  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.
