Evolution of energy distribution in a model system without conventional Lorentzian lifetime broadening

Abstract

Transition probabilities are generally assumed to be broadened owing to the finite width of the final states. That is, the energy-conserving \ensuremath{\delta} function of the golden rule is assumed to be spread into a Lorentzian whose width is of the order of the final-state's inverse lifetime, what I call conventional Lorentzian lifetime broadening (CLLB). But this paper concludes that CLLB greatly overestimates the broadening, even when the spectral density is Lorentzian. Numerical solution of the time-dependent Schr\"odinger equation for a model system shows that state occupations in a single band evolve with transition rates, which, for long times, reduce to the golden rule; and when the golden rule is inapplicable, as for transitions across a gap larger than the phonon energy, resonant multiphonon processes, rather than CLLB, give the proper description. The results are confirmed by fourth-order perturbation theory and in accord with self-consistent Green-function theory.