Phase-matched four wave mixing and quantum beam splitting of matter waves in a periodic potential

We show that the dispersion properties imposed by an external periodic potential ensure both energy and quasimomentum conservation such that correlated pairs of atoms can be generated by four wave mixing from a Bose-Einstein condensate moving in an optical lattice potential. In our numerical solution of the Gross-Pitaevskii equation, a condensate with initial quasimomentum $k_0$ is transferred almost completely $(>95\%)$ into a pair of correlated atomic components with quasimomenta $k_1$ and $k_2$, if the system is seeded with a smaller number of atoms with the appropriate quasimomentum $k_1$.