Single-photon nonlinearities in two-mode optomechanics

We present a detailed theoretical analysis of a weakly driven, multimode optomechanical system, in which two optical modes are strongly and near-resonantly coupled to a single mechanical mode via a three-wave mixing interaction. We calculate one- and two-time intensity correlations of the two optical fields and compare them to analogous correlations in atom-cavity systems. Nonclassical photon correlations arise when the optomechanical coupling $g$ exceeds the cavity decay rate $\ensuremath{\kappa}$, and we discuss signatures of one- and two-photon resonances as well as quantum interference. We also find a long-lived correlation that decays slowly with the mechanical decay rate $\ensuremath{\gamma}$, reflecting the heralded preparation of a single-phonon state after detection of a photon. Our results provide insight into the quantum regime of multimode optomechanics, with potential applications for quantum information processing with photons and phonons.