Figs. 2a-b demonstrate the breakdown of gauge invariance between Ĥ′ p•A (Eq. 66) and Ĥd•E (Eq. 67) for model systems with a square and double well potential, respectively. In both models, the energy eigenspectra using each gauge (Eqs. 66 and 67) for an election in a given potential is plotted as a function of coupling strength when truncated to only two matter levels, and the matrix elements of x and p are visualized. In Fig. , these results are shown for a square potential. In this case, the dipole gauge results outperform Coulomb gauge results but still fails to capture much of the physics of the full system. This is a consequence of the locality of x and p in the energy picture, shown by the matrix element visualizations in Fig. . The position matrix elements are much more localized than the momentum matrix elements. However, a two level truncation is still not a good approximation, since | ⟨ψ 1 | x |ψ 2 ⟩ | matrix elements are significant, meaning that at least three states are needed to accurately describe the first two states. These results can be contrasted with those of Fig. , where the potential is a double well potential. In this case, the matrix elements of x are more localized for the first two levels, such that it can be well approximated as a two-level system. For the momentum matrix elements, however, the first two states are strongly coupled to many high energy states. This disparity is apparent in the energy eigenspectra from the dipole and Coulomb gauges. The dipole gauge results follow the fully converged results, while the Coulomb gauge results diverge.