of the main text. In Fig. S-1(a) about $J=0$, the numerical calculation gives a perfect agreement with the known analytical result of a cavity with fixed size—the photon number tends to a steady value after a period in the order of $1/\kappa$. By intuition, a small $J$ would only lead to perturbative effect, but this is true only at the beginning period of the dynamical evolutions—the plots in Fig. S-1(b) first show a tendency of getting close to the corresponding steady value in Fig. S-1(a), but keep the growing oscillations due to the existence of optomechanical interaction. The finally stabilized photon number evolutions are stably oscillating ones as shown in the upper inset of Fig. S-1(b), in which the photon numbers tend to such a stable dynamical evolution phase together with the corresponding thermal phonon numbers in these cooling processes; compare the corresponding plots in the lower and upper insets of Fig. S-1(b). The photon number curves in the upper inset of Fig. S-1(b) look stuck together because the oscillation period is much shorter than the time scale to reach the finally stable oscillations. The corresponding dimensionless displacements $q_m^0(t)= \frac{1}{\sqrt{2}}\langle \hat{b}(t)+\hat{b}^\dagger(t)\rangle$ of the mechanical oscillator for the three different situations 