Laser excitation of multilevel systems: Limitations on the use of Wilcox-Lamb and other rate-equation approximations

Using the generalized master equation (GME) formalism, we analyze several approximations for the solutions of Boltzmann or optical Bloch equations. Included in our analysis are the Wilcox-Lamb equations, ordinary master equation (OME), and a new approximation which we designate as the renormalized small-eigenvalue (RSE) approximation. We demonstrate that in multilevel systems, the GME formalism allows direct systematic calculation of radiative-transition coefficients required in the Wilcox-Lamb equations, including all multilevel phase-coherent transitions. However, serious limitations in the applicability of the Wilcox-Lamb approximation are noted both in the high-pressure domain, as an asymptotically exact equation, and for low pressures, as a coarse-grained approximation. The Wilcox-Lamb equations, generally assumed to be valid in a high-pressure domain, are shown to fail in the presence of very efficient inelastic ($T_1$) relaxation processes regardless of the pressure. Under similar conditions the presence of efficient inelastic processes is also shown to invalidate the use of an OME even when inelastic cross sections are somewhat smaller and the Wilcox-Lamb equations are valid. In particular, we show that when energy losses to the heat bath are severe, an ordinary master equation, traditionally believed to provide a valid description of radiative contributions to level-population evolution when the ratio of Rabi to elastic collision frequency is small, is not valid under any conditions. On this ground we strongly question the recent use of a multilevel OME description to explain the existence of an energy threshold for laser-induced decomposition of S${\mathrm{F}}_6$. Finally, we show that for low gas pressures where Rabi oscillations are a prominent feature of the exact solutions, neither the OME nor the Wilcox-Lamb approximation provides an adequate coarse-grained solution. We show that the Wilcox-Lamb solution is in error primarily because of a circumstance not previously recognized, namely, a systematic large error in the way exponential terms contributing to the solution are normalized. A closely related approximation, the RSE approximation, which we present here for the first time, is shown to provide an accurate coarse-grained approximation to the fully coherent GME solutions.