Semiclassical Dynamics, Berry Curvature and Spiral Holonomy in Optical Quasicrystals

TL;DR

This paper develops a semiclassical theory for atom dynamics in 2D optical quasicrystals, revealing spiral holonomy, Berry curvature effects, and topological properties extending Chern number classification to quasicrystals.

Contribution

It introduces a semiclassical framework for quasicrystals, demonstrating spiral holonomy, Berry curvature effects, and topological invariants in non-periodic systems.

Findings

Verification of semiclassical equations with numerical solutions

Observation of spiral holonomy in cyclic trajectories

Extension of Chern number classification to quasicrystals

Abstract

We describe the theory of the dynamics of atoms in two-dimensional quasicrystalline optical lattices. We focus on a regime of shallow lattice depths under which the applied force can cause Landau-Zener tunneling past a dense hierarchy of gaps in the quasiperiodic energy spectrum. We derive conditions on the external force that allow for a "semiadiabatic" regime in which semiclassical equations of motion can apply, leading to Bloch oscillations between the edges of a pseudo-Brillouin-zone. We verify this semiclassical theory by comparing to the results of an exact numerical solution. Interesting features appear in the semiclassical dynamics for the quasicrystal for a particle driven in a cyclic trajectory around the corner of the pseudo-Brillouin-zone: The particle fails to return to its initial state, providing a realization of a "spiral holonomy" in the dynamics. We show that there can appear anomalous velocity contibutions, associated with nonzero Berry curvature. We relate these to the Berry phase associated with the spiral holonomy, and show how the Berry curvature can be accessed from the semiclassical dynamics. Finally, by identifying the pseudo-Brillouin-zone as a higher genus surface, we show that the Chern number classification for periodic systems can be extended to a quasicrystal, thereby determining a topological index for the system.