  We show that there are effective three- and higher-body interactions
generated by the two-body collisions of atoms confined in the lowest
vibrational states of a 3D optical lattice. The collapse and revival dynamics
of approximate coherent states loaded into a lattice are a particularly
sensitive probe of these higher-body interactions; the visibility of
interference fringes depend on both two-, three-, and higher-body energy
scales, and these produce an initial dephasing that can help explain the
surprisingly rapid decay of revivals seen in experiments. If inhomogeneities in
the lattice system are sufficiently reduced, longer timescale partial and
nearly full revivals will be visible. Using Feshbach resonances or control of
the lattice potential it is possible to tune the effective higher-body
interactions and simulate effective field theories in optical lattices.
