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Physical Review A

Exact steady-state Wigner function for a nondegenerate parametric oscillator

Karen V. Kheruntsyan, K. G. Petrosyan

Published Jun 7, 2000• 19 citations
Publisher

Abstract

We consider the model of a nondegenerate parametric oscillator, in the adiabatic limit of strongly damped pump mode, and find an exact analytic solution for the joint two-mode Wigner function in the steady-state regime. The one-mode Wigner functions for the individual signal and idler modes are derived as well. This gives an explicit and complete phase-space representation of this two-mode nonequilibrium quantum system. Simple examples are given illustrating the phase-space images of optical bistability and the phase diffusion effect.

DOI
10.1103/physreva.62.015801

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We derive the steady-state solution of the Fokker-Planck equation that describes the dynamics of the nondegenerate optical parametric oscillator in the truncated Wigner representation of the density operator. We assume that the pump mode is strongly damped, which permits its adiabatic elimination. When the elimination is correctly executed, the resulting stochastic equations contain multiplicative noise terms and do not admit a potential solution. However, we develop a heuristic scheme leading to a satisfactory steady-state solution. This provides a clear view of the intracavity two-mode entangled state valid in all operating regimes of the optical parametric oscillator. A non-Gaussian distribution is obtained for the above threshold solution.
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We study the nondegenerate parametric oscillator in the limit of small fluctuations where a linearized theory is valid. Interest is focused on solutions above threshold. It has been shown that the field undergoes phase diffusion above threshold,1 and the theory incorporates this effect. We calculate the spectrum of fluctuations in the intensity difference between the two modes,2 allowing for differing cavity mode decay rates. Possibilities for quantum nondemolition experiments and other quantum properties are discussed. In particular, the effect of phase diffusion on the long-time heterodyne squeezing spectra is calculated. The linearized results are compared to the full nonlinear steady-state solutions in the limit of a very fast decaying pump.