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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2023__361_G11_1761_0,
     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},
}

TY  - JOUR
AU  - Anne Boutet de Monvel
AU  - Lech Zielinski
TI  - Asymptotic formula for large eigenvalues of the two-photon quantum Rabi model
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 1761
EP  - 1766
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.515
LA  - en
ID  - CRMATH_2023__361_G11_1761_0
ER  -

%0 Journal Article
%A Anne Boutet de Monvel
%A Lech Zielinski
%T Asymptotic formula for large eigenvalues of the two-photon quantum Rabi model
%J Comptes Rendus. Mathématique
%D 2023
%P 1761-1766
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.515
%G en
%F CRMATH_2023__361_G11_1761_0

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/

Bibliographie
Cité par

[1] Julian Edward Spectra of Jacobi matrices, differential equations on the circle, and the $s u (1, 1)$ Lie algebra, SIAM J. Math. Anal., Volume 24 (1993) no. 3, pp. 824-831 | DOI | MR | Zbl

[2] Clive Emary; Raymond F. Bishop Bogoliubov transformations and exact isolated solutions for simple nonadiabatic Hamiltonians, J. Math. Phys., Volume 43 (2002) no. 8, pp. 3916-3926 | DOI | MR | Zbl

[3] Ilya D. Feranchuk; Alexey Ivanov; Van-Hoang Le; Alexander P. Ulyanenkov Non-perturbative description of quantum systems, Lecture Notes in Physics, 894, Springer, 2015 | DOI

[4] Ilya D. Feranchuk; L. I. Komarov; Alexander P. Ulyanenkov Two-level systems in a one-mode quantum field: Numerical solution on the basis of the operator method, J. Phys. A. Math. Gen., Volume 29 (1996) no. 14, pp. 4035-4047 | DOI | Zbl

[5] Christopher Gerry Two-photon Jaynes–Cummings model interacting with the squeezed vacuum, Phys. Rev. A, Volume 37 (1988) no. 7, pp. 2683-2686 | DOI

[6] Elinor K. Irish Generalized Rotating-Wave Approximation for Arbitrarily Large Coupling, Phys. Rev. Lett., Volume 99 (2007), 173601 | DOI

[7] Anne Boutet de Monvel; Serguei Naboko; Luis O. Silva Eigenvalue asymptotics of a modified Jaynes–Cummings model with periodic modulations, Comptes Rendus Math. Acad. Sci. Paris, Volume 338 (2004) no. 1, pp. 103-107 | DOI | Numdam | MR | Zbl

[8] Anne Boutet de Monvel; Serguei Naboko; Luis O. Silva The asymptotic behavior of eigenvalues of a modified Jaynes–Cummings model, Asymptotic Anal., Volume 47 (2006) no. 3-4, pp. 291-315 | MR | Zbl

[9] Anne Boutet de Monvel; Lech Zielinski Oscillatory behavior of large eigenvalues in quantum Rabi models, Int. Math. Res. Not., Volume 2021 (2019) no. 7, pp. 5155-5213 | DOI | Zbl

[10] Anne Boutet de Monvel; Lech Zielinski Asymptotic behavior of large eigenvalues of the two-photon Rabi model, Schrödinger operators, spectral analysis and number theory. In memory of Erik Balslev (Springer Proceedings in Mathematics & Statistics), Volume 348, Springer, 2021, pp. 89-115 | DOI | MR | Zbl

[11] Grigori V. Rozenblum Near-similarity of operators and the spectral asymptotic behavior of pseudodifferential operators on the circle, Tr. Mosk. Mat. O.-va, Volume 36 (1975), pp. 59-84

[12] Jaouad Sahbani Spectral theory of certain unbounded Jacobi matrices, J. Math. Anal. Appl., Volume 342 (2008) no. 1, pp. 663-681 | DOI | MR | Zbl

[13] Eduard A. Tur Jaynes–Cummings model without rotating wave approximation (2002) (https://arxiv.org/abs/math-ph/0211055)

[14] Eduard A. (Tur) Yanovich Asymptotics of eigenvalues of an energy operator in a problem of quantum physics, Operator methods in mathematical physics. Conference on operator theory, analysis and mathematical physics (OTAMP) 2010, Bedlewo, Poland (Operator Theory: Advances and Applications), Volume 227, Birkhäuser, 2013, pp. 165-177 | DOI | MR | Zbl

[15] Qiongtao Xie; Honghua Zhong; Murray T. Batchelor; Chaohong Lee The quantum Rabi model: solution and dynamics, J. Phys. A. Math. Theor., Volume 50 (2017) no. 11, 113001, 40 pages | DOI | MR | Zbl

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Many-body quantum electrodynamics networks: Non-equilibrium condensed matter physics with light

Karyn Le Hur; Loïc Henriet; Alexandru Petrescu; ...

C. R. Phys (2016)

Eigenvalue asymptotics of a modified Jaynes–Cummings model with periodic modulations

Anne Boutet de Monvel; Serguei Naboko; Luis O. Silva

C. R. Math (2004)

Electronically induced nuclear transitions – temperature dependence and Rabi oscillations

Jean-Jacques Niez

C. R. Phys (2002)