  We study the dissipative phase transition in a quantum oscillator with
two-photon drive and two-photon dissipation. Using the semi-classical Langevin
equation and the Fokker-Plank approach, we construct a theory of
non-perturbative quantum fluctuations and go beyond the semi-classical
approximation. We demonstrate the mapping of a two-photon quantum dissipative
oscillator onto a classical equilibrium model of a nonlinear classical
oscillator in a colored-noise environment. Then, we justify the applicability
of the Landau theory for a given dissipative phase transition. To do that, we
explicitly demonstrate the Boltzmann-like form of stationary distribution
function depending on the effective temperature, which is determined by the
frequency detuning and the rates of two-photon drive and dissipation. 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.
