$\widehat{\mathit{p}}·\widehat{\mathit{A}}$ vs $\widehat{\mathit{x}}·\widehat{\mathit{E}}$: Gauge invariance in quantum optics and quantum field theory

We compare the predictions of the fundamentally motivated minimal coupling ($\widehat{\mathit{p}}·\widehat{\mathit{A}}$) and the ubiquitous dipole coupling ($\widehat{\mathit{x}}·\widehat{\mathit{E}}$) in the light-matter interaction. By studying the light-matter interaction for hydrogenlike 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 gauge noninvariant nature of the minimal coupling, and we compare it with the explicitly gauge invariant dipole coupling.

Figure 1

Vacuum transition probability $1s\to 2p$ for $Z=1$ atom. The short time behavior, i.e., $t\in (0,40{\mathrm{\Omega }}^{-1})$, corresponds to all vacuum modes constructively contribution with very minor phase differences. In the long time limit all modes are dephased with one another, leading to a constant transition probability. The dashed lines correspond to analytic results obtained from (36) and (37) via $\underset{t\to \infty }{\lim }{\sin }^2((\omega +\mathrm{\Omega })\frac{t}{2})=\frac{1}{2}$. From these analytic expressions we know that the offset is $2.58\times {10}^{-4}$.

Figure 2

Long time transition probability as a function of $Z$ and the relative error between the two models. As $Z$ increases the atom becomes smaller; however, contrary to Scully and Zubairy, the models diverge. Minimal model is dashed.

Figure 3

Vacuum transition probability $2p\to 1s$ for $Z=1$ atom. The short time behavior corresponds to the region where the single mode approximation is invalid. During this time the dipole approximation is violated and the observed offset is generated. The dashed line corresponds to the single mode approximation. Note that for longer times the curve becomes linear as dictated by the single mode approximation.

Figure 4

Vacuum emission transition rates as a function of $Z$. Note that as $Z$ increases and the atom becomes smaller the two models diverge. Minimal model is dashed. By implementing the single mode approximation in (36) and (37) one can show that the transition rates are given by $R=\frac{6.26\times {10}^8{Z}^4}{{(1+3.33\times {10}^{-6})}^n}$, where $n=4$, 6 for the minimal and dipole coupling respectively.

Figure 5

Excitation probability ($P_{\varphi }$ only) for a Gaussian coherent pulse with central frequency $\mathrm{\Omega }$ (resonant). As the pulse arrives the transition probability increases to $3\times {10}^{-10}$. Once the pulse is far from the atom the transition probability remains roughly constant. Note that the relative error remains small throughout $2\times {10}^{-3}$. Also note, the bump at early times is believed to be a numerical imprecision.

Figure 6

Long time excitation probability ($P_{\varphi }$ only) as a function of $Z$. Unlike the vacuum cases the dominant frequency is dictated by the field excitation and not the single mode approximation; however, since we chose the field excitation to remain resonant with the atomic transition the relative error increases with $Z$ as $\mathrm{\Omega }$ ceases to satisfy the dipole approximation.

Figure 7

Difference in asymptotic behavior between dipole and minimal models for vacuum excitations, as a function of a hard UV cutoff. The x-axis is normalized to dipole approximation frequency cutoff. $\mathrm{\Lambda }\ll 1$ corresponds to an application of the dipole approximation. This reflects the conclusion of (42).