Abstract
We investigate an active-passive-cavity system where a membrane is quadratically coupled to the passive cavity. When the passive cavity is strong coherently driven to enhance the optomechanical coupling, the optical field interacts with the mechanical oscillator through the two-phonon process. By calculating the eigenfrequencies of system in Keldysh framework, we show the significant splitting of exceptional point in the presence of the optomechanical coupling. We also calculate the transmission rate to show how this system responds to an additional weak probe field. We find the gain effect will induce the two-phonon process amplification window. However, the gain effect becomes weak when the photon-tunneling strength increases. Then the transmission rate of the probe field changes from amplification to absorption. These results have potential applications in quantum metrology, quantum information processing, and quantum optical devices.
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