# Matrix diagonalization and exact solution of the k-photon Jaynes-Cummings model

## TL;DR

This paper presents an exact algebraic solution to the two-photon and k-photon Jaynes-Cummings models, deriving their energy spectra and eigenfunctions using a novel method based on $SU(2)$ algebra and coherent states.

## Contribution

It introduces a new algebraic approach employing Pauli matrices and $SU(2)$ tilting transformation to diagonalize the Hamiltonian of multi-photon Jaynes-Cummings models.

## Key findings

- Explicit energy spectra derived for the models
- Eigenfunctions explicitly obtained
- Method applicable to similar quantum optical systems

## Abstract

We study and exactly solve the two-photon and k-photon Jaynes-Cummings models by using a novelty algebraic method. This algebraic method is based on the Pauli matrices realization and the tilting transformation of the $SU(2)$ group and let us diagonalize the Hamiltonian of these models by properly choosing the coherent state parameters of the transformation. Finally, we explicitly obtain the energy spectrum and eigenfunctions for each model.