Direct measurement of topological invariants in optical lattices

TL;DR

This paper introduces an experimental method to directly measure topological invariants like the Chern number in optical lattices using combined time-of-flight and in situ detection, applicable to various topological phases.

Contribution

It presents a novel measurement technique for topological invariants in cold atom systems, extending topological charge pumping concepts to optical lattices.

Findings

Demonstrates direct measurement of Chern number in Hofstadter lattice

Method applicable to measuring Z2 topological invariants

Provides a general approach for topological classification in cold atoms

Abstract

We propose an experimental technique for classifying the topology of band structures realized in optical lattices, based on a generalization of topological charge pumping in quantum Hall systems to cold atom in optical lattices. Time-of-flight measurement along one spatial direction combined with in situ detection along the transverse direction provide a direct measure of the system's Chern number, as we illustrate by calculations for the Hofstadter lattice. Based on an analogy with Wannier functions techniques of topological band theory, the method is very general and also allows the measurement of other topological invariants, such as the $Z_2$ topological invariant of time-reversal symmetric insulators.