quantumness. Practically starting from its thermal equilibrium with the environment, such process is implemented by coupling the oscillator to a cavity field generated by a red-detuned external drive, to reduce the associated thermal phonon number to a low level, similar to cooling the oscillator to a lower temperature. An important feature we will illustrate is that the cooling result depends on how fast the optomechanical system (OMS) evolves to the finally stable quantum state.

So far numerous experiments have realized the cooling to a few and even less than one mechanical quanta [@ex1; @ex2; @ex3; @ex4; @ex5; @ex6; @ex7; @ex8; @ex9; @ex10; @ex11; @ex12; @ex13]. Following the earlier study of quantum fluctuations under radiation pressure [@00; @01], the theoretical description of such optomechanical cooling (see, e.g. [@c1; @c2; @c0; @ck; @c3; @c4; @c5; @c6]) was based on a linearization procedure as that described in [@rev]; that is to decompose the cavity field mode $\hat{a}$ into the sum of the classical mean value $\alpha$ and its quantum fluctuation $\delta \hat{a}$. The linearized Hamiltonian gives the cooling action as a beamsplitter (BS) type coupling between the mechanical mode $\hat{b}$ and the fluctuation $\delta \hat{a}$ with their coupling intensity $g$ magnified by $\alpha$, 