Exact solvability and two-frequency Rabi oscillation in cavity-QED setup with moving emitter

In this paper, we investigate the energy spectrum and coherent dynamical process in a cavity-QED setup with moving emitter, which is subject to harmonic potential. We find that vibration of emitter will induce effective Kerr optomechanical interactions. generalize Bogliubov operators approach dealt quantum Rabi model, our cavity-emitter-vibration system obtain exactly. With assistance approach, Furthermore, show dynamics exhibit two-frequency oscillation behavior. explain such behavior by interaction induced transition between emitter-cavity dressed states. hope cavity mode provide versatile platform explore more exotic effects potential applications scenario.