Critical fluctuations and entanglement in the nondegenerate parametric oscillator

Abstract

We present a fully quantum mechanical treatment of the nondegenerate optical parametric oscillator both below and near threshold. This is a nonequilibrium 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 Einstein-Podolsky-Rosen (EPR) correlations as well as higher order tripartite correlations, and show how these are modified as the critical point is approached. These results are compared with those obtained using two degenerate parametric oscillators, and we find that in the near-critical region the nondegenerate oscillator has stronger EPR correlations. In general, the critical fluctuations represent an ultimate limit to the possible entanglement that can be achieved in a nondegenerate parametric oscillator.

Semiclassical Wigner distribution for a two-mode entangled state generated by an optical parametric oscillator

“…(16) for the intracavity noise squeezing in the combined quadratures. As we mentioned before, at threshold we have perfect external squeezing while the unsqueezed combined quadratures blow up showing the failure of linearization [15]. Moreover, it is easy to characterize this state as entangled by using the Duan-Simon criterion [8,9].…”

“…The usual approach given to quantum noise in the literature is obtained from first order stochastic equations of motion [15,18]. These equations are often used to predict squeezing in a linearized fluctuation analysis.…”

“…They are defined as These quantities correspond to the squeezed and antisqueezed combined quadratures obtained in the linearized theory. From the first order stochastic equations one can easily obtain the steady state variances of the EPR variables [15,18]:…”

Resonant cascaded down-conversion

“…We note here that a linearized analysis is only the first stage in a stochastic diagram perturbation expansion [31], which in general needs to be taken to higher order to reveal nonGaussian behavior [13].…”

“…Here, we investigate a different type of source, which can produce non-Gaussian light directly. Theoretical studies of optical parametric oscillators (OPOs) which are based on a single second-order optical nonlinearity (χ (2) ) have shown non-Gaussian signatures to be rather scarce [12] except in the case of tripartite correlations between the three fields [13]. An interesting approach is to use an optical nonlinearity of, at least, third order.…”