mode and the oscillator mode, respectively. As before, $N_{\mathrm{tot}} = a^{\dagger}a+b^{\dagger}b$ and $[H_{\textsc{af}},N_{\mathrm{tot}}]=0$. As in the BEC model of the preceding section, the eigenvalues $E_{{\textsc{af}}}(N,k)$ and the common eigenstates $\ket{\phi_{N,k}}$ of these two operators are labelled by $N = 0,1,\ldots$ (the eigenvalue of $N_{\mathrm{tot}}$) and, within each $(N+1)$-dimensional subspace for a given $N$, by the index $k$ that runs from $0$ to $N$.

![(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. The curves correspond to $k=0$ (red solid), $1$ (blue dashed) and $2$ (green dotted). $\omega_{\mathrm{f}}=1.5, \,\omega_{\mathrm{a}}=1, \,
\gamma=1$.[]{data-label="fig:energySpectrumSVNEagarwalVaryLambdaDetuned"}](fig_energy_4_agarwal_vary_lambda_U_1_detuned.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. The curves correspond to $k=0$ (red solid), $1$ (blue dashed) and $2$ (green dotted). $\omega_{\mathrm{f}}=1.5, \,\omega_{\mathrm{a}}=1, \,
\gamma=1$.[]{data-label="fig:energySpectrumSVNEagarwalVaryLambdaDetuned"}](fig_entropy_4_agarwal_vary_lambda_U_1_detuned.eps "fig:"){width="45.00000%"}

We find $\ket{\phi_{N,k}}$ and $E_{{\textsc{af}}}(N,k)$ numerically. Figures \[fig:energySpectrumSVNEagarwalVaryLambdaDetuned\](a) and (b) show plots of $E_{{\textsc{af}}}(N,k)$ and $\xi_{\textsc{svne}}$ versus $g$ for $N=4$ and $k=0,1,2$ in the case $\omega_{\mathrm{f}}=1.5, \,\omega_{\mathrm{a}}=1$. Avoided crossings occur at $g=0$, with a corresponding minimum in $\xi_{\textsc{svne}}$ that drops down to zero for each of the three states $\ket{\phi_{4,0}},  \ket{\phi_{4,1}}$ and $\ket{\phi_{4, 2}}$. These states are therefore unentangled at $g=0$, i.e., in the absence of interaction between the 