$$\label{corr_8}
l_{\mathcal{B}} = \sqrt{\frac{\hbar c}{e B}} \,$$ providing a natural unit to measure the depth of the potential. Equations (\[corr\_6a\]) and (\[corr\_6b\]) take the form $$\label{corr_9}
\lbrace \mp i  \hbar  \partial_{t}  - c \sigma_{j} \hat{p}_{j} \rbrace \phi^{\pm}_{D} = \left\lbrace i\sqrt{2}eB l_{\mathcal{B}} \left( \hat{a} \sigma_{+} - \hat{a}^{\dagger} \sigma_{-} \right) \mp \frac{eE l_{\mathcal{B}}}{ \sqrt{2}} \left( \hat{a} + \hat{a}^{\dagger} \right)  \right\rbrace \phi^{\pm}_{D} \,$$ under these definitions. Since the eigenvalues of $\hat{p}_{2}$ and $\hat{p}_{3}$ remain constant through the evolution, we can consider the trivial scenario where they are zero, in which case the correspondence between equation (\[corr\_9\]) and the evolution under equation (\[back\_1\]) becomes transparent. The motion and spin state of the Dirac particle will map to the field mode and two-level system, respectively, in the quantum optical analogue. A similar correspondence between the undriven Jaynes–Cummings model and the Dirac oscillator was found in Ref. [@Rozmej_1999].

Eigenstates and eigenvalues: Quasienergy collapse {#secc:Lorentz}
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A method that has proven useful to determine the solutions of the Dirac equation in the presence of electromagnetic fields, equation (\[corr\_1\]), relies on considering the Dirac-Pauli equation as an auxiliary equation [@Landau_1971]. This allows us to extract a basis set for the spinors while giving a differential equation for the 