Exploring corrections to the Optomechanical Hamiltonian

We compare two approaches to refine the "linear model" of cavity optomechanics, in order to describe radiation pressure effects that are beyond first order in the coupling constant. We compare corrections derived from (I) a widely used phenomenological Hamiltonian that conserves the photon number and (II) a two-mode truncation of C. K. Law's microscopic model, which we take as the "true" Hamiltonian of the system. While these approaches agree at first order, the latter model does not conserve the photon number, resulting in challenging computations from second order onwards. Our numerics suggest that the phenomenological Hamiltonian significantly improves the linear model, yet it does not fully capture all second-order corrections arising from the C. K. Law model. We conclude that, even when the mechanical frequency is much lower than the cavity one, photon number conservation must be eventually given up to model cavity optomechanics with high accuracy.