when the critical point is reached.

![Quasi-energy level splitting and collapse at the critical point.[]{data-label="fig:collapse"}](tikz_collapse_2){width=".5\linewidth"}

Besides the collapse of the level structure of the driven JC Hamiltonian, the behaviour of the zero quasienergy state captures the nature of the transition at hand. Below the critical drive amplitude, the corresponding eigenstate in the position representation is $$\label{qpt_2}
\phi_{D}^{-}  = \exp\left[- \frac{\sqrt{1 - \beta_{\mathcal{B}}^{2}} }{2 l_{\mathcal{B}}^{2}} x_{1}^{2} \right] \left( \begin{array}{c}
\Xi_{\mathcal{B}}^{-}   \\
-i \Xi_{\mathcal{B}}^{+} \\
\end{array} \right) \, .$$ This separable product of field-mode and two-level system states is normalizable. The coherent drive determines the polarization of the two-level system through $$\label{qpt_3}
\langle \hat{\sigma}_{-} \rangle = i \beta_{\mathcal{B}} = i 2\epsilon/\lambda \, .$$ On the other hand, above critical drive, the eigenstate is $$\label{qpt_4}
\phi_{D}^{-} = \exp\left[- \frac{i \sqrt{1 - \beta_{\mathcal{E}}^{2}} }{2 l_{\mathcal{E}}^{2}} x_{1}^{2} \right] \left( \begin{array}{c}
\Xi_{\mathcal{E}}^{+} + i \Xi_{\mathcal{E}}^{-}  \\
\Xi_{\mathcal{E}}^{+} - i \Xi_{\mathcal{E}}^{-}  \\
\end{array} \right) \, ,$$ a separable product which can be delta-normalized [@Lo_1990]. Disregarding the normalization factor associated with the field mode, the polarization of the two-level system can be shown to settle in the equator of the Bloch sphere with its phase depending on the drive amplitude, $$\langle \hat{\sigma}_{-} \rangle = \sqrt{1- \beta_{\mathcal{E}}^{2} } + 