  We study theoretically the dynamics of multiple mechanical oscillators
coupled to a single cavity field mode via linear or quadratic optomechanical
interactions. We focus specifically on the strong coupling regime where the
cavity decays much faster than the mechanical modes, and the optomechanical
coupling is comparable to or larger than the mechanical frequency, so that both
the optical and mechanical systems operate in the deep quantum regime. Using
the examples of one and two mechanical oscillators we show that the system can
classically exhibit bistability and bifurcations, and we explore how these
manifest themselves in interference, entanglement, and correlation in the
quantum theory, while revealing the impact of decoherence of the mechanical
system due to cavity fluctuations and coherent driving.
