Optical bistability and four-wave mixing in a hybrid optomechanical system

We explore theoretically the optical bistability and four-wave mixing (FWM) in a hybrid optomechanical system, where the mechanical resonator is simultaneously coupled to a cavity field and a two-level system (qubit). We can use a strong control field driving the cavity to control the bistable behavior of the steady-state photon number, phonon number, and the population inversion. The impact of qubit-resonator coupling strength on the bistable behavior is discussed. Furthermore, the two-level system can significantly modify the output fields of the cavity, leading to double optomechanically induced transparency (OMIT) and the enhancement of the FWM intensity. We find that the distance between the two peaks in the FWM spectrum can be controlled by the qubit-resonator coupling strength, and the peak value of the FWM intensity can be adjusted by the Rabi frequency of the control field.