Asymptotic formula for large eigenvalues of the two-photon quantum Rabi model

Anne Boutet de Monvel; Lech Zielinski

doi:10.5802/crmath.515

Physique mathématique, Théorie spectrale

Asymptotic formula for large eigenvalues of the two-photon quantum Rabi model
[Formule asymptotique pour les grandes valeurs propres du modèle quantique de Rabi à deux photons]

Anne Boutet de Monvel ¹ ; Lech Zielinski ²

¹ Institut de Mathématiques de Jusieu, Université Paris Cité, 8 place Aurélie Nemours, 75013 Paris, France
² Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville UR 2597, Université du Littoral Côte d’Opale, 62228 Calais, France

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1761-1766.

Résumé
Abstract

Nous démontrons que le spectre de l’hamiltonien du modèle quantique de Rabi à deux photons est constitué de deux suites de valeurs propres ${(E_{m}^{+})}_{m = 0}^{\infty}$ , ${(E_{m}^{-})}_{m = 0}^{\infty}$ vérifiant une formule asymptotique à trois termes avec l’estimation de l’erreur $O (m^{- 1} ln m)$ quand $m$ tend vers l’infini. Par analogie avec le modèle quantique de Rabi à un photon, les trois termes dominants dans cette formule asymptotique décrivent l’approximation de l’onde tournante pour les grandes valeurs propres du modèle quantique de Rabi à deux photons.

We prove that the spectrum of the two-photon quantum Rabi Hamiltonian consists of two eigenvalue sequences ${(E_{m}^{+})}_{m = 0}^{\infty}$ , ${(E_{m}^{-})}_{m = 0}^{\infty}$ satisfying a three-term asymptotic formula with the remainder estimate $O (m^{- 1} ln m)$ when $m$ tends to infinity. By analogy to the one-photon quantum Rabi model, the leading three terms of this asymptotic formula, describe a generalized rotating-wave approximation for large eigenvalues of the two-photon quantum Rabi model.

Reçu le : 2023-05-11
Accepté le : 2023-05-16
Publié le : 2023-12-21

DOI : 10.5802/crmath.515

Affiliations des auteurs :

Anne Boutet de Monvel ¹ ; Lech Zielinski ²

¹ Institut de Mathématiques de Jusieu, Université Paris Cité, 8 place Aurélie Nemours, 75013 Paris, France
² Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville UR 2597, Université du Littoral Côte d’Opale, 62228 Calais, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Anne Boutet de Monvel and Lech Zielinski},
     title = {Asymptotic formula for large eigenvalues of the two-photon quantum {Rabi} model},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1761--1766},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.515},
     language = {en},
}

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Anne Boutet de Monvel; Lech Zielinski. Asymptotic formula for large eigenvalues of the two-photon quantum Rabi model. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1761-1766. doi : 10.5802/crmath.515. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.515/

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