Quantum optomechanics with quadratic cavity-membrane couplings

We study the dynamics of a quantum optomechanics system consisting of a membrane and an optical cavity. The mechanical membrane is inside the optical cavity and divides the cavity into two small cavities. We examine the tunnelling of the photon between the two cavities and the dynamics of the membrane with and beyond the mean-field approximation. The time dependence of the photon inversion and the mean position of the membrane are calculated and discussed. Effects of decoherence on the tunnelling and self-trapping of photons are examined.