Off-resonant transitions in the collective dynamics of multilevel atomic ensembles

TL;DR

This paper investigates how off-resonant transitions affect the collective dynamics of multilevel atomic ensembles interacting with the electromagnetic field, highlighting limitations of the Rotating Wave Approximation and proposing a method to accurately derive atom-atom interactions.

Contribution

It explicitly derives the atom-atom interaction for multilevel atoms coupled to the full vector electromagnetic field, extending the validity of previous approximations.

Findings

Off-resonant transitions significantly influence atomic dynamics.

Extending photon mode integrals to negative frequencies recovers correct interactions.

The derived formulas are valid for multilevel atoms with vector field coupling.

Abstract

We study the contributions of off-resonant transitions to the dynamics of a system of N multilevel atoms sharing one excitation and interacting with the quantized vector electromagnetic field. The Rotating Wave Approximation significantly simplifies the derivation of the equations of motion describing the collective atomic dynamics, but it leads to an incorrect expression for the dispersive part of the atom-atom interaction terms. For the case of two-level atoms and a scalar electromagnetic field, it turns out that the atom-atom interaction can be recovered correctly if integrals over the photon mode frequencies are extended to incorporate negative values. We explicitly derive the atom-atom interaction for multi-level atoms, coupled to the full vector electromagnetic field, and we recover also in this general case the validity of the results obtained by the extension to negative frequencies of the formulas derived with the Rotating Wave Approximation.