Entanglement between atomic condensates in an optical lattice: effects of interaction range

We study the area-dependent entropy and two-site entanglement for two state Bose-Einstein condensates in a 2D optical lattice. We consider the case where the array of two component condensates behave like an ensemble of spin-half particles with the interaction to its nearest neighbors and next nearest neighbors. We show how the Hamiltonian of their Bose-Einstein condensate lattice with nearest-neighbor and next-nearest-neighbor interactions can be mapped into a harmonic lattice. We use this to determine the entropy and entanglement content of the lattice.