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Journal of the Physical Society of Japan

Second Order Optical Process of a Three-Level System in Contact with a Nearly Gaussian-Markoffian Noise Bath

Yoshitaka Tanimura, Tetsuhiro Suzuki, Ryogo Kubo

Published May 15, 1989• 15 citations
Publisher

Abstract

A general formulation is given for the second order optical process of a three-level system in contact with a heat bath consisting of harmonic oscillators with a proper frequency spectrum producing a Markoffian random perturbation. Elimination of the bath using the influence functional method of Feynman and Vernon yields a continuous fraction expression and the result affords a basis to clarify the relation between the stochastic and dynamical approaches to treat the partial destruction of the quantum coherence of the optical process. As an example, a three-level system with intermediate-state interaction is treated in some detail.

DOI
10.1143/jpsj.58.1850

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A general formulation is given for the second order optical process of a three-level system in contact with a heat bath consisting of harmonic oscillators with a proper frequency spectrum producing a Markoffian random perturbation. Elimination of the bath using the influence functional method of Feynman and Vernon yields a continuous fraction expression and the result affords a basis to clarify the relation between the stochastic and dynamical approaches to treat the partial destruction of the quantum coherence of the optical process. As an example, a three-level system with intermediate-state interaction is treated in some detail.
A test system is assumed to interact with a heat bath consisting of harmonic oscillators or an equivalent bath with a proper frequency spectrum producing a Gaussian-Markoffian random perturbation. The effect of reaction of the test system to the bath is considered in the high temperature approximation. Elimination of the bath using the influence functional method of Feynman and Vernon yields a continuous fraction expression for the reduced density matrix of the test system. The result affords a basis to clarify the relationship between the stochastic and the dynamical approaches to treat the problem of partial destruction of quantum coherence of a system interacting with its environment.
Two-time correlation functions of physical operators of a system interacting with a heat bath producing a Gaussian-Markoffian noise are treated with the use of Feynman-Vernon's influence functional formalism. The Laplace transforms of correlation functions are shown to be expressed in terms of continued fractions of relevant hyperoperators. The result is valid to all orders of the system-bath interaction. In the limitation of motional narrowing, it reduces to those obtained in conventional methods.
A two-level system diagonally coupled to a heat bath consisting of harmonic oscillators with a proper frequency spectrum producing a Markoffian random perturbation is considered. Assuming relatively weak or fast modulation conditions, we calculate spectra from the system driven by a continuous wave and a pulsed laser of arbitrary strength by applying the physical spectrum proposed by Eberly and Wódkiewicz. The results give the coherent and the incoherent spectra; clarification of how dynamical evolution of these components is influenced by the change of physical parameters is obtained.
A stochastic model of a multi-level atom interacting with its environment randomly in its excited states is considered. Assuming a two state-jump model for the random intermediate-state interaction, analytical expressions of emission spectra of three- and four-level atoms in the second order optical process are obtained. In each case, the expression is shown to be separated into three types of terms. Two types of terms correspond to the pure Raman spectrum and the luminescence, respectively, while the other type of term bears a mixed character contributing to broadening of the Raman line when the random modulation is slow and merging into the luminescence when the modulation is fast. This characteristic behavior is discussed with the use of the analytical expressions.
A general theory of the intermediate state interaction in the second order optical process is developed by extending the stochastic theory of line shape. We introduce the stochastic models of the intermediate state interaction and discuss the general features of the secondary emission spectrum, focusing particular attention on how the correlation between the incident and the scattered photons are affected by the intermediate state interaction. It is clarified how the spectral features or the relative weights among the various components of the second order optical process are determined by a few physical parameters; the radiative damping rate and the modulation amplitude and rate. Further we discuss the relationships between the first and the second order optical processes.
We have considered the stochastic effect of a thermal reservoir on a two-level system interacting with a classical electromagnetic field. The collective effect of the bath is modelled in this work as a random Bohr frequency whose manifestation is the broadening of the upper level according to a prescribed random function. This assumption transforms the optical Bloch equations to a system of multiplicative stochastic differential equations whose solution is found. After obtaining analytical expressions for the linear absorption coefficient and linear refraction index, numerical results are presented. This is done in the form of tridimensional plots of the optical properties as a function of the pump-frequency detuning factor and the strength or relaxation time of the noise source.