Emergent equilibrium and quantum criticality in a two-photon dissipative oscillator

We study the dissipative phase transition (DPT) in a quantum oscillator with two-photon drive and two-photon dissipation. Using the semiclassical Langevin equation and the truncated Wigner approximation, we construct a theory of nonperturbative quantum fluctuations and go beyond the semiclassical approximation. We demonstrate the mapping of a two-photon quantum dissipative oscillator onto a classical equilibrium model of a nonlinear classical oscillator in a white-noise environment. Then we justify the applicability of the Boltzmann-Gibbs theory for a given DPT. To do that, we explicitly demonstrate the Boltzmann-Gibbs-like form of the stationary distribution function depending on the effective temperature, which is determined by the two-photon pump rate. In addition, we provide a description of the quantum critical region and obtain critical exponents that appear to be in very good agreement with numerical simulations.