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

Amandine Aftalion[1]; Francis Nier[2]

  • [1] CNRS & Université Versailles-Saint-Quentin-en-Yvelines, Laboratoire de Mathématiques de Versailles, CNRS UMR 8100, 45 avenue des États-Unis, 78035 Versailles Cedex, France
  • [2] IRMAR, Université de Rennes 1, 35042 Rennes Cedex, France. 2) CERMICS, INRIA project-team MICMAC

Annales de la faculté des sciences de Toulouse Mathématiques (2013)

  • Volume: 22, Issue: 1, page 43-131
  • ISSN: 0240-2963

Abstract

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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.

How to cite

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Aftalion, Amandine, and Nier, Francis. "Adiabatic approximation for a two-level atom in a light beam." Annales de la faculté des sciences de Toulouse Mathématiques 22.1 (2013): 43-131. <http://eudml.org/doc/275295>.

@article{Aftalion2013,
abstract = {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.},
affiliation = {CNRS & Université Versailles-Saint-Quentin-en-Yvelines, Laboratoire de Mathématiques de Versailles, CNRS UMR 8100, 45 avenue des États-Unis, 78035 Versailles Cedex, France; IRMAR, Université de Rennes 1, 35042 Rennes Cedex, France. 2) CERMICS, INRIA project-team MICMAC},
author = {Aftalion, Amandine, Nier, Francis},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {resonant atom-field interactions; vortices; synthetic gauge potentials},
language = {eng},
month = {6},
number = {1},
pages = {43-131},
publisher = {Université Paul Sabatier, Toulouse},
title = {Adiabatic approximation for a two-level atom in a light beam},
url = {http://eudml.org/doc/275295},
volume = {22},
year = {2013},
}

TY - JOUR
AU - Aftalion, Amandine
AU - Nier, Francis
TI - Adiabatic approximation for a two-level atom in a light beam
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2013/6//
PB - Université Paul Sabatier, Toulouse
VL - 22
IS - 1
SP - 43
EP - 131
AB - 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.
LA - eng
KW - resonant atom-field interactions; vortices; synthetic gauge potentials
UR - http://eudml.org/doc/275295
ER -

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