two modes of the bipartite system, as one might expect.

In order to examine what happens when there [*is*]{} a crossing of energy levels, we introduce a degeneracy by setting $\omega_{\mathrm{f}}= \omega_{\mathrm{a}}$.

![(a) $E_{{\textsc{af}}}(N,k)$ and (b) $\xi_{\textsc{svne}}$ vs. $g$ for $N=4$, $k=0,1,2$ in the atom-field interaction model, in the degenerate case $\omega_{\mathrm{f}}=\omega_{\mathrm{a}}
= 1$. The curves correspond to $k=0$ (red solid), $1$ (blue dashed) and $2$ (green dotted). $\gamma=1$.[]{data-label="fig:energySpectrumSVNEagarwalVaryLambda"}](fig_energy_4_agarwal_vary_lambda_U_1.eps "fig:"){width="45.00000%"} ![(a) $E_{{\textsc{af}}}(N,k)$ and (b) $\xi_{\textsc{svne}}$ vs. $g$ for $N=4$, $k=0,1,2$ in the atom-field interaction model, in the degenerate case $\omega_{\mathrm{f}}=\omega_{\mathrm{a}}
= 1$. The curves correspond to $k=0$ (red solid), $1$ (blue dashed) and $2$ (green dotted). $\gamma=1$.[]{data-label="fig:energySpectrumSVNEagarwalVaryLambda"}](fig_entropy_4_agarwal_vary_lambda_U_1.eps "fig:"){width="45.00000%"}

Figures \[fig:energySpectrumSVNEagarwalVaryLambda\] (a) and (b) are plots of $E_{{\textsc{af}}}(N,k)$ and $\xi_{\textsc{svne}}$ versus $g$ for $N=4$ and $k = 0,1, 2$, with $\gamma$, $\omega_{\mathrm{a}}$ and $\omega_{\mathrm{f}}$ set equal to $1$. Both a level crossing and an avoided crossing are seen to occur at $g=0$, signalled by a minimum in $\xi_{\textsc{svne}}$ for each of the three states concerned. The crossing of $E_{{\textsc{af}}}(4,0)$ and $E_{{\textsc{af}}}(4,1)$ arises as follows. Let $\ket{p,4-p}$ denote the product state $\ket{p}\otimes\ket{4-p}$, where $\ket{p}$ is a photon number state of the field mode and $\ket{4-p}$ is an oscillator state of the atom 