Low-noise dissipative quantum limit-cycles using parametric driving

There has been a resurgence of interest in driven-dissipative quantum systems exhibiting limit cycle physics, a kind of spontaneous symmetry breaking that serves as a dissipative analogue of time-crystal behavior. The standard route to realizing this physics is to construct dissipative dynamics with an incoherent pump and nonlinear loss with a U(1) symmetry; this symmetry is then spontaneously broken. Here, we show that limit cycle behavior can be achieved starting from dynamics that involves only coherent pumping and that would seem to explicitly pick a phase and break U(1) symmetry. We show that coupling two (or more) parametrically driven bosonic modes, subject to a weak Kerr nonlinearity (e.g. a Hubbard like interaction) yields a new kinds of limit cycle phenomena. The coherent nature of the driving leads to suppressed phase diffusion, below the standard Schawlow-Townes (ST) value. Generalizing our model to the N mode case, the model has a manifest O(N) weak symmetry and admits an exactly solvable steady-state. The resulting steady-manifold S^{N-1} supports 2N-3 fluctuation directions that pairs into N-1 type-B Goldstone modes. Our model is experimentally realizable in both quantum optical setups and superconducting circuits, and provides a new platform for studying studying dissipative spontaneous symmetry breaking and quantum synchronization.