$\hat p \cdot \hat A$ vs $\hat x \cdot \hat E$: Gauge invariance in quantum optics

TL;DR

This paper compares two fundamental models of light-matter interaction, revealing conditions under which the dipole approximation remains valid even without a dominant wavelength, and discusses gauge invariance issues.

Contribution

It demonstrates the validity of the dipole approximation over long interaction times in regimes lacking a dominant wavelength, and clarifies gauge invariance considerations in light-matter models.

Findings

Dipole approximation valid for interaction times longer than light-crossing time.

No a-priori justification for dipole approximation in vacuum excitation physics.

Differences in gauge invariance properties of the two models.

Abstract

We compare the predictions of the fundamentally motivated minimal coupling ($\hat{\boldsymbol{p}}\cdot\hat{\boldsymbol{A}}$) and the ubiquitous dipole coupling ($\hat{\boldsymbol{x}}\cdot\hat{\boldsymbol{E}}$) in the light-matter interaction. By studying the light-matter interaction for hydrogen-like atoms we find that the dipole approximation cannot be a-priori justified to analyze the physics of vacuum excitations (a very important phenomenon in relativistic quantum information) since a dominant wavelength is absent in those problems, no matter how small (as compared to any frequency scale) the atom is. Remarkably, we show that the dipole approximation in those regimes can still be valid as long as the interaction time is longer than the light-crossing time of the atoms, which is a very reasonable assumption. We also highlight some of the subtleties that one has to be careful with when working with the explicitly non-gauge invariant nature of the minimal coupling and we compare it with the explicitly gauge invariant dipole coupling.