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Physical Review A

Dynamics of stochastic systems in nonlinear optics. I. General formalism

A. A. Villaeys, Alex Boeglin, S. H. Lin

Published Oct 1, 1991• 11 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.44.4660

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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.
A stochastic approach to the quantum dynamics randomly modulated in time by a discrete state non-Markovian noise, which possesses an arbitrary nonexponential distribution of the residence times, is developed. The formally exact expression for the Laplace-transformed quantum propagator averaged over the stationary realizations of such N -state non-Markovian noise is obtained. The theory possesses a wide range of applications. It includes some previous Markovian and non-Markovian theories as particular cases. In the context of the stochastic theory of spectral line shape and relaxation, the developed approach presents a non-Markovian generalization of the Kubo-Anderson theory of sudden modulation. In particular, the exact analytical expression is derived for the spectral line shape of optical transitions described by a Kubo oscillator with randomly modulated frequency which undergoes jumplike non-Markovian fluctuations in time.
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.
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.
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.
A complete stochastic theory of relaxation is developed in terms of a homogeneous equation for the averaged density matrix of a system immersed in a thermal bath.\nThis theory is then used as the basis of a new stochastic approach to the phenomenon of collisional broadening of spectral line shapes. Single photon and multiphoton processes are studied.\nThe features of a line shape are linked by simple expressions to the statistical properties of a stochastic hermitian Hamiltonian. The ordinary line shape predicted by Kubo's approach is generalized. The present approach predicts broadening as well as asymmetry and shift. A representation of line shapes in multiphoton processes by diagrams is also developed.
Half century has past since the pioneering works of Anderson and Kubo on the stochastic theory of spectral line shape were published in J. Phys. Soc. Jpn. 9 (1954) 316 and 935, respectively. In this review, we give an overview and extension of the stochastic Liouville equation focusing on its theoretical background and applications to help further the development of their works. With the aid of path integral formalism, we derive the stochastic Liouville equation for density matrices of a system. We then cast the equation into the hierarchy of equations which can be solved analytically or computationally in a nonperturbative manner including the effect of a colored noise. We elucidate the applications of the stochastic theory from the unified theoretical basis to analyze the dynamics of a system as probed by experiments. We illustrate this as a review of several experimental examples including NMR, dielectric relaxation, Mössbauer spectroscopy, neutron scattering, and linear and nonlinear laser spectroscopies. Following the summary of the advantage and limitation of the stochastic theory, we then derive a quantum Fokker–Planck equation and a quantum master equation from a system–bath Hamiltonian with a suitable spectral distribution producing a nearly Markovian random perturbation. By introducing auxiliary parameters that play a role as stochastic variables in an expression for reduced density matrix, we obtain the stochastic Liouville equation including temperature correction terms. The auxiliary parameters may also be interpreted as a random noise that allows us to derive a quantum Langevin equation for non-Markovian noise at any temperature. The results afford a basis for clarifying the relationship between the stochastic and dynamical approaches. Analytical as well as numerical calculations are given as examples and discussed.