the limit of $\omega_m/\kappa\rightarrow \infty$. In this limit the stabilized thermal phonon number given by $\Delta=\omega_m$ can be found analytically from Eqs. (2)-(3) as $$\begin{aligned}
n_{m,f}
&=& \Gamma_m n_{th}\frac{1}{|\eta_{+}-\eta_{-}|^2}\nonumber\\
&\times & \big\{4\mbox{Re} \{\frac{\eta_{+}^\ast \eta_{-}}{(\lambda_+^\ast+\lambda_-)}\}-\frac{|\eta_{+}|^2}{\mbox{Re}\lambda_{-}}-\frac{|\eta_{-}|^2}{\mbox{Re}\lambda_{+}}\big\},
\label{l}\end{aligned}$$ where $\lambda_{\pm}=\frac{1}{2}(-1-\Gamma_m\pm \sqrt{(1-\Gamma_m)^2-4J^2})$ and $\eta_{\pm}=-1+\Gamma_m\pm \sqrt{(1-\Gamma_m)^2-4J^2}$ with $\Gamma_m=\gamma_m/\kappa$. It is solely from the contribution of the second term in Eq. (\[n\]), since the squeezing action with its intensity $J\Gamma_s(\kappa t)$ has been completely averaged out by an infinite $\omega_m/\kappa$. Analogous to a first-order phase transition, this limiting value exhibits a discontinuous jump across the point $J=1/2(1-\Gamma_m)$ (in Fig. 4 it is around $J=0.5$), which separates the weak coupling regime from the strong coupling one that has been experimentally observed [@ex8; @ex10; @strong]. In the strong coupling regime, the limiting value becomes the constant $n_{m,f}=\Gamma_m n_{th}$. Inside the weak coupling regime the stabilized phonon numbers for different $\omega_m/\kappa$ can be close to the limiting value and drop quickly with increased $J$ (see the inset in Fig. 4). With the parameters of the experimental setup in [@ex9] (equivalent to $J\approx 0.16$), for example, an ideal cooling process would not reach a phonon number lower than the limit of $n_{m,f}\approx 0.26$.

The parametric conditions for achieving good cooling 