Interferometric approach to measuring band topology in 2D optical lattices

Recently, optical lattices with non-zero Berry's phases of Bloch bands have been realized. New approaches for measuring Berry's phases and topological properties of bands with experimental tools appropriate for ultracold atoms need to be developed. In this paper, we propose an interferometric method for measuring Berry's phases of two dimensional Bloch bands. The key idea is to use a combination of Ramsey interference and Bloch oscillations to measure Zak phases, i.e. Berry's phases for closed trajectories corresponding to reciprocal lattice vectors. We demonstrate that this technique can be used to measure Berry curvature of Bloch bands, the \pi-Berry's phase of Dirac points, and the first Chern number of topological bands. We discuss several experimentally feasible realizations of this technique, which make it robust against low-frequency magnetic noise.