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Physical Review A

Oscillations and quantized second-harmonic generation

Craig M. Savage

Published Jan 1, 1988• 24 citations
Publisher

Abstract

We investigate quantum-mechanical second-harmonic generation for parameters such that classical electrodynamics predicts oscillations. Specifically we calculate the Q distribution function in a Gaussian approximation about the classical limit cycle. In the classical limit initial rapid collapse of the Q distribution into the neighborhood of the limit cycle is followed by diffusion around the limit cycle. The experimental significance of this quantum diffusion is discussed.

DOI
10.1103/physreva.37.158

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We investigate quantum-mechanical second-harmonic generation for parameters such that classical electrodynamics predicts oscillations. Specifically we calculate the Q distribution function in a Gaussian approximation about the classical limit cycle. In the classical limit initial rapid collapse of the Q distribution into the neighborhood of the limit cycle is followed by diffusion around the limit cycle. The experimental significance of this quantum diffusion is discussed.
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Nonlinear optical effects where quantum mechanical predictions are at variance with the results of classical calculations are studied. Certain initial conditions, those of unstable equilibrium or those that lead to unstable equilibrium, classically, give rise to an aperiodicity in the intensity of the emitted radiation. One example of such a system is second-harmonic generation, which is studied in detail because of the potential amenability of the aperiodicity to experimental observation. Numerical and analytic predictions for the evolution of the intensity and photon statistics of the second harmonic light are presented. The photon statistics of the second harmonic light are predicted to undergo a sharp transition following the first maximum of the intensity.
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Stochastic electrodynamics (SED) predicts a Gaussian probability distribution for a classical harmonic oscillator in the vacuum field. This probability distribution is identical to that of the ground state quantum harmonic oscillator. Thus, the Heisenberg minimum uncertainty relation is recovered in SED. To understand the dynamics that give rise to the uncertainty relation and the Gaussian probability distribution, we perform a numerical simulation and follow the motion of the oscillator. The dynamical information obtained through the simulation provides insight to the connection between the classic double-peak probability distribution and the Gaussian probability distribution. A main objective for SED research is to establish to what extent the results of quantum mechanics can be obtained. The present simulation method can be applied to other physical systems, and it may assist in evaluating the validity range of SED.
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Summary form only given. The situation when nonlinear interactions involve a large number of photons, N, is quite typical of many of the problems of quantum and nonlinear optics. To describe the dynamics of such quantum systems a classical approach is often used. In this theoretical work we compare classical and quantum dynamics of several dissipative quantum optical systems, which have a limit cycle in the classical limit N/spl rarr//spl infin/. In particular, we consider the dynamics of intracavity second harmonic generation (SHG) described by the Hamiltonian.