Bloch oscillations on two dimensional lattices

Abstract

Summary form only given. The evolution of an excitation propagating through a discrete lattice shows common features regardless of the physical nature of the underlying system. Provided that a linearly growing potential is applied Bloch oscillations are observed in semiconductor superlattices, in Bose condensates trapped in optical potentials, on molecular chains or in optical waveguide arrays, etc. These systems have in common that their eigensolutions are localized and that the corresponding eigenvalues are evenly spaced (Wannier-Stark states). Therefore every excitation results in a periodic motion (Bloch oscillations). Until now exact expressions for higher dimensional lattices have only been found for the simple cubic case without diagonal interaction, which separates into orthogonal 1D systems. Using a discrete model we find Bloch oscillations in realistic two-dimensional lattices as e.g. bundles of optical fibers. We derive analytical solutions for cubic lattices including diagonal interaction as well as for hexagonal structures. Likewise our scheme can be extended to a wider interaction range and is also applicable to 3D lattices.