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Optics and Spectroscopy

The shape of the statistical wing of an absorption line: Derivation by the Green’s function method

Е. А. Титов

Published Mar 1, 2006• 0 citations
Publisher

Abstract

A formula is derived for the absorption coefficient in the statistical wing of an absorption line in the case in which both levels of a two-level system are involved in collisions. For this purpose, the kinetic equations for two-level particles in a strong field at an arbitrary detuning of the field frequency from the transition frequency are used, which were derived previously with the Green’s function method for nonequilibrium systems.

DOI
10.1134/s0030400x06030027

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A formula is derived for the absorption coefficient in the statistical wing of an absorption line in the case in which both levels of a two-level system are involved in collisions. For this purpose, the kinetic equations for two-level particles in a strong field at an arbitrary detuning of the field frequency from the transition frequency are used, which were derived previously with the Green’s function method for nonequilibrium systems.
Kinetic quantum equations are derived for a density matrix with collision integrals describing nonlinear effects in spectra line wings. These equations take into account the earlier established inequality of the spectral densities of Einstein coefficients for absorption and stimulated radiation emission by a two-level quantum system in the far wing of a spectral line in the case of frequent collisions. The relationship of the absorption and stimulated emission probabilities with the characteristics of radiation and an elementary scattering event is found.
Using the eikonal approximation, we have calculated effective collision frequencies in density-matrix kinetic equations describing nonlinear effects in the wings of spectral lines. We have established the relation between the probabilities of absorption and stimulated emission and the characteristics of the radiation and elementary scattering event. The example of the power interaction potential shows that quantum mechanical calculation of the collision frequencies in the eikonal approximation and previously known spectral line wing theory give similar results for the probability of radiation absorption.
A general expression is found for the Green's function of a two-level system interacting with a classical monochromatic field of arbitrary frequency and intensity. The Green's function is first used to calculate the transition probability from the lower to the upper state as a function of the frequency and intensity of the field and a comparison of the results is made with experiment. It is then used to study the problem of scattering. It is found that the spectrum of the scattered radiation consists of a line at the frequency of the field (Rayleigh line) and two satellites symmetrically displaced from the Rayleigh line by an amount depending both on the intensity and frequency of the field. In addition there are emissions at three, five, etc., times the frequency of the field, and accompanying satellites. The cross sections for the production of the Rayleigh line and its satellites, and for the third-harmonic line and its satellites have been calculated for a range of values of the frequency and intensity of the incident field.
We develop an expansion for the absorption line shape of a two-level system in a strong radiation field being perturbed by dephasing collisions with a thermal bath of foreign atoms. We assume that the bath atoms do not interact with the radiation field and with each other. The microscopic information relevant for line broadening in a strong field is expressed in terms of a hierarchy of dipole correlation functions. The latter may be calculated with the use of the Franck-Condon factors for the nuclear wave functions. The present theory reduces in the weak-field limit to the well-known linear unified theory of line broadening and in the impact limit we recover the Karplus-Schwinger formula.