&= e E l_{\mathcal{B}}/\sqrt{2} \, ,\end{aligned}$$ such that the $F_{\mu \nu} F^{\mu \nu} =0$ condition refers to the transition point given in equation (\[back\_2\]). The inclusion of $l_{\mathcal{B}}$ in these definitions already hints at the importance of the magnetic field to the existence of discrete states. It is instructive to use the above relations to obtain the discrete quasi-energies of the driven JC model from equation (\[eige\_10\]): $$\upvarepsilon_{n, \pm}^{0} = \pm \hbar \lambda \left(1 - \frac{4 \epsilon^{2}}{\lambda^{2}}  \right)^{3/4} \sqrt{n +1} \, .$$ As mentioned above, the same result has been obtained through an algebraic method by Alsing, Guo and Carmichael (see equation (50) of Ref. [@Alsing_1992_2]) where the effect of the external driving field is interpreted as a dynamic Stark shift of the JC quasienergies. This can be extended to include counter-rotating interactions through equation (\[back\_4\]), leading to $$\upvarepsilon_{n, \pm}^{\eta} = \pm \hbar \lambda \left(1 - \frac{4 \epsilon^{2}}{(1+\eta)^{2}\lambda^{2}}  \right)^{3/4} \sqrt{n +1} \, ,$$ and a displaced transition point. In Figure \[fig:collapse\], the discrete quasienergies of the driven JC model are displayed as a function of the driving amplitude. The solutions were obtained by computing the eigenvalues of $\mathcal{H}_{0}$ in a truncated basis and show the characteristic $\sqrt{n}$-dependence and ensuing collapse 