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Physical Review A

Molecular dynamics and a simplified master equation for spectral line shapes

A. D. May

Published May 1, 1999• 37 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.59.3495

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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 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.
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.
A complete stochastic theory of relaxation is developed in terms of a homogeneous equation for the averaged density matrix of a system immersed in a thermal bath.\nThis theory is then used as the basis of a new stochastic approach to the phenomenon of collisional broadening of spectral line shapes. Single photon and multiphoton processes are studied.\nThe features of a line shape are linked by simple expressions to the statistical properties of a stochastic hermitian Hamiltonian. The ordinary line shape predicted by Kubo's approach is generalized. The present approach predicts broadening as well as asymmetry and shift. A representation of line shapes in multiphoton processes by diagrams is also developed.
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.
We propose a formulation to obtain the line shape of a magnetic response with dissipative effects that directly reflects the nature of the environment. Making use of the fact that the time evolution of a response function is described by the same equation as the reduced density operator, we formulate a full description of the complex susceptibility. We describe the dynamics using the equation of motion for the reduced density operator, including the term for the initial correlation between the system and a thermal bath. In this formalism, we treat the full description of non-Markovian dynamics including the initial correlation. We present an explicit and compact formula up to the second order of cumulants, which can be applied in a straightforward way to multiple-spin systems. We also take into account the frequency shift by the system-bath interaction. We study the dependence of the line shape on the type of interaction between the system and the thermal bath. We demonstrate that the present formalism is a powerful tool for investigating various kinds of systems and we show how it is applied to spin systems, including those with up to three spins. We distinguish the contributions of the initial correlation and the frequency shift and make clear the role of each contribution in the Ohmic coupling spectral function. As examples of applications to multispin systems, we obtain the dependence of the line shape on the spatial orientation in relation to the direction of the static field (Nagata-Tazuke effect), including the effects of the thermal environment, in a two-spin system, along with the dependence on the arrangement of a triangle in a three-spin system.