  We propose a possible scheme for generating spin-J geometric phases using a
coupled two-mode Bose-Einstein condensate (BEC). First we show how to observe
the standard Berry phase using Raman coupling between two hyperfine states of
the BEC. We find that the presence of intrinsic interatomic collisions creates
degeneracy in energy that allows implementation of the non-Abelian geometric
phases as well. The evolutions produced can be used to produce interference
between different atomic species with high numbers of atoms or to fine control
the difference in atoms between the two species. Finally, we show that errors
in the standard Berry phase due to elastic collisions may be corrected by
controlling inelastic collisions between atoms.
