Chaos and band structure in a three-dimensional optical lattice

Classical chaos is known to affect wave propagation because it signifies the presence of broken symmetries. The effect of chaos has been observed experimentally for matter waves, electromagnetic waves, and acoustic waves. When these three types of waves propagate through a spatially periodic medium, the allowed propagation energies form bands. For energies in the band gaps, no wave propagation is possible. We show that optical lattices provide a well-defined system that allows a study of the effect of chaos on band structure. We have determined the band structure of a body-centered-cubic optical lattice for all theoretically possible couplings, and we find that the band structure for those lattices realizable in the laboratory differs significantly from that expected for the bands in an "empty" body-centered-cubic crystal. However, as coupling is increased, the lattice becomes increasingly chaotic and it becomes possible to produce band structure that has behavior qualitatively similar to the "empty" body-centered-cubic band structure, although with fewer degeneracies.