  We present a fully quantum mechanical treatment of the nondegenerate optical
parametric oscillator both below and near threshold. This is a non-equilibrium
quantum system with a critical point phase-transition, that is also known to
exhibit strong yet easily observed squeezing and quantum entanglement. Our
treatment makes use of the positive P-representation and goes beyond the usual
linearized theory. We compare our analytical results with numerical simulations
and find excellent agreement. We also carry out a detailed comparison of our
results with those obtained from stochastic electrodynamics, a theory obtained
by truncating the equation of motion for the Wigner function, with a view to
locating regions of agreement and disagreement between the two. We calculate
commonly used measures of quantum behavior including entanglement, squeezing
and EPR correlations as well as higher order tripartite correlations, and show
how these are modified as the critical point is approached. In general, the
critical fluctuations represent an ultimate limit to the possible entanglement
that can be achieved in a nondegenerate parametric oscillator.
