diagonalized. In this case, all sites $j$ decouple and become independent. Since the total number of excitations $\hat N_j = \hat a_j^\dagger \hat a_j + \hat \sigma^+_j \hat \sigma^-_j$ on every site $j$ is a constant of motion, the local JC Hamiltonian block diagonalizes. Within each two-dimensional excitation subspace, the eigenstates can easily be found. Adapting the notation from [@Makin2008], these are given by $$\begin{aligned}
 \left| \pm , n\right \rangle &= \frac{\left[ \chi_n\mp(\omega-\Delta)\right]   \left| \uparrow , 
n-1\right\rangle \pm 2 g\sqrt n   \left| \downarrow , n\right\rangle}{\sqrt{2}\sqrt{\chi_n^2\mp (\omega-\Delta)\chi_n}}\label{eq:JCGroundState}\\
 &:= \alpha_n^\pm  \left| \uparrow , n-1\right\rangle   \pm \beta_n^\pm \left| \downarrow , n\right\rangle\end{aligned}$$ with $\chi_n=\sqrt{(\Delta-\omega)^2+4 n g^2}$ and $n>0$, and the eigenenergies are $$E_n^\pm = n \omega +\frac{\Delta-\omega}2 \pm \frac12\chi_n. \label{eq:JCEnergy}$$ For $n=0$, the ground state is non-degenerate and given by $\left|-,0\right\rangle =\left|\downarrow ,0\right \rangle$ with $E_0=0$. Here, the state $ \left| \uparrow , n-1\right\rangle$ describes an atomic excitation together with $n-1$ bosonic excitations; $\left| \downarrow , n\right\rangle$ is the state with the atom in the ground state and $n$ bosonic excitations. In the strong interaction limit $g\gg |\Delta-\omega|$, the energy gap $\Delta E_n = E_n^+ - E_n^- = \chi_n\sim 2 g\sqrt n$ is large compared to any other energy scale in the system and 