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

In a recent experiment Campbell et al. [Phys. Rev. Lett. 96, 020406 (2006)] observed degenerate four-wave mixing of matter-waves in a one-dimensional optical lattice, a process with potential for generating entanglement among atoms. We analyse 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) effect. We demonstrate a significant violation of entanglement inequalities that is robust to a moderate level of coherent seeding. The system is thus a promising candididate for generating macroscopically entangled atomic samples.