Generation of two-dimensional cluster states by using high-finesse bimodal cavities

We propose two novel schemes to generate the two-dimensional 2xN and 3xN cluster states by using a chain of (two-level) Rydberg atoms in the framework of cavity QED. These schemes work in a completely deterministic way and are based on the resonant interaction of the atoms in a chain with a bimodal cavity that supports two independent modes of the photon field. We demonstrate that a 2xN cluster state can be generated efficiently with only a single bimodal cavity, while two such cavities are needed in order to construct a 3xN cluster state. It is shown, moreover, how these schemes can be extended towards the generation of MxN two-dimensional cluster states.