resonance frequency of the two-level transition, field mode frequency, and drive frequency share the same value, it is possible to move to an interaction picture where the dynamics are simplified. In this interaction picture, the evolution of the system is ruled by the time-independent Hamiltonian $$\label{back_1}
\mathcal{H}_{0} = i\hbar \lambda \left( \hat{a} \hat{\sigma}_{+} - \hat{a}^{\dagger} \hat{\sigma}_{-} \right) + \hbar \epsilon \left( \hat{a} + \hat{a}^{\dagger} \right) \, ,$$ with $\hat{a}^{\dagger}$ and $\hat{a}$ the creation and annihilation operators of the field mode, $\hat{\sigma}_{+}$ and $\hat{\sigma}_{-}$ the raising and lowering operators of the two-level system, and parameters $\lambda$, $\epsilon$ to determine the dipole coupling strength and external field amplitude, respectively. This model describes an idealized scenario where a phase transition of light is found [@Carmichael_2015; @Alsing_1992_2; @Alsing_1992_1], caused by the competition between coupling strength and driving field amplitude, with the former advocating for a definite excitation number while the latter advocates for a definite phase. The phase transition is characterized by a change in the level structure of the Hamiltonian, equation (\[back\_1\]), at the critical drive amplitude $$\label{back_2}
\epsilon_{0} = \lambda / 2 \, .$$ For driving amplitudes below $\epsilon_{0}$ the Hamiltonian displays a discrete level structure, with level spacings that collapse to 