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Quantum Optomechanical system serves as an interface for coupling between photons and phonons due to mechanical oscillations. We use the Heisenberg-Langevin approach to study a quadratically coupled optomechanical system which contains a thin dielectric membrane coupled to the cavity field. A decorrelation method beyond mean field approximation is employed to get a closed set of equations which are further used to study temporal dynamics. Transient mean number of cavity photons and phonons has been studied under different coupling regime. We have also studied the two-boson correlation functions for the cavity field, membrane and the cross-correlation between them. Furthermore, we use this approach to study a hybrid optomechanical system that contains a single undoped semiconductor quantum well in a cavity, where one of its mirrors is a thin dielectric membrane having quadratic response to the cavity fields. The nonclassical quantum statistical properties provide useful insights for the coherent control of photon statistics as well as photon correlations in the quadratic optomechanical systems.
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