From the Bloch model to the rate equations II: the case of almost degenerate energy levels

Abstract

Bloch equations give a quantum description of the coupling between an atom and a driving electric force. In this article, we address the asymptotics of these equations for high frequency electric fields, in a weakly coupled regime. We prove the convergence towards rate equations (i.e. linear Boltzmann equations, describing the transitions between energy levels of the atom). We give an explicit form for the transition rates. This has already been performed in [BFCD03] in the case when the energy levels are fixed, and for different classes of electric fields, quasi or almost periodic, KBM, or with continuous spectrum. Here, we extend the study to the case when energy levels are possibly almost degenerate. However, we need to restrict to quasiperiodic forcings. The techniques used stem from manipulations on the density matrix and the averaging theory for ordinary differential equations. Possibly perturbed small divisor estimates play a key rôle in the analysis. In the case of a finite number of energy levels, we also precisely analyze the initial time-layer in the rate aquation, as well as the long-time convergence towards equilibrium. We give hints and counterexamples in the infinite dimensional case.