Eigenstates of two-level systems in a single-mode quantum field: from quantum Rabi model to $N$-atom Dicke model

TL;DR

This paper presents an analytical approximation method for diagonalizing the Hamiltonian of N two-level systems interacting with a single-mode quantum field, applicable to both the quantum Rabi and Dicke models, enhancing understanding of their eigenstates.

Contribution

It introduces a simple basis set for accurate analytical approximation of eigenstates and eigenvalues in the quantum Rabi and Dicke models, improving upon numerical solutions.

Findings

Accurate analytical eigenstates and eigenvalues across a broad coupling range.

A regular basis for calculating corrections and convergence.

Applicability to both single-atom and multi-atom models.

Abstract

In the present paper we show that the Hamiltonian describing the resonant interaction of $N$ two-level systems with a single-mode electromagnetic quantum field in the Coulomb gauge can be diagonalized with a high degree of accuracy using a simple basis set of states. This allows one to find an analytical approximation for the eigenvectors and eigenvalues of the system, which interpolates the numerical solution in a broad range of the coupling constant values. In addition, the introduced basis states provide a regular way of calculating the corrections and estimating the convergence to the exact numerical solution. The obtained results are valid for both quantum Rabi model ($N = 1$) and the Dicke model for $N \geq 2$ atoms.