which was generally treated as a constant of steady-state value. In an actual cooling process, however, the cavity mean field $\langle \hat{a}(t)\rangle=\alpha(t)$ is built up from zero (when the mechanical oscillator is in thermal equilibrium with its environment) and takes time to evolve to stable value. Then the effective coupling strength $g|\alpha|$ used in the previous studies should be more appropriately taken as a variable, since $\alpha(t)$ keeps changing during a cooling process. Due to the impossibility of finding the time-dependent $\alpha(t)$ analytically, it is difficult to study the cooling as a dynamical process if adopting the above-mentioned linearization.

In the present work we put forward a quantum dynamical theory for optomechanical cooling. Using this completely quantum approach that linearizes a weakly coupled OMS’s dynamics without resorting to its classical mean values such as $\alpha(t)$, one can numerically predict the involved physical quantities that evolve with time. The residual occupation of the thermal excitation of mechanical oscillator, as found in our approach, is connected with the whole cooling process. Compared with the previous theoretical predictions, this dynamical approach provides richer information about how good a quantum OMS can be fully cooled down to.

Fig. 1(a) illustrates an exemplary optomechanical cooling 