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

Abstract

We compare the predictions of the fundamentally motivated minimal coupling ($\hat{\mathbf{p}}\cdot \hat{\mathbf{A}}$) and the ubiquitous dipole coupling ($\hat{\mathbf{x}}\cdot \hat{\mathbf{E}}$) 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.

arXiv comment19 pages: 11 pages + 8 pages of appendices. 7 figures. RevTeX 4.1

References in corpus (5)

• Multiphoton Quantum Optics and Quantum State Engineering
• Violating Bell's inequalities in the vacuum
• Harvesting correlations from the quantum vacuum
• Entanglement harvesting from the electromagnetic vacuum with hydrogenlike atoms
• The Fermi problem with artificial atoms in circuit QED

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