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arXiv:1710.10945 · Journal of Mathematical Physics

Algebraic approach to the Tavis-Cummings model with three modes of oscillation

E. Choreño, D. Ojeda-Guillén, V. D. Granados

Published Jul 1, 2018• 6 citations
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Abstract

We study the Tavis-Cummings model with three modes of oscillation by using four different algebraic methods: the Bogoliubov transformation, the normal-mode operators, and the tilting transformation of the SU(1, 1) and SU(2) groups. The algebraic method based on the Bogoliubov transformation and the normal-mode operators lets us obtain the energy spectrum and eigenfunctions of a particular case of the Tavis-Cummings model, while with the tilting transformation we are able to solve the most general case of this Hamiltonian. Finally, we compute some expectation values of this problem by means of the SU(1, 1) and SU(2) group theory. The authors of the article agree to the retraction of the article effective 21 September 2023

DOI
10.1063/1.5012910

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We study the Tavis-Cummings model with three modes of oscillation by using four different algebraic methods: the Bogoliubov transformation, the normal-mode operators, and the tilting transformation of the SU(1, 1) and SU(2) groups. The algebraic method based on the Bogoliubov transformation and the normal-mode operators lets us obtain the energy spectrum and eigenfunctions of a particular case of the Tavis-Cummings model, while with the tilting transformation we are able to solve the most general case of this Hamiltonian. Finally, we compute some expectation values of this problem by means of the SU(1, 1) and SU(2) group theory. The authors of the article agree to the retraction of the article effective 21 September 2023
An algebraic method is introduced for an analytical solution of the eigenvalue problem of the Tavis-Cummings Hamiltonian, based on polynomially deformed su(2), i.e., sun(2){\mathrm{su}}_{n}(2)sun​(2) algebras. In this method the eigenvalue problem is solved in terms of a specific perturbation theory, developed here up to third order. Generalization to the N-atom case of the Rabi frequency and dressed states is also provided. A remarkable enhancement of spontaneous emission of N\mathcal{N}N atoms in a resonator is found to result from collective effects.
We present the exact solution of the Richardson–Gaudin models associated with the SU (3) Lie algebra. The derivation is based on a Gaudin algebra valid for any simple Lie algebra in the rational, trigonometric and hyperbolic cases. For the rational case additional cubic integrals of motion are obtained, whose number is added to that of the quadratic ones to match, as required from the integrability condition, the number of quantum degrees of freedom of the model. We discuss different SU (3) physical representations and elucidate the meaning of the parameters entering in the formalism. By considering a bosonic mapping limit of one of the SU (3) copies, we derive new integrable models for three level systems interacting with two bosons. These models include a generalized Tavis–Cummings model for three level atoms interacting with two modes of the quantized electric field.
Abstract Using the quantum group technique we construct a one-parametric family of integrable modifications of the n -level, n − 1 mode Tavis–Cummings Hamiltonian possessing an additional Stark-type term. We show that in the ‘quasiclassical’ limit the constructed Hamiltonian transforms into the integrable Hamiltonian of the quantum n -level, n − 1 mode Tavis–Cummings model with the equal interaction strengths considered in Skrypnyk (2008 J. Phys. A: Math. Theor . 41 475202, 2009 J. Math. Phys . 50 103523). We diagonalize the constructed ‘modified’ Tavis–Cummings Hamiltonian and its second order integrals of motion using the nested Bethe ansatz.