Abstract
We theoretically investigate the phonon statistics of a quadratically coupled optomechanical system, in which an effective second-order nonlinear interaction between an optical mode and a mechanical mode is induced by a strong optical driving field on two-phonon red-sideband resonance. We show that strong phonon antibunching can be observed even if the strength of the effective second-order nonlinear interaction is much weaker than the decay rates of the system, by driving the optical and mechanical modes with weak driving fields respectively. Moreover, the phonon statistics can be dynamically controlled by tuning the strengths and the phase difference of the weak driving fields. The scheme proposed here can be used to realize tunable single-phonon sources with quadratically optomechanical coupling.