Atom-mediated effective interactions between modes of a bimodal cavity

We show a procedure for engineering effective interactions between two modes in a bimodal cavity. Our system consists of one or more two-level atoms, excited by a classical field, interacting with both modes. The two effective Hamiltonians have a similar form of a beam-splitter and quadratic beam-splitter interactions, respectively. We also demonstrate that the nonlinear Hamiltonian can be used to prepare an entangled coherent state, also known as multidimensional entangled coherent state, which has been pointed out as an important entanglement resource. We show that the nonlinear interaction parameter can be enhanced considering N independent atoms trapped inside a high-finesse optical cavity.