Mechanical cooling in single-photon optomechanics with quadratic nonlinearity

Abstract

In the paper we study the nonlinear mechanical cooling processes in an intrinsic quadratically optomechanical coupling system without linearizing the optomechanical interaction. We apply scattering theory to calculate the transition rates between different mechanical Fock states using the resolvent of the Hamiltonian, which allows for a direct identification of the underlying physical processes, where only even-phonon transitions are permitted and odd-phonon transitions are forbidden. We verify the feasibility of the approach by comparing the steady-state mean phonon number obtained from transition rates with the simulation of the full quantum master equation, and also discuss the analytical results in the weak coupling limit that coincide with two-phonon mechanical cooling processes. Furthermore, to evaluate the statistical properties of steady mechanical state, we respectively apply the Mandel $Q$ parameter to show that the oscillator can be in nonclassical mechanical states, and the phonon number fluctuations $F$ to display that the even-phonon transitions favor suppressing the phonon number fluctuations compared to the linear coupling optomechanical system.