An analysis is presented of the instantaneous emission rate from an atomic level that is populated by arbitrary conditions of both collisional excitation and cascade repopulation. The general solution to the rate equations is obtained in terms of transition probabilities, excitation cross sections, and a simple integral operator, and applications are given for the specific examples of impulsive, stepwise, rectangular, gaussian, and modulated excitation conditions.
A semiclassical discussion is given of a simple case of the Hanle effect when the population of the level of interest varies with time because of decays to it from higher levels excited by collisions. It is found that the width of the intensity-vs-applied-field curve is no longer simply related to the lifetime of this single level, but depends on the relative populations and lifetimes of all other participating levels. It is shown by examples that, unless these effects are taken into account, the lifetimes deduced from the measurements can be greatly in error.
We present a formalism to analyze the effects of collisions and intramolecular energy transfer and dephasing processes (
T1 and
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
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
2N [or
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 $W(\ensuremath{\lambda}=0)$ 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.
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.
Explicit calculations demonstrate that cascade repopulation effects are a possible explanation of the ${t}^{\ensuremath{-}1.5}$ dependence in the decay of Ly-$\ensuremath{\alpha}$ radiation at long times after excitation observed recently by Braithwaite, Matthews, and Moore. Good agreement with all available experimental data is obtained with reasonable population models; excitation proportional to ${n}^{\ensuremath{-}m}$ with $m=2.2\ensuremath{-}4$ is indicated.
The formal structure of the time dependence of the state of an atom in a perturber bath in an intense classical monochromatic radiation field is studied. The authors allow the cross sections of binary collisions to be affected by the radiation field. The result is used to derive an expression for the collision-broadened saturated fluorescence excitation spectrum and absorption rate in terms of a collisional relaxation operator. This operator can be related to the rate of collisionally aided radiative excitation.