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Soviet Journal of Quantum Electronics

Some features of the optical manifestation of relaxation

Pavel A Apanasevich, Alexander P. Nizovtsev

Published Aug 31, 1975• 2 citations

Abstract

It is shown that, in general, the interaction between relaxing quantum systems and radiation is described by non-Markov transport equations for the density matrix ρ p and the relaxation matrix depends on the properties of the radiation. The usual Markov-type equations for ρ p are obtained from these equations if γτ c ≪1 and (∊ 2 +4 V 2 ) 1/2 τ c ≪1 (γ is the width of the investigated transition in a quantum system, rc is the correlation time of the perturbations causing relaxation, e is the frequency detuning, and V is the energy of the interaction between the system and the radiation field). When the second inequality is disobeyed, the relaxation process deviates strongly from the Markov type. In the case of small values of V , this results in a non-Lorentzian profile of the absorption lines characterized by a steep descent in the wings. In the range of high values of V , when 2 V τ c 1, the whole relaxation process becomes greatly modified: the dependence of the relaxation coefficients on V becomes strong and a coupling appears between the relaxations of the diagonal and the nondiagonal elements of the density matrix. This is manifested, in particular, in the saturation effect: the term describing saturation in the expression for the polarizability changes considerably and the expression for the line width loses the component due to the adiabatic (elastic) perturbation.

DOI: 10.1070/qe1975v005n08abeh011646

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It is shown that, in general, the interaction between relaxing quantum systems and radiation is described by non-Markov transport equations for the density matrix ρ p and the relaxation matrix depends on the properties of the radiation. The usual Markov-type equations for ρ p are obtained from these equations if γτ c ≪1 and (∊ 2 +4 V 2 ) 1/2 τ c ≪1 (γ is the width of the investigated transition in a quantum system, rc is the correlation time of the perturbations causing relaxation, e is the frequency detuning, and V is the energy of the interaction between the system and the radiation field). When the second inequality is disobeyed, the relaxation process deviates strongly from the Markov type. In the case of small values of V , this results in a non-Lorentzian profile of the absorption lines characterized by a steep descent in the wings. In the range of high values of V , when 2 V τ c 1, the whole relaxation process becomes greatly modified: the dependence of the relaxation coefficients on V becomes strong and a coupling appears between the relaxations of the diagonal and the nondiagonal elements of the density matrix. This is manifested, in particular, in the saturation effect: the term describing saturation in the expression for the polarizability changes considerably and the expression for the line width loses the component due to the adiabatic (elastic) perturbation.

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Numerical integration of a system of equations describing the coherent interaction between radiation and a two-level medium (e.g., a semiconductor) is employed to analyze the effect of relaxation processes on the energy characteristics and dynamics of radiation pulses of duration τpΣ T '2 (T '2 is the irreversible polarization relaxation time). It is shown that the interaction of these pulses with the medium is strongly dependent on their area on entry to the medium and the attenuation of the pulses as a result of relaxation processes decreases with increasing pulse input power. This may be utilized for probing the medium in the resonant absorption range for short polarization relaxation times, in solution 42.65.Gv.

Relaxation of a three-level atom with non-equidistant spectrum interacting with a thermostat and an external stochastic field is studied. The master equation for the reduced density matrix and its solution in the first order of the perturbation theory are obtained. Using quantum regression theorem, explicit formulas for two-time correlation functions and shapes of radiation lines are derived for all types of three-level atoms and the following stochastic processes: a delta-correlated process and a Kubo-Anderson process. The influence of stochastic perturbations of atom’s energy levels and damping constants of adjacent transitions on width of radiation lines is shown.

Decoupling of equations for the diagonal and nondiagonal elements of the density matrix of laser radiation is used in an analysis of the interaction of the radiation field with an active medium. An algorithm for the calculation of the eigenvalues and eigenvectors is proposed for the diagonal elements describing the radiation field statistics. The first minimal (in respect of the magnitude) eigenvalues are calculated and nonequilibrium distribution functions are found for the dynamics of a transition from a vacuum to an equilibrium state; the results agree well with those obtained in machine (numerical) experiments. The equations for the nondiagonal elements of the density matrix are used to find the characteristic times of changes in these elements and to determine the emission line width.

The resonant response of a multilevel atomic system to strong coherent radiation is analyzed using an extension of the methods of time-independent scattering theory, adapted to the density-matrix formalism in order to include thermal bath effects. General steady-state expressions are derived for self-attenuation and resonance scattering rates in strong saturating fields, as functions of the coupling strengths, relaxation parameters, and frequency detunings. Formal expressions are given for radiative-damping and collision-broadening relaxation parameters, relating the latter to scattering amplitudes. Modifications of collision-broadening parameters by the presence of very strong fields are discussed. The Bloch equations are derived by transformation to the time domain, under specified conditions.