zero when this critical value is reached; the level structure becomes continuous above it [@Alsing_1992_2].

The critical behaviour persists if the interaction between the two-level system and field mode is extended to include counter-rotating coupling [@Gutierrez_2017b]. In this extended model the dynamics in the interaction picture are ruled by the generalized Jaynes–Cummings–Rabi Hamiltonian $$\begin{aligned}
\label{back_3}
\mathcal{H}_{\eta} = i\hbar \lambda^{\prime} \left[ \left( \hat{a} + \eta \hat{a}^{\dagger} \right) \hat{\sigma}_{+} - \left( \hat{a}^{\dagger} + \eta \hat{a} \right) \hat{\sigma}_{-} \right]
+ \hbar \epsilon^{\prime} \left( \hat{a} + \hat{a}^{\dagger} \right) \, ,\end{aligned}$$ with $\eta$ a continuous parameter ranging from $\left[ 0 ,1 \right]$. The expected dynamics under this Hamiltonian can be realized using a cavity QED architecture [@Dimer_2007]. In this realization the rotating and counter-rotating interactions are controlled independently by promoting externally driven Raman transitions between a pair of ground states via a manifold of far-detuned excited states; the time-independent Hamiltonian, equation (\[back\_3\]), is realized in an appropriate interaction picture (see Sections IIIC and D of Reference [@Gutierrez_2017b]). Considering this open system perspective clarifies the origin of the photons injected into the field mode through the counter-rotating coupling. In fact, equations (\[back\_1\]) and (\[back\_3\]) are connected through a squeezing transformation, $$\label{back_6}
\mathcal{S}(z) \mathcal{H}_{0} \mathcal{S}^{\dagger}(z) = \mathcal{H}_{\eta} 