Distributed entanglement in the two-photon two-mode non-linear process

In this paper we consider a quantum optics model where two-mode quantum light cavity with Kerr-like medium is coupled to an atom via two-photon process. The dynamical evolution of the system is studied in terms of entanglement measured by quantum relative entropy. The entanglements for the different bipartite partitions of the system, i.e., atom-two modes, mode-mode, mode-(atom+mode), are calculated explicitly and interesting trade-off relations between the different kinds of entanglement can be observed in different cases. The results show the entanglement between mode-mode is generally out of phase with that between atom and two modes, even though the two modes do not interact directly, and the Kerr-like medium prevents the atom and two modes from entangling.