rgence is observed. Divergences of the NL can be understood, in general, as a consequence of the distinguishability between the post-work state and the reference thermal state used to evaluate the final free energy $F(\lambda_f, \beta)$. As previously commented, the NL can be written in terms of the relative entropy between those two states [@daffner]. As so, the smaller the NL, the more indistinguishable the two states are and, for orthogonal states, it diverges. In a quench process, as considered here, the initial state does not change after the work protocol [@fusco2014]. Consequently, the post-work state is a Gibbs state defined with inverse temperature $\beta$ and Hamiltonian (\[h0\]). When $\beta\to \infty$, this state is basically $|0, g \rangle$. In the same limit, the reference thermal state used to evaluate $F(\lambda_f, \beta)$ will be given by the ground state of either the JC Hamiltonian or the AJC Hamiltonian, depending on the chosen $\omega_L$. For the physical parameters used in the simulations, the ground state of the JC Hamiltonian coincides with the post-work state which is $|0, g \rangle$, while the ground state of the AJC Hamiltonian will be a superposition of $|0,e\rangle$ and $|m,g\rangle$. We can then see that NL will be smaller for the JC than for the AJC because the post-work state is more indistinguishable from the ground state of the former than from the ground state of the latter. All eigenstates and eigenvalues for AJC and JC Hamiltonians can be found in the Appendix.

We may wonder under which parameters choice the JC case can present divergences in the NL. In order words, how the system parameters can be chosen to cause at least one of the $\Phi_n^m$ in (\[limZ-T\]) to be negative. The analysis of Eq. (\[funcphi\]) reveals that this is the case provided $$\label{limi1}
\left|f_n^m\right| > \frac{2}{\Omega} \sqrt{\nu (\omega_0+n\nu) (n+m)},$$ for a fixed $m$ (sideband) and some value of $n$. Now, in order to see this effect, one needs to go a bit beyond the current experimental set