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Physical Review A

Quantum-theoretical treatments of three-photon processes

M. K. Olsen, Lev I. Plimak, Michael Fleischhauer

Published Apr 22, 2002• 12 citations
Publisher

Abstract

We perform and compare different analyses of triply degenerate four-wave mixing in the regime where three fields of the same frequency interact via a nonlinear medium with a field at three times the frequency. As the generalized Fokker-Planck equation (GFPE) for the positive-$P$ function of this system contains third-order derivatives, there is no mapping onto genuine stochastic differential equations. Using techniques of quantum field theory, we are able to write stochastic difference equations that we may integrate numerically. We compare the results of this method with those obtained by the use of approximations based on semiclassical equations, and on truncation of the GFPE leading to stochastic differential equations. In the region where the difference equations converge, the stochastic methods agree for the field intensities, but give different predictions for the quantum statistics.

DOI
10.1103/physreva.65.053806

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We present an introduction to phase-space techniques (PST) based on a quantum-field-theoretical (QFT) approach. In addition to bridging the gap between PST and QFT, our approach results in a number of generalizations of the PST. First, for problems where the usual PST do not result in a genuine Fokker-Planck equation (even after phase-space doubling) and hence fail to produce a stochastic differential equation (SDE), we show how the system in question may be approximated via stochastic difference equations $(\mathrm{S}\ensuremath{\Delta}\mathrm{E}).$ Second, we show that introducing sources into the SDE's (or $\mathrm{S}\ensuremath{\Delta}\mathrm{E}\mathrm{'}\mathrm{s})$ generalizes them to a full quantum nonlinear stochastic response problem (thus generalizing Kubo's linear reaction theory to a quantum nonlinear stochastic response theory). Third, we establish general relations linking quantum response properties of the system in question to averages of operator products ordered in a way different from time normal. This extends PST to a much wider assemblage of operator products than are usually considered in phase-space approaches. In all cases, our approach yields a very simple and straightforward way of deriving stochastic equations in phase space.
We demonstrate a method which allows the stochastic modelling of quantum systems for which the generalised Fokker-Planck equation in the phase space contains derivatives of higher than second order. This generalises quantum stochastics far beyond the quantum-optical paradigm of three- and four-wave mixing problems to which these techniques have so far only been applicable. To verify our method, we model a full Wigner representation for the optical parametric oscillator, a system where the correct results are well known and can be obtained by other methods.
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The study of quantum stochastic processes presents severe difficulties, both on the theory level as well as on technical grounds. The numerically exact solution remains prohibitive even today. In this paper we review and present new results for three different methods used for the modelling of quantum stochastic processes. These include a mixed quantum classical approach, semiclassical initial value representations of the quantum propagator and the reduced density matrix approach as typified by the quantum Wigner-Fokker-Planck equation. A new semiclassical initial value representation that does away with cumbersome prefactors which depend on the monodromy matrix elements but is exact for a harmonic oscillator is presented and its properties analysed. A recently proposed systematic method for improving semiclassical initial value representations is reviewed. The generalization of the Wigner-Fokker-Planck equation to stochastic processes with memory is obtained by using a novel integral equation representation.
The Fokker-Planck equation approach to forward four-wave mixing developing from the initial coherent states is adopted to obtain a set of recursive relations for the expectation values of the normally ordered field operators. This set of equations is closed, in accordance with the model of the generalized superposition of coherent and chaotic fields, to provide differential equations for unknown quantities in this model. A numerical solution of these equations enables one to determine the photon number distribution, its factorial moments, squeezing variances, and entropy of radiation under discussion. Fully quantum-mechanical features of radiation such as its sub-Poisson behavior and squeezing of vacuum fluctuations are preserved in this approach. The validity of this approximation is discussed and numerical results are provided.