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Physical Review A

Dicke narrowing as an example of line mixing

S. Dolbeau, Richard Berman, James R. Drummond, A. D. May

Published May 1, 1999• 10 citations
Publisher

Abstract

In the preceeding paper it was shown that the general problem of spectral line shapes in multiline, IR spectra may be determined by solving a transport relaxation equation for the off-diagonal elements of the density matrix. This is a semiclassical equation at the Wang-Chang--Uhlenbeck level, i.e., it treats the quantized internal states as nondegenerate. Here we apply the master equation to the case of Dicke narrowing, and by discretizing the velocity distribution show that Dicke narrowing of a single line may be treated in exactly the same manner as line mixing. Both effects lead to a narrowing of a spectral distribution. We indicate how the numerical technique can be extended and used to calculate profiles in the general case of spectra with speed-dependent broadening, shifting, and line mixing.

DOI
10.1103/physreva.59.3506

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Physical Review A•Aug 1, 1998•151 citations
In the preceeding paper it was shown that the general problem of spectral line shapes in multiline, IR spectra may be determined by solving a transport relaxation equation for the off-diagonal elements of the density matrix. This is a semiclassical equation at the Wang-Chang--Uhlenbeck level, i.e., it treats the quantized internal states as nondegenerate. Here we apply the master equation to the case of Dicke narrowing, and by discretizing the velocity distribution show that Dicke narrowing of a single line may be treated in exactly the same manner as line mixing. Both effects lead to a narrowing of a spectral distribution. We indicate how the numerical technique can be extended and used to calculate profiles in the general case of spectra with speed-dependent broadening, shifting, and line mixing.
Borrowing heavily from the literature, we develop a transport relaxation equation for the off-diagonal elements of the density matrix. It is applicable to a large range of topics in laser theory, nonlinear optics and spectral line shapes, etc. Applying this single equation to the case of infrared electric dipole transitions permits us to recover, quickly and simply, all of the well-known spectral line shapes found in the literature. It also offers further insight into the process of line mixing, particularly in the weak mixing limit. An outline of a new treatment of Dicke narrowing and other speed-dependent contributions to spectral profiles is also given in terms of the transport relaxation equation. The treatment also provides a numerical method for including the effects of statistical correlation between the evolution of the internal and translational degrees of freedom.
In this paper, we present a formalism for the exact calculation of spectral profile of an isolated line undergoing speed-dependent collisional broadening and shifting and Dicke narrowing. It is couched in the language of the density matrix as in the closely related fields of transport phenomena, laser physics, and nonlinear optics. The formalism is ab intio in spirit, permitting the calculation of the line shape starting only from a given interaction potential. The line shape is obtained by solving a transport/relaxation equation for the off-diagonal element of the density matrix, a macroscopic quantity. The collision operator in the transport/relaxation equation, which is determined from the (microscopic) interaction between the active molecule and the perturber, is presumed known.
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.
The purpose of this work is to verify the possibility of the observation of the Dicke narrowing of atomic spectral lines in the optical domain. As an example we have chosen experimental results obtained by means of a laser-induced fluorescence method for the 114 Cd 326.1 nm line perturbed by xenon. Experimental results were carefully reanalysed using a line shape model which takes into account the speed dependence of collisional broadening and shifting, the velocity-changing collisions and the collision-time asymmetry. We have found no convincing evidence for the occurrence of Dicke narrowing of the 326.1 cadmium line perturbed by xenon. The Dicke narrowing (if it occurs at all) appears to be smaller than that resulting from the estimation based on the accessible data describing the diffusion of Cd in Xe. As a source of this behaviour the correlation between velocity-changing and dephasing collisions was indicated.
The effect of the velocity ( v ) dependence of the transport collision frequency ν tr v on the Dicke line narrowing is analyzed in terms of the strong-collision model generalized to velocity-dependent collision frequencies (the so-called kangaroo model). This effect has been found to depend on the mass ratio of the resonance ( M ) and buffer ( M b ) particles, β = M b / M : it is at a minimum for β ≪ 1 and reaches a maximum for β ≳ 3. A power-law particle interaction potential, U ( r ) ∝ r − n , is used as an example to show that, compared to ν tr v ( v ) = const ( n = 4), the line narrows if ν tr v ( v ) decreases with increasing v ( n < 4) and broadens if ν tr v ( v ) increases with v ( n > 4). At β ≳ 3, the line width can increase [compared to ν tr v ( v ) = const] by 5 and 12% for the potentials with n = 6 and n ≳ 10, respectively; for the potentials with n = 1 (Coulomb potential) and n = 3, it can decrease by more than half and 6%, respectively. The line profile I (Ω) has been found to be weakly sensitive to ν tr v ( v ) at some detuning Ω c of the radiation frequency Ω. Dicke line narrowing is used as an example to analyze the collisional transport of nonequilibrium in the resonance-particle velocity distribution in a laser field. The transport effect is numerically shown to be weak. This allows simpler approximate one-dimensional quantum kinetic equations to be used instead of the three-dimensional ones to solve spectroscopic problems in which it is important to take into account the velocity dependence of the collision frequency when the phase memory is preserved during collisions.
The Rautian-Sobelman approach describing the collision-Doppler shape of spectral line that takes into account both soft and hard velocity-changing collisions is extended to the case when correlation between Doppler and collision broadening as well as the dispersion line asymmetry should be considered. Numerical calculations show that speed-dependent effects can produce the additional line narrowing as well as broadening and also the line asymmetry. It is shown that speed-dependent profiles based on the soft and hard collision models differ much more one from another than those which omit the speed-dependent effects.