Dissipative dynamical Casimir effect in terms of complex spectral analysis in the symplectic Floquet space

Abstract

Abstract The dynamical Casimir effect of the optomechanical cavity interacting with a one-dimensional photonic crystal is theoretically investigated in terms of complex spectral analysis of the Floquet–Liouvillian in the symplectic Floquet space. The quantum vacuum fluctuation of the intra-cavity mode is parametrically amplified by a periodic motion of the mirror boundary, and the amplified photons are spontaneously emitted to the photonic band. We have derived the non-Hermitian effective Floquet–Liouvillian from the total system Liouvillian using the Brillouin–Wigner–Feshbach projection method in the symplectic Floquet space. The microscopic dissipation process of the photon emission from the cavity has been taken into account by the energy-dependent self-energy. We have obtained the discrete eigenmodes of the total system by nonperturbatively solving the nonlinear complex eigenvalue problem of the effective Floquet–Liouvillian, where the eigenmodes are represented by the multimode Bogoliubov transformation. Based on the microscopic dynamics, the nonequilibrium stationary eigenmodes are identified as the eigenmodes with vanishing values of their imaginary parts due to the balance between the parametric amplification and dissipation effects. We have found that the nonlocal stationary eigenmode appears when the mixing between the cavity mode and the photonic band is caused by indirect virtual transition, where the external field frequency causing the dynamical Casimir effect can be largely reduced by using the finite-bandwidth photonic band.