Excitations of optomechanically driven Bose-Einstein condensates in a cavity: photodetection measurements

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.