Nonlocality under Jaynes-Cummings evolution: beyond pseudospin operators

We re-visit the generation and evolution of (Bell) nonlocality in hybrid scenarios whose dynamics is determined by the Jaynes-Cummings Hamiltonian, a relevant example of which is the atom-cavity system. Previous approaches evaluate the nonlocality through the well-known qubit-qubit CHSH formulae, using combinations of pseudospin operators for the electromagnetic (EM) field observables. While such approach is sensible, it is far from optimal. In the present work we have used recent results on the optimal Bell violation in qubit-qudit systems, showing that the nonlocality is much greater than previously estimated, both with and without noise. We perform also an optimal treatment of the noise, so our results are optimal in this sense as well. We illustrate the results using different initial states for the EM field, including squeezed and coherent states. In addition, we study the asymptotic behavior of the entanglement. Remarkably, starting with a generic separable (pure) coherent state, the asymptotic (mixed) state is entangled, though does not violate Bell inequalities.