Entanglement properties of degenerate four-wave mixing of matter waves in a periodic potential

In a recent experiment, Campbell [Phys. Rev. Lett. 96, 020406 (2006)] observed degenerate four-wave mixing of matter waves in a one-dimensional optical lattice. We analyze the essential quantum features of the experiment to show that entanglement is created between the quadratures of the two scattered atomic clouds and is a true many-body (rather than two-body) phenomenon. We demonstrate that well-known bipartite entanglement inequalities are significantly violated and that this is robust to a moderate level of coherent seeding. The system is thus a promising candidate for generating spatially separated macroscopically entangled atomic samples.