  Dynamical Casimir effect of the optomechanical cavity interacting with
one-dimensional photonic crystal is theoretically investigated in terms of the
complex spectral analysis of 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 with the use of 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
non-perturbatively 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 the indirect virtual transition, where the
external field frequency to cause the DCE can be largely reduced by using the
finite bandwidth photonic band.
