the coexisting squeezing (heating) effect stronger. These two tendencies strike a balance somewhere in the parameter space, so that the best cooling under a fixed drive intensity $E$ takes place at an optimum $\omega_m/\kappa$ \[see Fig. 3(b)\]. The latter tendency will dominate when the parameter $\omega_m/\kappa$ is continuously lowered, which will increase $n_{m,f}$ significantly on the left of the optimum $\omega_m/\kappa$. On the other hand, the suppression of the factor $\Gamma_s(t)$ by faster oscillations due to larger $\omega_m/\kappa$ will diminish the squeezing effect, leading to the tendency of the stabilized $n_m$ in Fig. 2(b). Among the three different terms in Eq. (7), the first one independent of $n_{th}$ monotonically decreases with increased $\omega_m/\kappa$, as illustrated in Figs. 3(c) and 3(d). The experimental investigation of the effect related to this contribution was recently reported in [@ex12]. Such pure cavity noise contribution decided by the squeezing coupling intensity $J\Gamma_s(\kappa t)$ increases significantly with the drive intensity $E$, which distinguishes our results fundamentally from the previous predictions; see the comparisons in Fig. 3(d) and Fig. S-4(b) in [@supp].

Another important issue is how to reach the best cooling from the initial thermal equilibrium of an OMS. To answer the question, we first examine 