Artificial Relativity with Optical Lattices

A driving force behind the study of ultracold atoms is the idea of ``quantum simulation" of other physical systems, including systems which may not be accessible in their original manifestations. In this talk, we discuss how to use optical lattice setups to generate a variety of effective Hamiltonians which resemble the Dirac Hamiltonian for a relativistic electron. Engineering such Hamiltonians suggests the possibility of a variety of experiments, including cold atom versions of \emph{Zitterbewegung} \footnote{J. Y. Vaishnav, Charles W. Clark. \emph{Phys. Rev. Lett.}, {\bf 100}, 153002 (2008).}, spintronic transistors \footnote{J. Y. Vaishnav, Julius Ruseckas, Charles W. Clark, Gediminas Juzeliunas. \emph{Phys. Rev. Lett.}, {\bf 101}, 265302 (2008).}, and topological insulators \footnote{T. D. Stanescu, V. Galitski, J. Y. Vaishnav, Charles W. Clark. Preprint.}.