are for a genuine quantum OMS rather than the quantum fluctuations around the trajectories of a classical one; this approach can also be used for blue-detuned drives [@s-laser]. This quantum dynamical picture of cooling applies to an OMS truly approaching its macroscopic quantum states.

We acknowledge the funding supports from NSFC (Grant No. 11574093 and Grant No. 61435007). L. Y. is sponsored by the Fundamental Research Funds for the Central Universities (No. HEUCFJ170402). This research is also supported by the Arkansas High Performance Computing Center and the Arkansas Economic Development Commission.

B. H. and L. Y. contributed equally to this work.

[Supplementary Material for “Radiation Pressure Cooling as a Quantum Dynamical Process"]{} {#supplementary-material-for-radiation-pressure-cooling-as-a-quantum-dynamical-process .unnumbered}
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I. DYNAMICAL EQUATIONS OF OPTOMECHANICAL SYSTEMS IN A ROTATING FRAME
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A quantum optomechanical system (OMS) is modeled by the coupled cavity mode $\hat{a}$ with the frequency $\omega_c$ and mechanical mode $\hat{b}$ with the frequency $\omega_m$, having the self-oscillation Hamiltonians $\omega_c\hat{a}^\dagger\hat{a}$ and $\omega_m\hat{b}^\dagger\hat{b}$ ($\hbar=1$) for the respective mode. Initially, when there is no external drive field, the mechanical oscillator is in a thermal equilibrium with the environment at a temperature corresponding to the occupation $n_{th}$. An external drive with the intensity $E$ and frequency $\omega_l$, 