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Full length article Wigner function for a generalized model of a parametric oscillator: phase-space tristability, competition and nonclassical effects

Karen V. Kheruntsyan, Gagik Yu. Kryuchkyan, K.G. Petrossian

Published Jan 1, 1997• 0 citations
OpenAlex

Abstract

We present an exact quantum treatment of the generalized model of a degenerate parametric oscillator, in which we allow for self-phase modulation of the signal mode. Using the steady-state solution of the Fokker-Planck equation in the complex P-representation we obtain an exact analytical result for the Wigner quasiprobability distribution function. The obtained Wigner function allows us to give an explicit phase-space description of the nonlinear system under consideration, including the critical transition behavior in the monostable and bistable (with respect to the signal mode intensity versus the pump field intensity) operation regimes and in the threshold region. The competitive effects influenced by the self-phase modulation and the phase-space tristability are analyzed. Nonclassical effects of quadrature squeezing and quantum superposition are discussed as well.

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We present an exact quantum treatment of the generalized model of a degenerate parametric oscillator, in which we allow for self-phase modulation of the signal mode. Using the steady-state solution of the Fokker-Planck equation in the complex P-representation we obtain an exact analytical result for the Wigner quasiprobability distribution function. The obtained Wigner function allows us to give an explicit phase-space description of the nonlinear system under consideration, including the critical transition behavior in the monostable and bistable (with respect to the signal mode intensity versus the pump field intensity) operation regimes and in the threshold region. The competitive effects influenced by the self-phase modulation and the phase-space tristability are analyzed. Nonclassical effects of quadrature squeezing and quantum superposition are discussed as well.
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.
We consider the quantum model of a driven anharmonic oscillator, in the presence of dissipation, and present an exact analytic solution for the corresponding Wigner function in the steady-state regime. This provides explicit phase-space images of the resulting state of the cavity mode, and allows us to understand how the quantum interference is built up into it. The photon number probability distribution is calculated and analysed as well. We monitor the transition from the semiclassical to extreme quantum regime of operation, and identify qualitative changes, where the conventional characteristics of the model, such as bistability or turning points, become meaningless.
The positive Prepresentation has proved a useful tool for analyzing optical systems which exhibit squeezing. The Glauber-Sudarshan phase-space representation gives a Fokker-Planck equation without positive semidefinite diffusion for systems which generate nonclassical fields. The positive Prepresentation provides a method for guaranteeing positive semidefinite diffusion and enables a quantum- dynamical model to be formulated in terms of classical stochastic differential equations. In previous work we had some success in applying this approach to describe the decay of the unstable state above threshold in a degenerate parametric oscillator.1 Below threshold a linearized analysis of fluctuations recovers the predictions of squeezing obtained by alternative methods.
We present an exact quantum treatment of intracavity parametric four-wave mixing accompanied by self- and cross-phase modulation effects. A steady-state solution of the Fokker-Planck equation in a generalized P-representation is found for a generated signal mode. A general expression for arbitrary moments of the signal field operators is obtained and qualitative effects caused by non-linear treatment of quantum fluctuations are discussed. The behaviour of the exact steady-state intensity, as well as of the second-order correlation function and of the photon number fluctuations of the signal mode, is studied. A sub-Poissonian photon statistic is found in the above-threshold region and a critical increase in the photon number fluctuations is shown in the transition region for the case of relatively small non-linearities. Characteristic properties of the threshold behaviour are discussed in the case of large non-linearities.