Physical background for parameters of the quantum Rabi model

TL;DR

This paper analyzes the validity of common approximations in the quantum Rabi model, such as single-mode and rotating wave approximations, by deriving conditions and relations from a multi-mode Hamiltonian.

Contribution

It provides a systematic method to determine when the single-mode approximation is valid and offers approximate relations for eigenstates and dynamics beyond the rotating wave approximation.

Findings

Derived conditions for single-mode approximation validity

Identified simple relations for eigenstates and eigenfunctions

Extended analysis beyond the rotating wave approximation

Abstract

We investigate the applicability of the two major approximations which are most commonly employed in the study of the quantum Rabi model, namely the description of a resonant cavity mode as a single-mode quantized field and the use of the rotating wave approximation. Starting from the Hamiltonian of a two-level system interacting with a multi-mode quantized field, we perform the canonical transformation of the field operators. This allows one to partition the Hamiltonian of the system into two parts. The first part is the interaction of the two-level system with a single collective field mode, while the second one describes the interaction with field fluctuations. The first part is usually associated with the resonant cavity mode. This division enables us to determine the applicability condition of the single-mode approximation. In addition we identify simple approximate relations for the description of the eigenstates, eigenfunctions and the time evolution of the quantum Rabi model beyond the rotating wave approximation.