FROM 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 atoms and a driving electric force. It is commonly used in optics to describe the interaction of a laser beam with a sample of atoms. In this paper, we address the asymptotics of these equations for a high frequency electric field, in a weak coupling regime. The electric forcing is taken quasiperiodic in time. We prove the convergence towards a rate equation, i.e. a linear Boltzmann equation, recovering in this way the physically relevant asymptotic model. It describes the transitions amongst the various energy levels of the atoms, governed by the resonances between the electric forcing and the atoms' eigenfrequencies. We also give the explicit value for the transition rates. The present task has already been addressed in Ref. 5 in the case when the energy levels are fixed, and for different classes of electric fields. Here, we extend the study in two directions. First, we consider almost degenerate energy levels, a natural situation in practice. In this case, almost resonances might occur. Technically, this implies that the small divisor estimates needed in Ref. 5 are false, due to the fact that the Diophantine condition is unstable with respect to small perturbations. We use an appropriate ultraviolet cutoff to restore the analysis and to sort out the asymptotically relevant frequencies. Second, since the asymptotic rate equation may be singular in time, we completely analyze the initial time-layer, as well as the associated convergence towards an equilibrium state.