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arXiv:0906.5316

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

K. Dechoum, Marcelo Dumas Hahn, Antonio Zelaquett Khoury, Raúl O. Vallejos

Published Jun 29, 2009• 0 citations
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Abstract

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. The adiabatic limit of strong pump damping is assumed. This phase space image provides a clear view of the intracavity two-mode entangled state valid in all operating regimes of the OPO. A nongaussian distribution is obtained for the above threshold solution.

DOI
10.48550/arxiv.0906.5316

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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. The adiabatic limit of strong pump damping is assumed. This phase space image provides a clear view of the intracavity two-mode entangled state valid in all operating regimes of the OPO. A nongaussian distribution is obtained for the above threshold solution.
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The degenerate parametric oscillator is an important source of squeezed light. If the pump mode is adiabatically eliminated, its operation is governed by two dimensionless parameters: the pump parameter λ and the coupling constant g. The pump parameter controls the threshold behavior predicted by semiclassical theory (threshold is reached for λ = 1). The coupling constant g determines the strength of the quantum fluctuations. The optical parametric oscillators used to generate squeezed light have g ~ 10–4 ≪ 1.1 Therefore, in these systems the quantum fluctuations only produce very small perturbations about semiclassical states. We have recently shown how the steady state of the degenerate parameter oscillator changes as a function of coupling strength.2 In the strong coupling limit, major departures from semiclassical states occur; for g2 ≫ 1 and 2λ2/g2 ≪ 1 the steady state approches the coherent state superposition | ψ + 〉 = | λ / g 〉 + | − λ / g .
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.