i \beta_{\mathcal{E}} = \sqrt{1- \frac{\lambda^{2}}{4\epsilon^{2}} } + i \frac{\lambda}{2 \epsilon} \, ,$$ as displayed in Figure \[fig:dipole\]. We have considered the zero quasi-energy states to underline the subtle differences between states above and below the critical point. The value of the polarization is, however, only dependent of the phase of the system and remains the same for each quasi-energy state. The same polarization values are found under the semiclassical approximation when dissipation and $N$ two-level systems are considered, but with finite photon number expectations [@Alsing_1992_1]. In fact, the divergences encountered above the critical point help to exemplify the role of dissipation in optical systems. Without dissipation, once the system is driven above the critical point, the field radiated by the two-level system is no longer strong enough to interfere destructively with the coherent drive \[see equation (\[qpt\_3\])\]. This leads to stationary states displaying an infinite photon number. Dissipation through the cavity walls would overcome the effect of the driving field, thus avoiding the divergences in photon number. While adding dissipation to the model goes beyond the scope of the current report, detailed studies of its effects in the driven Jaynes-Cummings model are presented in Refs. [@Carmichael_2015] and [@Alsing_1992_1]. The authors 