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

Boutet de Monvel, Anne; Zielinski, Lech

Comptes Rendus. Mathématique, Tome 361 (2023) no. G11, pp. 1761-1766

We prove that the spectrum of the two-photon quantum Rabi Hamiltonian consists of two eigenvalue sequences ${\left(E_m^{+}\right)}_{m=0}^{\infty }$, ${\left(E_m^{-}\right)}_{m=0}^{\infty }$ satisfying a three-term asymptotic formula with the remainder estimate $O(m^{-1}lnm)$ 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.

Nous démontrons que le spectre de l’hamiltonien du modèle quantique de Rabi à deux photons est constitué de deux suites de valeurs propres ${\left(E_m^{+}\right)}_{m=0}^{\infty }$, ${\left(E_m^{-}\right)}_{m=0}^{\infty }$ vérifiant une formule asymptotique à trois termes avec l’estimation de l’erreur $O(m^{-1}lnm)$ 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.

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