# Comparison of the Iterated Equation of Motion Approach and the Density Matrix Formalism for the Quantum Rabi Model

## TL;DR

This paper compares the density matrix formalism and the equation of motion approach in modeling the quantum Rabi system, analyzing their accuracy, advantages, and limitations in capturing non-equilibrium dynamics.

## Contribution

It provides a detailed comparison of two semi-analytical methods applied to the quantum Rabi model, highlighting their respective accuracies and physical consistency.

## Key findings

- Both methods yield exact results for bilinear Hamiltonians.
- Differences in truncation schemes affect the accuracy for non-bilinear Hamiltonians.
- The study identifies conditions under which each method produces physically consistent results.

## Abstract

The density matrix formalism and the equation of motion approach are two semi-analytical methods that can be used to compute the non-equilibrium dynamics of correlated systems. While for a bilinear Hamiltonian both formalisms yield the exact result, for any non-bilinear Hamiltonian a truncation is necessary. Due to the fact that the commonly used truncation schemes differ for these two methods, the accuracy of the obtained results depends significantly on the chosen approach. In this paper, both formalisms are applied to the quantum Rabi model. This allows us to compare the approximate results and the exact dynamics of the system and enables us to discuss the accuracy of the approximations as well as the advantages and the disadvantages of both methods. It is shown to which extent the results fulfill physical requirements for the observables and which properties of the methods lead to unphysical results.