process for a mechanical oscillator to evolve from its initial thermal state to another state with higher purity, which takes time. Cooling starts after turning on a red-detuned drive that realizes a BS type coupling between the fast damping cavity mode and the slowly decaying mechanical mode, so that the oscillator’s thermal excitation being converted to cavity photons could be totally eliminated after a time controlled by the parameter $J$. Only with this scenario giving the phonon number in Eq. (\[s\]), the mechanical oscillator would be cooled down to a vacuum state modified by its inevitable motion under radiation pressure to a coherent state. The simultaneous quantum noise actions, however, cause the evolution to deviate from going to such a pure quantum state and add to the thermal phonon number in Eq. (\[n\]). The thermal occupation left in the end is decided by how soon the coupled cavity and mechanical modes evolve together to a dynamical stability, after which the system cannot be cooled down further. Like the fully quantum mechanical treatment of OMS in [@law] and recent studies of the quantum dynamical features of other physical systems (see, e.g. [@dd1; @dd2]), the properties in a cooling process illustrated here 