High frequency behaviour of the Maxwell-Bloch model with relaxations: convergence to the Schrödinger-Boltzmann system

Abstract

We study the Maxwell-Bloch model, which describes the propagation of a laser through a material and the associated interaction between laser and matter (polarization of the atoms through light propagation, photon emission and absorption, etc.). The laser field is described through Maxwell's equations, a classical equation, while matter is represented at a quantum level and satisfies a quantum Liouville equation known as the Bloch model. Coupling between laser and matter is described through a quadratic source term in both equations. The model also takes into account partial relaxation effects, namely the trend of matter to return to its natural thermodynamic equilibrium. The whole system involves $6$+$N \, (N+1)/2$ unknowns, the six-dimensional electromagnetic field plus the $N \, (N+1)/2$ unknowns describing the state of matter, where $N$ is the number of atomic energy levels of the considered material. We consider at once a high-frequency and weak coupling situation, in the general case of anisotropic electromagnetic fields that are subject to diffraction. Degenerate energy levels are allowed. The whole system is stiff and involves strong nonlinearities. We show the convergence to a nonstiff, nonlinear, coupled Schrö\-din\-ger-Boltzmann model, involving $3$+$N$ unknowns. The electromagnetic field is eventually described through its envelope, one unknown vector in $\C^3$. It satisfies a Schrö\-din\-ger equation that takes into account propagation and diffraction of light inside the material. Matter on the other hand is described through a $N$-dimensional vector describing the occupation numbers of each atomic level. It satisfies a Boltzmann equation that describes the jumps of the electrons between the various atomic energy levels, as induced by the interaction with light. The rate of exchange between the atomic levels is proportional to the intensity of the laser field. The whole system is the physically natural nonlinear model. In order to provide an important and explicit example, we completely analyze the specific (two dimensional) Transverse Magnetic case, for which formulae turn out to be simpler. Technically speaking, our analysis does not enter the usual mathematical framework of geometric optics: it is more singular, and requires an \emph{ad hoc} Ansatz.