]) to $U_{nl}(t,\tau)\hat{c}U^\dagger_{nl}(t,\tau)$, unlike the general form $U^{\dagger}(t)\hat{o}U(t)$ for a unitary operation on an operator $\hat{o}$. According to Eq. (\[NL\]), the modified operators differ from the original ones $\hat{c}$ by the corrections in the orders of $g/\omega_m\ll 1$, so these corrections can be well neglected for a weakly coupled OMS. This step noted with an approximation sign on the last line of Eq. (\[expect\]) is the only theoretical approximation made in our approach, 