  We study generation of non-local correlations by atomic interactions in a
pair of bi-modal Bose-Einstein Condensates in state-dependent potentials
including spatial dynamics. The wave-functions of the four components are
described by combining a Fock state expansion with a time-dependent
Hartree-Fock Ansatz, so that both the spatial dynamics and the local and
non-local quantum correlations are accounted for. We find that despite the
spatial dynamics, our protocole generates enough non-local entanglement to
perform an EPR steering experiment with two spatially separated con-densates of
a few thousands of atoms.
