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Physical Review A

Laser-induced rate processes in gases: Intermolecular and intramolecular relaxation in a polyatomic molecule

James P. Stone, Myron F. Goodman

Published Dec 1, 1978• 25 citations
Publisher

Abstract

We present a formalism to analyze the effects of collisions and intramolecular energy transfer and dephasing processes (${T}_{1}$ and ${T}_{2}$ relaxation) on level-population dynamics and unimolecular decay of laser-driven gas molecules. Utilizing a Boltzmann or optical Bloch equation for the density matrix to describe laser excitation of a relaxing multilevel absorber, we develop a generalized master equation (GME) to describe the dynamics of level population evolution for multilevel absorbers having arbitrary numbers of quantum levels $N$. Because the resulting level-population equations explicitly retain phase memory, the effect of complete or partial phase-coherent driving or incoherent driving by the laser field are all included within the GME solutions. The formalism provides straightforward means to calculate time-dependent solutions for arbitrary $N$, where a single expression is valid on transient, post transient, approach to steady state, and steady-state time scales. We illustrate the behavior of these solutions for the special case of three- and eight-level systems, but the method is not restricted with regard to number of levels. In both this and the following paper, we use specific examples and general arguments to show that an ordinary master equation (OME) is invalid even under apparently favorable conditions including saturating field strengths. We demonstrate that, for an $N$-level absorber with given coupling elements, there are $\frac{N}{2}$ [or $\frac{(N+1)}{2}$] independent saturation solutions and that special conditions on the relaxation times are required to produce the ordinary equal-population solution. Explicit forms for the steady-state radiative-transition matrix $W(\ensuremath{\lambda}=0)$ are derived for arbitrary $N$, and general arguments are given to show that the correct description of the dynamics given by the GME results from matrix singularities in $W(\ensuremath{\lambda}=0)$ which can never occur in an OME or similar rate equations.

DOI
10.1103/physreva.18.2618

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We present a formalism to analyze the effects of collisions and intramolecular energy transfer and dephasing processes (T1{T}_{1}T1​ and T2{T}_{2}T2​ relaxation) on level-population dynamics and unimolecular decay of laser-driven gas molecules. Utilizing a Boltzmann or optical Bloch equation for the density matrix to describe laser excitation of a relaxing multilevel absorber, we develop a generalized master equation (GME) to describe the dynamics of level population evolution for multilevel absorbers having arbitrary numbers of quantum levels NNN. Because the resulting level-population equations explicitly retain phase memory, the effect of complete or partial phase-coherent driving or incoherent driving by the laser field are all included within the GME solutions. The formalism provides straightforward means to calculate time-dependent solutions for arbitrary NNN, where a single expression is valid on transient, post transient, approach to steady state, and steady-state time scales. We illustrate the behavior of these solutions for the special case of three- and eight-level systems, but the method is not restricted with regard to number of levels. In both this and the following paper, we use specific examples and general arguments to show that an ordinary master equation (OME) is invalid even under apparently favorable conditions including saturating field strengths. We demonstrate that, for an NNN-level absorber with given coupling elements, there are N2\frac{N}{2}2N​ [or (N+1)2\frac{(N+1)}{2}2(N+1)​] independent saturation solutions and that special conditions on the relaxation times are required to produce the ordinary equal-population solution. Explicit forms for the steady-state radiative-transition matrix W(\ensuremathλ=0)W(\ensuremath{\lambda}=0)W(\ensuremathλ=0) are derived for arbitrary NNN, and general arguments are given to show that the correct description of the dynamics given by the GME results from matrix singularities in W(\ensuremathλ=0)W(\ensuremath{\lambda}=0)W(\ensuremathλ=0) which can never occur in an OME or similar rate equations.
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We have recast a linear quantum-mechanical Boltzmann equation, describing the dynamics of a gas of NNN-level atoms absorbing light between an arbitrary pair of levels, into the form of a generalized master equation. The atoms are permitted to undergo both phase-changing and energy-changing collisions with a heat bath of inert molecules. The possibility of unimolecular decay is incorporated into the Boltzmann equation. The solutions of the generalized master equation give an accurate description of the system for short as well as long times and for arbitrary light-field intensity. Eigenvalues obtained from the solution of the NNN-level master equation are shown to have particular significance with regard to the scattered electromagnetic field, with the imaginary parts of the complex eigenvalues providing the location and the real parts of the eigenvalues giving the widths of scattered radiation bands. The dynamics and scattered field for a two-level system is discussed in detail. Two collision parameters giving the frequency of phase-changing and energy-changing collisions are important. For special values of these parameters, our results reduce to scattered field spectra calculated previously. When unimolecular decay is included, one identifies the smallest eigenvalue of the generalized master equation with the macroscopic rate constant. Observation of the scattered field for different incident field intensities would permit separation of radiative and thermal effects in an enhanced reaction rate, without requiring an accurate temperature measurement.
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Excitation dynamics for pulsed optical excitation are described with the density-matrix equations and the rate equations for a two-level system. A critical comparison of the two descriptions is made with complete and consistent formalisms that are amenable to the modeling of applied laser-diagnostic techniques. General solutions, resulting from numerical integration of the differential equations describing the excitation process, are compared for collisional conditions that range from the completely coherent limit to the steady-state limit, for which the two formalisms are identical. This analysis demonstrates the failure of the rate equations to correctly describe the transient details of the excitation process outside the steady-state limit. However, reasonable estimates of the resultant population are obtained for nonsaturating (linear) excitation. This comparison provides the laser diagnostician with the means to evaluate the appropriate model for excitation through a simple picture of the breakdown of the rate-equation validity.