mechanical treatment for an OMS entering its macroscopic quantum states, indicates that a small coupling $J$ can still affect the cavity photon number considerably. Enhanced cavity photon numbers in oscillation is a signature of macroscopic quantum states for the OMS.

There are two intrinsic time scales for reaching the dynamical stability in the cooling of an OMS, $1/\kappa$ for the cavity mode and $1/\gamma_m$ for the mechanical mode. When these two modes are uncoupled ($J=0$), the cavity mode tends to a steady state within a time in the order of $1/\kappa$ as in Fig. S-1(a), while the mechanical mode under a thermal equilibrium with the environment satisfies the equation $$\begin{aligned}
\dot{\hat{b}}&=&-\gamma_m\hat{b}+\sqrt{2\gamma_m}\hat{\xi}_m(t),\end{aligned}$$ which is from Eq. (\[e1\]). According to the above equation, the mechanical mode takes the time of $1/\gamma_m\gg 1/\kappa$ to damp to zero, a stable value, while the second noise drive term keeps the thermal phonon number $n_{th}$ invariant in this situation. Once these two system modes are coupled ($J\neq 0$), the time scale $t_s$ for the OMS to approach the stable evolution phase will be decided by their effective coupling intensity $J$ and be reduced from $1/\gamma_m$ due to the coupling to the cavity mode with a faster 