Classical and quantum linearized descriptions of degenerate optomechanical parametric oscillators

Recent advances in the development of modern quantum technologies have opened the possibility of studying the interplay between spontaneous parametric down-conversion and optomechanics, two of the most fundamental nonlinear optical processes. Apart from practical reasons, such scenario is very interesting from a fundamental point of view, because it allows exploring the optomechanical interaction in the presence of a strongly quantum-correlated field, the down-converted mode. In this work we analyze such problem from two approximate but valuable perspectives: the classical limit and the limit of small quantum fluctuations. We show that, in the presence of optomechanical coupling, the well-known classical phase diagram of the optical problem gets modified by the appearance of new dynamical instabilities. As for the quantum-mechanical description, we prove the ability of the squeezed down-converted field to cool down the mechanical motion not only to thermal but also to squeezed mechanical states, and in a way that can be much less sensitive to parameters (e.g., detuning of the driving laser) than standard sideband cooling.