\, ,$$ with squeeze parameter given by $$\begin{aligned}
\label{back_7_t}
\cosh z = \lambda^{\prime}/ \lambda \, , \label{back_7_1} \\ \sinh z =  \eta \lambda^{\prime} / \lambda   \, , \label{back_7_2}\end{aligned}$$ which introduces the scaled driving field amplitude $$\label{back_7_3}
\epsilon^{\prime} = \epsilon ( \lambda^{\prime} / \lambda ) \left( 1 + \eta \right) \, .$$ The addition of the counter-rotating interaction not only affects the mode photon number, but also displaces the critical drive towards the value $$\label{back_4}
\epsilon^{\prime}_{\eta} =  \lambda^{\prime} (1 + \eta) / 2 \, .$$ These transformations establish a simple connection between the on-resonance JC and Jaynes–Cummings–Rabi models. We emphasize, in particular, that they allow us to target the treatment in the following sections to the JC interaction without loss of generality.

Previously the quasienergies and eigenstates of the system below the critical amplitude have been obtained through a Bogoliubov transformation [@Alsing_1992_2; @Gutierrez_2017b]; this method ceases to be applicable, however, for the continuous spectrum above the critical amplitude. We overcome this problem in the following, where we develop a new method for obtaining the quasienergies and eigenstates, both below and above critical drive, for the driven Jaynes–Cummings model. The solutions of the generalized model are then readily available through the squeezing 