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

TL;DR

This paper develops algebraic methods to solve and analyze the Tavis-Cummings model with three oscillation modes, deriving energy spectra, eigenfunctions, and expectation values using group theory transformations.

Contribution

It introduces algebraic techniques, including Bogoliubov and tilting transformations, to solve the general three-mode Tavis-Cummings model, extending previous approaches.

Findings

Derived energy spectrum and eigenfunctions for specific cases.

Solved the general Hamiltonian using tilting transformation.

Computed expectation values via $SU(1,1)$ and $SU(2)$ group theory.

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 let 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.