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We describe the theory of the dynamics of atoms in two-dimensionalquasicrystalline optical lattices. We focus on a regime of shallow latticedepths under which the applied force can cause Landau-Zener tunneling past adense hierarchy of gaps in the quasiperiodic energy spectrum. We deriveconditions on the external force that allow for a "semiadiabatic" regime inwhich semiclassical equations of motion can apply, leading to Blochoscillations between the edges of a pseudo-Brillouin-zone. We verify thissemiclassical theory by comparing to the results of an exact numericalsolution. Interesting features appear in the semiclassical dynamics for thequasicrystal for a particle driven in a cyclic trajectory around the corner ofthe pseudo-Brillouin-zone: The particle fails to return to its initial state,providing a realization of a "spiral holonomy" in the dynamics. We show thatthere can appear anomalous velocity contibutions, associated with nonzero Berrycurvature. We relate these to the Berry phase associated with the spiralholonomy, and show how the Berry curvature can be accessed from thesemiclassical dynamics. Finally, by identifying the pseudo-Brillouin-zone as ahigher genus surface, we show that the Chern number classification for periodicsystems can be extended to a quasicrystal, thereby determining a topologicalindex for the system.

Semiclassical Dynamics, Berry Curvature and Spiral Holonomy in Optical Quasicrystals