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