\Xi_{\mathcal{E}}^{+} - \Xi_{\mathcal{E}}^{-} \sigma_{2} \right] \left( \begin{array}{c}
 D_{-a}(\xi) -  b_{-}D_{-a-1}(\xi) \\
 D_{-a}(\xi) +  b_{-}D_{-a-1}(\xi) \\
\end{array} \right) \, , \label{eige_21b}\end{aligned}$$ with the transformed variables $$\begin{aligned}
& \xi = \sqrt{2} e^{i \pi /4} \left(1- \beta_{\mathcal{E}}^{2} \right)^{1/4} \left[  \frac{\left( \beta_{\mathcal{E}} \hbar c k_{2} -  \upvarepsilon \right) l_{\mathcal{E}}}{  \hbar c \left(1 - \beta_{\mathcal{E}}^{2} \right) } - \frac{x_{1}}{l_{\mathcal{E}}} \right] \, , \label{eige_22} \\
& a = -\frac{i l_{\mathcal{E}}^{2}}{2} \left[ \frac{\left( k_{2} - \beta_{\mathcal{E}} \upvarepsilon / \hbar c \right)^{2}}{(1- \beta_{\mathcal{E}}^{2})^{3/2}} + \frac{k^{2}_{3}}{(1 - \beta_{\mathcal{E}}^{2})^{1/2}} \right] \, , \\
& \Xi^{\pm}_{\mathcal{E}} = \sqrt{1 \pm \left[1 - \beta_{\mathcal{E}}^{2} \right]^{1/2}}  \, . \label{eige_23}\end{aligned}$$

Back to light {#secc:Jaynes_cummings}
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The quantized motion and discrete eigenvalues found when the magnetic field dominates \[equations (\[eige\_10\]) and (\[eige\_11a\])-(\[eige\_11b\])\] contrast the continuous behaviour encountered when the electric field dominates \[equations (\[eige\_21a\])-(\[eige\_21b\])\]. As mentioned in the introduction, in the simplified case where $k_{2}$ and $k_{3}$ both equal zero, the $\phi^{-}_{D}$ spinor of these two complementary cases corresponds to the eigenstates of the driven Jaynes–Cummings Hamiltonian below and above critical drive. A direct comparison between equation (\[back\_1\]) and (\[corr\_9\]) shows that the dipole coupling constant and driving amplitude relate to the electromagnetic fields through $$\begin{aligned}
\label{qpt_1}
\lambda &= \sqrt{2}e B l_{\mathcal{B}} \, , \\
\epsilon 