# Berry phase of the Tavis-Cummings model with three modes of oscillation

## TL;DR

This paper presents a new method to calculate the Berry phase for time-dependent Hamiltonians with $SU(1,1)$ and $SU(2)$ structures, applied to a three-mode Tavis-Cummings model, revealing geometric phase properties.

## Contribution

It introduces a general approach using similarity transformations of displacement operators to diagonalize Hamiltonians with $SU(1,1)$ and $SU(2)$ symmetry, and applies it to a trilinear Tavis-Cummings model.

## Key findings

- Derived explicit formulas for Berry phases in the model.
- Demonstrated the method's effectiveness on a three-mode system.
- Provided insights into geometric phases in quantum optics.

## Abstract

In this paper we develop a general method to obtain the Berry phase of time-dependent Hamiltonians with a linear structure given in terms of the $SU(1,1)$ and $SU(2)$ groups. This method is based on the similarity transformations of the displacement operator performed to the generators of each group, and let us diagonalize these Hamiltonians. Then, we introduce a trilinear form of the Tavis-Cummings model to compute the $SU(1,1)$ and $SU(2)$ Berry phases of this model.