Triple correlations in non-degenerate parametric oscillators

We calculate the triple quadrature correlations in non-degenerate quantum parametric amplifiers and oscillators. Firstly, at low signal intensities, a highly non-classical triple correlation can emerge in a parametric amplifier. This cannot be represented using the positive Wigner function approximation, sometimes called the semiclassical method. For this reason, measurements of triple quadrature correlations are a very strong test of stochastic electrodynamics and other hidden variable theories. As the corresponding local oscillator measurements involve a macroscopic particle flux at the detector, a high detector efficiency is possible. We show that triple correlations also feature in the critical fluctuations of the non-degenerate quantum parametric oscillator at threshold. This result allows us to asymptotically calculate the critical squeezing produced for large system size.