Tunable transparency and amplification in a hybrid optomechanical system with quadratic coupling

We theoretically study the optical response of a hybrid optomechanical system with quadratic coupling, where an optomechanical cavity with a movable membrane in the middle is coupled to an auxiliary cavity with a tunable gain rate. We show that the optical transmission of the system can be controlled by a variety of parameters, including power of the control field, coupling strength between the two cavities, the gain rate of the auxiliary cavity, reflectivity of the membrane, and the temperature of the environment. In particular, tunable transparency and amplification can be realized by modifying the gain rate of the auxiliary cavity. Under the condition of two-phonon resonance, we find that (i) for the no-loss-gain cavity, a broad transparency window appears in the probe transmission due to the coupling between the two cavities; (ii) for the passive cavity, an extremely narrow transparency window occurs within a broad transmission window due to quadratic optomechanical coupling; (iii) for the active cavity, the probe field can be amplified in a wide frequency regime. Our proposal provides a new platform to control the propagation of light.