Dissipative phase transitions and chaos in two-photon driven quantum optomechanics

This paper investigates the nonequilibrium dynamics of a quantum optomechanical system driven by a two-photon pump. Unlike standard single-photon drives, the two-photon drive preserves a discrete Z2 symmetry (cavity parity), allowing the system to host rich critical phenomena. The authors combine semiclassical stability analysis, Liouvillian spectral theory, and stochastic quantum trajectories to map the system's behavior across different parameter regimes.

The study identifies two distinct types of dissipative phase transitions (DPTs) depending on the cavity-pump detuning. At negative detuning, the system undergoes a second-order DPT, where the cavity-parity symmetry is spontaneously broken in the thermodynamic limit. This is evidenced by the closing of the Liouvillian gap and the emergence of an odd-parity eigenoperator. Conversely, at positive detuning, the system exhibits a first-order DPT. This transition is marked by a discontinuous jump in photon and phonon populations and the coexistence of metastable states, signaled by the closing of an additional symmetric Liouvillian mode.

Beyond phase transitions, the authors explore the high-pump-power regime where the mean-field dynamics lose all stable fixed points. In this region, the system develops limit cycles and chaotic attractors characterized by positive Lyapunov exponents. By analyzing the spectral statistics of quantum trajectories and the delocalization of the stochastic wavefunction, the authors demonstrate that this regime hosts quantum signatures of chaos, including enhanced steady-state entropy and complex, chaotic-like motion in the quantum state space.

This work establishes two-photon-driven optomechanics as a versatile platform for studying the interplay between dissipative criticality, symmetry breaking, and quantum chaos. By providing a clear mapping of these phenomena in an experimentally accessible setup, the study offers a framework for future investigations into non-equilibrium quantum matter and the control of complex dynamical systems in quantum technologies.