# Generalized James' effective Hamiltonian method

## TL;DR

This paper extends James' effective Hamiltonian method to higher orders, enabling more accurate analysis of detuned quantum systems beyond second-order perturbation, simplifying calculations and broadening applicability.

## Contribution

The paper generalizes James' effective Hamiltonian method to higher-order perturbation theory, allowing for more precise and versatile analysis of quantum systems.

## Key findings

- Results match third-order perturbation theory and adiabatic elimination.
- Method simplifies calculations for specific quantum problems.
- Effective Hamiltonian is more general and easier to compute.

## Abstract

James' effective Hamiltonian method has been extensively adopted to investigate largely detuned interacting quantum systems. This method is just corresponding to the second-order perturbation theory, and cannot be exploited to treat the problems which should be solved by using the third or higher-order perturbation theory. In this paper, we generalize James' effective Hamiltonian method to the higher-order case. Using the method developed here, we reexamine two examples published recently [Phys. Rev.Lett. 117, 043601 (2016), Phys. Rev A 92, 023842 (2015)], our results turn out to be the same as the original ones derived from the third-order perturbation theory and adiabatic elimination method respectively. For some specific problems, this method can simplify the calculating procedure, and the resultant effective Hamiltonian is more general.