  We present a detailed study to analyse the Dicke quantum phase transition
within the thermodynamic limit for an optomechanically driven Bose-Einstein
condensates in a cavity. The photodetection-based quantum optical measurements
have been performed to study the dynamics and excitations of this
optomechanical Dicke system. For this, we discuss the eigenvalue analysis,
fluorescence spectrum and the homodyne spectrum of the system. It has been
shown that the normal phase is negligibly affected by the mechanical mode of
the mirror while it has a significant effect in the superradiant phase. We have
observed that the eigenvalues and both the spectra exhibit distinct features
that can be identified with the photonic, atomic and phononic branches. In the
fluorescence spectra, we further observe an asymmetric coherent energy exchange
between the three degrees of freedom of the system in the superradiant phase
arising as a result of optomechanical interaction and Bloch-Siegert shift.
