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arXiv:quant-ph/0201126 · Physical Review A

Algebraic approach to the Tavis-Cummings problem

Ilya P Vadeiko, George P. Miroshnichenko, Andrei V. Rybin, JUSSI TIMONEN

Published May 28, 2003• 34 citations
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Abstract

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., ${\mathrm{su}}_{n}(2)$ 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 $\mathcal{N}$ atoms in a resonator is found to result from collective effects.

DOI
10.1103/physreva.67.053808

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A Comment on "Algebraic approach to the Tavis-Cummings model with three modes of oscillation" [J. Math. Phys. 59, 073506 (2018)]

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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 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
Abstract This chapter extends the algebraic Bethe ansatz to the quantum Tavis–Cummings model, an N atom generalization of the Jaynes–Cummings model to describe the strong interaction between light and quantum matter. In the case of the quantum Tavis–Cum- mings model there is no underlying vertex model to suggest the constituent building blocks of the algebraic Bethe ansatz approach, e.g.like the L-matrix or ultimately the transfer matrix. The algebraic Bethe ansatz is then first applied to the Tavis–Cummings Hamiltonian with an added Stark term using a conjecture for the transfer matrix. The original Tavis–Cummings model and its algebraic Bethe ansatz are obtained in the limit of vanishing Stark term, which requires considerable care.
Quantum inverse methods are developed for the exact solution of models which describe N two-level atoms interacting with one mode of the quantized electromagnetic field containing an arbitrary number of excitations M. Either a Kerr-type nonlinearity or a Stark-shift term can be included in the model, and it is shown that these two cases can be mapped from one to the other. The method of solution provides a general framework within which many related problems can similarly be solved. Explicit formulae are given for the Rabi splitting of the models for some N and M, on- and off-resonance. It is also shown that the solution of the pure Tavis - Cummings model can be reduced to solving a homogeneous ordinary differential equation of second order. Generalization of the method to the case of several cavity modes is indicated.
Applying a perturbation method, constructed in terms of SU(2) group representations, we study spontaneous emission of a system of N identical two-level atoms immersed in a single-mode ideal cavity with s atoms initially inverted. For the cases s\ensuremath{\ge}3, characterized by nonequidistant spectra of the eigenvalues of the Hamiltonian, the phenomenon of collective collapses and revivals manifests itself. For s=3 the approximate solution is compared with the exact one and the limit of good applicability of our method is established.