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We compare the predictions of the fundamentally motivated minimal coupling($\hat{\boldsymbol{p}}\cdot\hat{\boldsymbol{A}}$) and the ubiquitous dipolecoupling ($\hat{\boldsymbol{x}}\cdot\hat{\boldsymbol{E}}$) in the light-matterinteraction. By studying the light-matter interaction for hydrogen-like atomswe find that the dipole approximation cannot be a-priori justified to analyzethe physics of vacuum excitations (a very important phenomenon in relativisticquantum 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 bevalid as long as the interaction time is longer than the light-crossing time ofthe atoms, which is a very reasonable assumption. We also highlight some of thesubtleties that one has to be careful with when working with the explicitlynon-gauge invariant nature of the minimal coupling and we compare it with theexplicitly gauge invariant dipole coupling.

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