  We present a three-cavity network model with two modes in each cavity and a
non-linear medium that generates a Kerr type interaction via both self-phase
and cross-phase modulation processes. We have two main goals. The first one is
to generate a multipartite Maximally Entangled State (MES), starting from the
ground state of the system. We address the problem both without and with
dissipation. Secondly, we want to protect the MES from decoherence. While
studying the MES, we analyze different bipartite and multipartite entanglement
measures. We also study the effect of an Avoided Level Crossing (ALC)
identified by the critical behavior of the entanglement measures, thus showing
that the quantum correlations act as a witness for such phenomena. Our findings
provide the quantum tools to perform the operation of generation and protection
of a maximally entangled state in a cavity QED environment.
