The properties of Stokes and anti-Stokes processes in a double-cavity optomechanical system

We study the nonlinear Stokes and anti-Stokes processes of a weak probe field relevant to normal mode splitting (NMS) in a double-cavity optomechanical system where a membrane oscillator is shared by two identical cavities. The two cavity modes experience an optomechanical coupling of same amplitudes but opposite signs when the membrane deviates from its equilibrium position due to the radiation pressures arising from two strong pump fields. Our calculations show that the critical power of left-cavity pump field above which the double-cavity system enters the NMS regime can be easily controlled by adjusting the right-cavity pump field in power. In addition, we show that various NMS features can be well examined by focusing on the spectral structure of an anti-Stokes signal generated in the four-wave-mixing process arising from optomechanical coupling. Last but not least we note that the anti-Stokes signal's generation is accompanied by the Stokes signal's amplification (absorption) in the absence (presence) of right-cavity pump field.