HAL : hal-00641565, version 2

arXiv : 1111.3811

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v1 (16-11-2011)

v2 (23-01-2012)

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

Amandine Aftalion 1, Francis Nier 2, 3

(16/11/2011)

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.

1 :	Laboratoire de Mathématiques de Versailles (LM-Versailles)
	CNRS : UMR8100 – Université de Versailles Saint-Quentin-en-Yvelines
2 :	Institut de Recherche Mathématique de Rennes (IRMAR)
	CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – INSA Rennes – Université Rennes II
3 :	MICMAC (INRIA Paris - Rocquencourt)
	Ecole des Ponts ParisTech – INRIA

Équipe de recherche : Equations aux dérivées partielles

Domaine

:

Mathématiques/Equations aux dérivées partielles

Mots Clés : adiabatic approximation – ground state – non linear problem – rotating Bose Einstein condensate

hal-00641565, version 2

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Contributeur : Francis Nier <>

Soumis le : Lundi 23 Janvier 2012, 11:52:20

Dernière modification le : Lundi 23 Janvier 2012, 13:14:40