  The promise of innovative applications has triggered the development of many
modern technologies capable of exploiting quantum effects. But in addition to
future applications, such quantum technologies have already provided us with
the possibility of accessing quantum-mechanical scenarios that seemed
unreachable just a few decades ago. With this spirit, in this work we show that
modern optomechanical setups are mature enough to implement one of the most
elusive models in the field of open system dynamics: degenerate parametric
oscillation. The possibility of implementing it in nonlinear optical resonators
was the main motivation for introducing such model in the eighties, which
rapidly became a paradigm for the study of dissipative phase transitions whose
corresponding spontaneously broken symmetry is discrete. However, it was found
that the intrinsic multimode nature of optical cavities makes it impossible to
experimentally study the model all the way through its phase transition. In
contrast, here we show that this long-awaited model can be implemented in the
motion of a mechanical object dispersively coupled to the light contained in a
cavity, when the latter is properly driven with multi-chromatic laser light. We
focus on membranes as the mechanical element, showing that the main signatures
of the degenerate parametric oscillation model can be studied in
state-of-the-art setups, thus opening the possibility of studying spontaneous
symmetry breaking and enhanced metrology in one of the cleanest dissipative
phase transitions.
