Resolvent method on the single-photon optomechanical cooling

We apply the method of the resolvent of Hamiltonian to investigate the mechanical cooling behavior in the single-photon optomechanical regime. This approach allows for a direct identification of the underlying physics processes, and the obtained transition rates clearly show the multiphonon involved processes. We resort to the Mandel Q parameter to study the statistical properties of the steady state of mechanical oscillator, and find that when the optomechanical coupling is weak, the oscillator is in a thermal state, while when the optomechanical coupling is strong, the multiple-phonon transitions begin to work, leading the oscillator to be in a nonthermal state.