  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.
