A Green's function approach to the linear response of a driven dissipative optomechanical system

We discuss the generalization of the linear response theory (LRT) to encompass the theory of open quantum systems and then apply the so-called generalized LRT to investigate the linear response of a driven-dissipative optomechanical system (OMS) to a weak time-dependent perturbation. To our knowledge, there are elements of ambiguities in the literature in unification of the standard LRT which has been basically formulated for closed systems and the theory of open quantum systems. In this paper, we try to shed light on this matter through the reformulation of the LRT of open quantum systems in the Heisenberg picture. It is shown how the Green's function equations of motion of a standard OMS as an open quantum system can be obtained from the quantum Langevin equations (QLEs) in the Heisenberg picture. The obtained results explain a wealth of phenomena, including the anti-resonance, normal mode splitting and the optomechanically induced transparency (OMIT). Furthermore, the reason why the Stokes or anti-Stokes sidebands are amplified or attenuated in the red or blue detuning regimes is clearly determined which is in exact coincidence, especially in the weak-coupling regime, with the Raman-scattering picture.