Adiabatic approximation for a two-level atom in a light beam

TL;DR

This paper rigorously justifies the adiabatic approximation for a two-level atom in a laser beam, revealing insights into vortex formation and addressing mathematical challenges in the adiabatic limit.

Contribution

It provides a rigorous derivation of the asymptotic model for a two-level atom in a laser beam, clarifying the role of linear Hamiltonian analysis and frequency partitioning.

Findings

Asymptotic model confirms vortex presence.

Linear Hamiltonian analysis is the main difficulty.

Provides insights into adiabatic approximation validity.

Abstract

Several misprints and small mistakes were in the initial version. They have been corrected. Following the recent experimental realization of synthetic gauge magnetic forces, Jean Dalibard adressed the question whether the adiabatic ansatz could be math- ematically justified for a model of an atom in 2 internal states, shone by a quasi resonant laser beam. In this paper, we derive rigorously the asymptotic model guessed by the physicists, and show that this asymptotic analysis contains the in- formation about the presence of vortices. Surprisingly the main difficulties do not come from the nonlinear part but from the linear Hamiltonian. More precisely, the analysis of the nonlinear minimization problem and its asymptotic reduction to simpler ones, relies on an accurate partition of low and high frequencies (or mo- menta). This requires to reconsider carefully previous mathematical works about the adiabatic limit. Although the estimates are not sharp, this asymptotic analysis provides a good insight about the validity of the asymptotic picture, with respect to the size of the many parameters initially put in the complete model.