  We study the dynamical Casimir effect using a fully quantum-mechanical
description of both the cavity field and the oscillating mirror. We do not
linearize the dynamics, nor do we adopt any parametric or perturbative
approximation. By numerically diagonalizing the full optomechanical
Hamiltonian, we show that the resonant generation of photons from the vacuum is
determined by a ladder of mirror-field {\em vacuum Rabi splittings}. We find
that vacuum emission can originate from the free evolution of an initial pure
mechanical excited state, in analogy with the spontaneous emission from excited
atoms. By considering a coherent drive of the mirror, using a master-equation
approach to take losses into account, we are able to study the dynamical
Casimir effect for optomechanical coupling strengths ranging from weak to
ultrastrong. We find that a resonant production of photons out of the vacuum
can be observed even for mechanical frequencies lower than the cavity-mode
frequency. Since high mechanical frequencies, which are hard to achieve
experimentally, were thought to be imperative for realizing the dynamical
Casimir effect, this result removes one of the major obstacles for the
observation of this long-sought effect. We also find that the dynamical Casimir
effect can create entanglement between the oscillating mirror and the radiation
produced by its motion in the vacuum field, and that vacuum Casimir-Rabi
oscillations can occur.
