  We in this paper derive the analytical expressions of ground-state energy,
average photon-number, and the atomic population by means of the
spin-coherent-state variational method for arbitrary number of atoms in an
optomechanical cavity. It is found that the existence of mechanical oscil-
lator does not affect the phase boundary between the normal and superradiant
phases. However, the superradiant phase collapses by the resonant damping of
the oscillator when the atom-field coupling increases to a so-called turning
point. As a consequence the system undergoes at this point an additional phase
transition from the superradiant phase to a new normal phase of the atomic
population-inversion state. The region of superradiant phase decreases with the
increase of photon-phonon coupling. It shrinks to zero at a critical value of
the coupling and a direct atomic population transfer appears between two
atom-levels. Moreover we find an unstable nonzero-photon state, which is the
counterpart of the superradiant state. In the absence of oscillator our result
re- duces exactly to that of Dicke model. Particularly the ground-state energy
for N = 1 (i.e. the Rabi model) is in perfect agreement with the numerical
diagonalization in a wide region of coupling constant for both red and blue
detuning. The Dicke phase transition remains for the Rabi model in agreement
with the recent observation.
