parameter $J=1.0$ used there has been on the right side of the turning point for $\omega_m/\kappa=8$.

The evolving Gaussian states of the weakly coupled OMSs, which follow the linear dynamical equations, can be depicted with their Wigner functions. A cooling process starts from a mechanical oscillator’s thermal state with its Wigner function being $W(q_m, p_m)=[1/\pi(1+2n_{th})]\exp\{-[1/(1+2n_{th})](q_m^2+p_m^2)\}$. After turning on a drive, the OMS will evolve under optomechanical coupling to two-mode Gaussian states, whose Wigner functions can be found by numerically calculating the correlation matrix of the two system modes. As illustrated in Fig. 1(b), a perfect “ground-state cooling" to $n_{m,f}=0$ evolves the mechanical mode’s Wigner function to $W(q_m, p_m)=\frac{1}{\pi}e^{-(q_m-q_m^0)^2-(p_m-p_m^0)^2}$ of a coherent state, which differs from that of a vacuum state only by the displacements $q_m^0(t)$, $p_m^0(t)$ determined by the coherent drive term $\vec{\lambda}(t)$ in Eq. (4). Without impairing the purity [@purity] of the target coherent state, the contribution from the coherent drives should be excluded from the thermal phonon number $n_{m}(t)$ to be reduced from the occupation $n_{th}$ of an initial thermal state.

In summary, we have developed a quantum dynamical approach to optomechanical cooling. The motivation for the development is to reflect the fact that such cooling is a 