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Physical Review A

Dynamics of stochastic systems in nonlinear optics. II. The case of a stochastic configuration interaction

A. A. Villaeys, Alex Boeglin

Published Oct 1, 1991• 2 citations
Publisher

Abstract

In the preceding paper we have developed a formal theoretical approach to describe the dynamics of stochastic systems in nonlinear optics. The main emphasis has been on the description of the equation of motion of the free system, since the optical fields are usually treated by perturbation expansion. Here we explicitly develop the dynamics of complex models, which should be helpful in realistic situations. Two different models are considered. The dynamics of a stochastic three-level system is introduced first. Then the case of two configurations stochastically coupled is explicitly treated. This last model is of interest in order to understand the effects of nonradiative transitions on the nonlinear-optical properties of stochastic systems.

DOI
10.1103/physreva.44.4671

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In the preceding paper we have developed a formal theoretical approach to describe the dynamics of stochastic systems in nonlinear optics. The main emphasis has been on the description of the equation of motion of the free system, since the optical fields are usually treated by perturbation expansion. Here we explicitly develop the dynamics of complex models, which should be helpful in realistic situations. Two different models are considered. The dynamics of a stochastic three-level system is introduced first. Then the case of two configurations stochastically coupled is explicitly treated. This last model is of interest in order to understand the effects of nonradiative transitions on the nonlinear-optical properties of stochastic systems.
A method for treating nonlinear stochastic systems is described which it is hoped will be useful in both the quantum-mechanical many-body problem and the theory of turbulence. In this method the true problem is replaced by models that lead to closed equations for correlation functions and averaged Green's functions. The model solutions are exact descriptions of possible dynamical systems, and, as a result, they display certain consistency properties. For example, spectral components of Green's functions which must be positive-definite in the true problem automatically are so for the models. The models involve a new stochastic element: Random couplings are introduced among an infinite collection of similar systems, the true problem corresponding to the limit where these couplings vanish. The method is first applied to a linear oscillator with random frequency parameter. The mean impulse-response function of the oscillator is obtained explicitly for two successive models. The results suggest the existence of a sequence of model solutions which converges rapidly to the exact solution of the true problem. Applications then are made to the Schrödinger equation of a particle in a random potential and to Burgers' analog for turbulence dynamics. For both problems, closed model equations are obtained which determine the average Green's function, the amplitude of the mean field, and the covariance of the fluctuating field. The model solutions can be expressed as sums of infinite classes of terms from the formal perturbation expansions of the solutions to the true problems. It is suggested that correspondence to stochastic models may be a useful criterion to help judge the validity of partial summations of perturbation series.
Stochastic theories have been used for a long time to describe relaxation as well as line-shape functions. Since the pioneering work of Kubo, these theories have gained considerable interest in the application to optical processes. While previous theories focused on stochastic descriptions of the relaxation, our emphasis is on the dynamics induced by stochastic interactions in a system undergoing relaxation and dephasing processes, including pure dephasing. All the theories based either on the Hilbert-space representation, or on the Liouvillian-space representation, but requiring spectral decomposition methods, are unable to describe these effects simultaneously. In this paper, the implications of the statistical properties of the stochastic variables on the dynamics are elucidated, and the general time evolution of a stochastic N-level system is established in the particular Markovian or weakly colored noise cases.
We present a theoretical approach to describe the dynamics of a system induced by stochastic couplings and undergoing dephasing processes. We consider a model frequently encountered in the study of non-radiative transitions in molecular systems, the case of two configurations stochastically coupled. In this model, we use a system whose excited radiant state is coupled stochastically to a non-radiant state of a lower electronic configuration. We show the line shape for stochastic non-diagonal perturbations to remain Lorentzian. This result is different from the case of frequency modulation where a transition from a Gaussianto a Loren tzianlin e shape canbe observed.
We present a theoretical approach to describe the dynamics of a system induced by stochastic couplings and undergoing dephasing processes. We consider a model frequently encountered in the study of non-radiative transitions in molecular systems, the case of two configurations stochastically coupled. In this model, we use a system whose excited radiant state is coupled stochastically to a non-radiant state of a lower electronic configuration. We show the line shape for stochastic non-diagonal perturbations to remain Lorentzian. This result is different from the case of frequency modulation where a transition from a Gaussian to a Lorentzian line shape can be observed.
For pt.I see ibid. vol.22, p.4877 (1989). Previously (I) the author considered a second-quantised Hamiltonian which can be used to model a large number of important strongly interacting many-body systems. He has shown that the dynamics of such systems can be described exactly using quantum fields. The equations of motion derived in this way are highly nonlinear partial differential equations (PDE). He now uses a standard field theoretical approach where initially the fields are treated as classical functions. He studies the solutions of these equations which remarkably, may be obtained making full use of very recent mathematical discoveries (the symmetry reduction method). He finds exact solutions of the equations of motion for the classical field for each of the four main cases separately found in the preceding paper. These cases may correspond to different physical situations and the last, the second-order case, corresponds to the most general physical situation. Subsequently he outlines possible quantisation procedures for these fields. He then briefly discusses the type of boundary conditions which may be applied to both the classical field and the quantum fluctuations which arise.
The Langevin-equation (1) for the fluctuating light-field E in a nonlinear semiconductor resonator has been derived in Ref.[1]: All quantities are dimensionless, εi being proportional to the amplitude of the incoming field, κ is a damping constant, ω" is the ratio of imaginary and real part of the dielectric function and Δ is proportional to the detuning. The additive term f(τ) is the classical noise source representing a Gauss-Markov process. The staionary properties of eq.(1) have been investigated in Ref.[1] and the details of optical bistable operation has been determined.