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Nonlinear quantum optical systems are of paramount relevance for modernquantum technologies, as well as for the study of dissipative phasetransitions. Their nonlinear nature makes their theoretical study verychallenging and hence they have always served as great motivation to developnew techniques for the analysis of open quantum systems. In this article weapply the recently developed self-consistent projection operator theory to thedegenerate optical parametric oscillator to exemplify its general applicabilityto quantum optical systems. We show that this theory provides an efficientmethod to calculate the full quantum state of each mode with high degree ofaccuracy, even at the critical point. It is equally successful in describingboth the stationary limit and the dynamics, including regions of the parameterspace where the numerical integration of the full problem is significantly lessefficient. We further develop a Gaussian approach consistent with our theory,which yields sensibly better results than the previous Gaussian methodsdeveloped for this system, most notably standard linearization techniques.

Self-Consistent Projection Operator Theory in Nonlinear Quantum Optical Systems: A case study on Degenerate Optical Parametric Oscillators