  We develop a systematic theory of quantum fluctuations in the driven
parametric oscillator (OPO), including the region near threshold. This allows
us to treat the limits imposed by nonlinearities to quantum squeezing and noise
reduction, in this non-equilibrium quantum phase-transition. In particular, we
compute the squeezing spectrum near threshold, and calculate the optimum value.
We find that the optimal noise reduction occurs at different driving fields,
depending on the ratio of damping rates. The largest spectral noise reductions
are predicted to occur with a very high-Q second-harmonic cavity. Our analytic
results agree well with stochastic numerical simulations. We also compare the
results obtained in the positive-P representation, as a fully quantum
mechanical calculation, with the truncated Wigner phase space equation, also
known as semiclassical theory.
