We propose an interferometric method to measure ${\mathbb{Z}}_2$ topological invariants of time-reversal invariant topological insulators realized with optical lattices in two and three dimensions. We suggest two schemes which both rely on a combination of Bloch oscillations with Ramsey interferometry and can be implemented using standard tools of atomic physics. In contrast to topological Zak phase and Chern number, defined for individual one-dimensional and two-dimensional Bloch bands, the formulation of the ${\mathbb{Z}}_2$ invariant involves at least two Bloch bands related by time-reversal symmetry which one must keep track of in measurements. In one of our schemes this can be achieved by the measurement of Wilson loops, which are non-Abelian generalizations of Zak phases. The winding of their eigenvalues is related to the ${\mathbb{Z}}_2$ invariant. We thereby demonstrate that Wilson loops are not just theoretical concepts but can be measured experimentally. For the second scheme we introduce a generalization of time-reversal polarization which is continuous throughout the Brillouin zone. We show that its winding over half the Brillouin zone yields the ${\mathbb{Z}}_2$ invariant. To measure this winding, our protocol only requires Bloch oscillations within a single band, supplemented by coherent transitions to a second band which can be realized by lattice shaking.

• Received 18 February 2014

DOI:https://doi.org/10.1103/PhysRevA.89.043621