eigenstates $\ket{\phi_{4,k}}, 
 \,0 \leqslant k \leqslant 4$ in the atom-field interaction model. $\omega_{\mathrm{f}}=\omega_{\mathrm{a}}=
 \gamma=1$.[]{data-label="fig:CorrPlotN4agarwalVaryLambda"}](correlation_matrix_plot_4_agarwal_vary_lambda_U_1_xx.eps "fig:"){width="30.00000%"}

\[sec:work2\]Avoided crossings in multipartite HQ systems
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Tavis-Cummings model
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As our third and final example, we consider hybrid quantum systems comprising several qubits interacting with an external field. These systems are described by the class of Tavis-Cummings models [@tavis] in a variety of diverse physical situations which include inherent field nonlinearities and inter-qubit interactions. The model we consider below is generic, applicable to a system of several two-level atoms with nearest-neighbour couplings interacting with an external radiation field in the presence of a Kerr-like nonlinearity, or to a chain of $M$ superconducting qubits interacting with a microwave field of frequency $\Omega_{\mathrm{f}}$. In the latter case, the model Hamiltonian (setting $\hbar = 1$) is given by [@QuantMeta2; @QuantMeta] $$\begin{aligned}
\nonumber H_{\textsc{tc}}=\Omega_{\mathrm{f}} 
a^{\dagger} a + \chi a^{\dagger \,2} a^{2}  + \sum_{p=1}^{M} &\Omega_{p} \sigma_{p z} + \Lambda ( a^{\dagger} \sigma_{p}^{-} + a \sigma_{p}^{+})\\
&+ \sum_{p=1}^{M-1} \Lambda_{s} ( \sigma_{p}^{-} \sigma_{(p+1)}^{+} + \sigma_{(p+1)}^{-} \sigma_{p}^{+}).
\label{eqn:HTC}\end{aligned}$$ Here, $\chi$ is the strength of the field nonlinearity, $\Lambda$ is the coupling strength between the field and each of the $M$ qubits, $\sigma_{p}^{\pm}$ are the ladder operators of the $p^{\rm th}$ qubit, and 