stable value. So it is more appropriate to use an evolving field $\alpha(t)$, which develops from zero to stable value throughout a cooling process, in that linearization approach. Our quantum dynamical approach totally dispenses with such time-dependent classical mean values, which should be found by solving the nonlinear classical dynamical equations \[those obtained by taking the averages of Eq. (\[e1\]) or Eq. (\[e2\])\]. The linearization of the system dynamics is realized by a unitary transformation $\hat{a}\rightarrow \hat{a}-i\frac{e^{i\Delta t}-1}{\Delta}E$ of the cavity mode, which is completely inequivalent to the previous linearization by the shift $\hat{a}\rightarrow  \hat{a}+\alpha$ with a time-independent classical steady-state value $\alpha$.

III. FURTHER DISCUSSION ON THE EFFECTS OF THE COHERENT DRIVES
-------------------------------------------------------------

A majority of physical effects in the concerned cooling processes come from the coherent drive term $\vec{\lambda}(t)$ and the noise drive term $\vec{\hat{\eta}}(t)$ in Eq. (\[dynamics\]). Here we discuss more of their properties that are complementary to those described in the main text. We first look at the former, which has been less explored by the previous studies. In terms of the five dimensionless parameters $G_m=g/\kappa, {\cal E}=E/\kappa, s_m=\omega_m/\kappa, Q=\omega_m/\gamma_m, \delta=\Delta/\omega_m$, as well as the dimensionless time $\tau$ going from $0$ to $\kappa t$, the evolving cavity 