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

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 $\frac{\mathrm{}(N+1\mathrm{})\mathrm{}}{2}$] 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 $W(\lambda =0)$ 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 $W(\lambda =0)$ which can never occur in an OME or similar rate equations.