Adiabatic approximation for a two-level atom in a light beam
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 22 (2013) no. 1, p. 43-131

Following the recent experimental realization of synthetic gauge potentials, Jean Dalibard addressed the question whether the adiabatic ansatz could be mathematically 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 information 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 momenta). 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.

Suite à la réalisation expérimentale de champs de jauge artificiels, Jean Dalibard a soulevé la question de l’approximation adiabatique pour un modèle d’atome à deux niveaux, éclairé par un faisceau laser résonnant. Dans cet article, nous dérivons rigoureusement le modèle asymptotique deviné par les physiciens et montrons que cette analyse contient l’information sur la présence de vortex. Les difficultés, et c’est une surprise, ne viennent pas du terme non linéaire. Plus précisément, l’analyse du problème non linéaire, et la réduction asymptotique à un modèle plus simple, reposent sur une séparation précise des grandes et basses fréquences (ou grands et bas moments). Cela nécessite de reconsidérer avec soin les résultats mathématiques existants sur la limite adiabatique. Bien que les estimations ne soient pas optimales, elles fournissent une bonne intuition sur la validité du modèle asymptotique, par rapport aux tailles des différents paramètres initialement mis dans le modèle.

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     author = {Aftalion, Amandine and Nier, Francis},
     title = {Adiabatic approximation for a two-level atom in a light beam},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 22},
     number = {1},
     year = {2013},
     pages = {43-131},
     doi = {10.5802/afst.1367},
     zbl = {06190674},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2013_6_22_1_43_0}
}
Aftalion, Amandine; Nier, Francis. Adiabatic approximation for a two-level atom in a light beam. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 22 (2013) no. 1, pp. 43-131. doi : 10.5802/afst.1367. http://www.numdam.org/item/AFST_2013_6_22_1_43_0/

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