Spontaneous emission in Casimir-Rabi oscillations through a weak optomechanical coupling

When the mechanical oscillator is initially in an excited state, its free evolution, driven by the dynamical Casimir effect (DCE), leads to the generation of radiation in the vacuum, analogous to the spontaneous emission of an excited atom. In the regime where the loss rates are lower than the corresponding frequency splittings, reversible energy exchange can occur between the excited mechanical oscillator and the vacuum cavity field (i.e., vacuum Casimir-Rabi oscillations). By using the quantum trajectory approach, we show that n-photon/n-phonon emission can be achieved by opening dissipative channels for these Casimir-Rabi oscillations. Simulation results from 10⁵ quantum trajectories demonstrate that the purities of two-photon emission, two-phonon emission, and three-phonon emission can reach 0.92, 0.94, and 0.95, respectively. These values are observed in Casimir-Rabi oscillations between the three-phonon state and the two-photon state in a weak optomechanical coupling. This phenomenon could provide insights into the physical characteristics of photon and phonon emission in the DCE.