Quantum correlations of quadratic optomechanical oscillator

The quantum optomechanical system serves as an interface for coupling between photons, excitons, and mechanical oscillations. We use the quantum Langevin 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. A decorrelation method is employed to solve for a large number of coupled equations. Transient mean numbers of cavity photons, moving membranes, and excitons that provide dynamical behavior are computed. We obtain the two-boson second-order correlation functions for the cavity field and the membrane oscillator, and their cross correlations that provide nonclassical quantum statistical properties and useful insights into the quadratic optomechanical system.