  Degenerate optomechanical parametric oscillators are optical resonators in
which a mechanical degree of freedom is coupled to a cavity mode that is
nonlinearly amplified via parametric down-conversion of an external pumping
laser. Below a critical pumping power the down-converted field is purely
quantum-mechanical, making the theoretical description of such systems very
challenging. Here we introduce a theoretical approach that is capable of
describing this regime, even at the critical point itself. We find that the
down-converted field can induce significant mechanical cooling and identify the
process responsible of this as a cooling-by-heating mechanism. Moreover, we
show that, contrary to naive expectations and semi-classical predictions,
cooling is not optimal at the critical point, where the photon number is
largest. Our approach opens the possibility for analyzing further hybrid
dissipative quantum systems in the vicinity of critical points.
