Temporal Dynamics and Nonclassical Photon Statistics of Quadratically Coupled Optomechanical Systems

Quantum optomechanical system serves as an interface for coupling between photons and phonons due to mechanical oscillations. We used the Heisenberg-Langevin approach under Markovian white noise approximation to study a quadratically coupled optomechanical system which contains a thin dielectric membrane quadratically coupled to the cavity field. A decorrelation method is employed to solve for a larger number of coupled equations. Transient mean numbers of cavity photons and phonons that provide dynamical behaviour are computed for different coupling regime. We have also obtained the two-boson second-order correlation functions for the cavity field, membrane oscillator and their cross correlations that provide nonclassical properties governed by quadratic optomechanical system.