a sufficiently large $\omega_m/\kappa$, because in Eq. (5) it suppresses the integrals of the matrix elements carrying the oscillating factor $\Gamma_s(t)$.

Our dynamical approach directly gives the picture of how the changes of the system parameters will turn the system from heating to cooling; see the time evolutions of the thermal phonon numbers in Figs. 2(a) and 2(b). In the transitional regimes where the cooling (BS) and heating (squeezing) effect compete with each other, the phonon numbers exhibit oscillations with time, as those in the insets of the figures. The corresponding cavity photon numbers in Figs. 2(c) and 2(d) evolve synchronously with the thermal phonon numbers, and the phenomenon is discussed further in [@supp].

New understanding of optomechanical cooling can be obtained from this dynamical picture, as it brings about another important parameter for cooling in addition to the widely concerned sideband resolution $\omega_m/\kappa$. A relevant phenomenon shown in Fig. 3(a) is that a higher sideband resolution beyond a certain value will be actually worse for cooling if the drive intensity $E$ is fixed. Our predicted thermal phonon number $n_{m,f}$ comes from the noise contributions as the integrals in Eq. (\[n\]), and is therefore determined by the whole process from 