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Physical Review A

Quantum Boltzmann-Lorentz model approach to the line-shape problem

Daniel Loss, A. Thellung, Łukasz A. Turski

Published Mar 1, 1990• 1 citation
Publisher

Abstract

Following our earlier work on quantum kinetic equations and the use of the Boltzmann-Lorentz model to describe the collisional broadening of spectra, we develop a new approach to that problem in which the center-of-mass motion of the emitter as well as the interaction during the collision between the emitter and the buffer gas are treated quantum mechanically. Possible further extensions of our model are discussed.

DOI
10.1103/physreva.41.3005

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Following our earlier work on quantum kinetic equations and the use of the Boltzmann-Lorentz model to describe the collisional broadening of spectra, we develop a new approach to that problem in which the center-of-mass motion of the emitter as well as the interaction during the collision between the emitter and the buffer gas are treated quantum mechanically. Possible further extensions of our model are discussed.
We present a theory of pressure effects in which the atomic center-of-mass motion is treated quantum mechanically. The quantum-mechanical calculation treats the perturber-induced energy-level variations and the velocity changes of the emitter caused by collisions on an equal basis. Specifically, we shall treat the problem of a laser to first order in the laser field, allowing for the fact that the laser atoms are undergoing collisions, but our results will also be applicable to the cases of stimulated emission or absorption. It will be sufficient to carry out a perturbation solution of the problem assuming that the laser atoms undergo at most one collision in their lifetime, since such a restricted calculation reveals the salient features of the theory. We shall find that, in general, there is no classical limit for our results. Thus, previous treatments employing a classical Boltzmann-equation approach for the atomic center-of-mass motion are invalid, and a quantum-mechanical description is necessary to correctly treat cases where both the modified Doppler effect (modified by collisions) and perturber-induced energy-level variations are present.
The shape of an isolated spectral transition is analyzed in terms of an approximation to the Waldmann–Snider kinetic equation. This equation is written in the form of a drift and collision operator acting on a density matrix. With the use of the spherical approximation, the collision operator is subdivided into an elastic Boltzmann-like collision term, an inelastic loss term, and a dephasing term. The Boltzmann-like term is responsible for Dicke narrowing of spectral lines, the inelastic loss term leads to line broadening and the dephasing term may contribute both to line broadening and shifting. Simple approximations to these terms are powerful enough to account for some of the details of experimental line shapes such as asymmetrical deviations from a Lorentzian line shape. Model numerical calculations are carried out assuming classical scattering potentials of the form 1/rn in the Boltzmann-like term and single complex frequencies for the other two. It was found that as long as the density was scaled to give the same diffusion constant, the exact form of the scattering cross section had little effect on the final line shape.
Inelastic-collisional effects are incorporated into a quantum-mechanical transport equation (QMTE) that was developed in an earlier paper. The QMTE enables one to follow the quantum-state evolution of moving atoms which are interacting with some external fields while undergoing collisions with perturber atoms. The collisional processes are treated quantum mechanically and then reinterpreted in terms of classical variables so that the QMTE becomes an integrodifferential equation for atomic density-matrix elements containing both well-defined quantum-mechanical collision parameters (i.e., scattering amplitudes) and densitymatrix elements which are functions of classical position and velocity variables. Solutions of the QMTE enable one to derive line-shape formulas, and a discussion will be given of the general features to be expected for spectral profiles, Hanle-effect line shapes, and laser output curves, as well as the manner in which these features differ from those predicted by theories that neglect some quantum-mechanical aspects of the collision events. The inclusion of inelastic-collisional effects does not cover cases in which the frequency spacing of the atomic levels under consideration is comparable to the inverse duration time of a collision; nevertheless, the QMTE to be derived will be applicable to the analysis of a large number of atomic systems.
Investigations in the field of quantum optics very often requires a molecular simulation of transport phenomena. This paper presents a discrete kinetic model to deal with the quantum optical problem of hole burning. Based on a full discretization of the velocity space, Boltzmann-like equations are established to govern the temporal evolution of heavily disturbed Maxwell distributions of optically excited particles diffusing in a buffer gas. These model equations involve elastic collisions and the interaction of photons with three-level atoms by taking into account the Doppler effect. Important features of the developed model are the realistic simulation of the relaxation process and the existence of a stable and unique equilibrium solution.
Quantum mechanics is used to treat the motion of the perturbers broadening a line. Several simplifying assumptions are made, such as the Born-Oppenheimer approximation, and the assumption of no degeneracy. These assumptions can be removed. Also made is the "impact approximation," which is essential for the validity of the results and cannot be removed. With it, the line has a Lorentz shape, and its width and shift are expressed in terms of the two scattering amplitudes of the perturbers by the atom in its upper and lower state. The case where the perturbers do not interact appreciably with the lower state is particularly simple. Then, the width and shift are proportional to the imaginary part and the real part of the forward scattering amplitude, respectively. A quantity called the "collision volume" is defined. It is shown that the impact approximation is valid only if the collision volume is much smaller than the volume per perturber. There is a second validity condition, which has no classical analog.