manifest more clearly with Fig. 4, in which the limiting value in Eq. (\[l\]) gives the boundary any cooling process cannot surpass. For a setup with the built-in sideband resolution $\omega_m/\kappa$, increasing the drive power can make its cooling better and faster, moving the achieved thermal phonon number along a curve of fixed $\omega_m/\kappa$ to larger $J$, though the enhanced power may change the phonon number less obviously for a system with $\omega_m/\kappa\gg 1$ (in the strong coupling regime its phonon curve asymptotically approaches the limit line). The higher the parameter $\omega_m/\kappa$ is, the stronger the drive can be applied to achieve a better cooling; this explains the phonon number tendency on the right of the optimum points in Fig. 3(b). However, the cooling cannot be improved further if $J$ arrives at a turning point (the ending dots of the phonon curves) where the coexisting squeezing effect begins to be significant. After crossing the point, the thermal phonon number will increase from the minimum value there, and its evolution will continue to become oscillating and then growing with time as in Fig. 2(a). Such a reference point indicates that the cooling processes in Fig. 2(b) are not the optimum ones—the 