Tomography of Floquet topological singularities: How to measure topological invariants in periodically driven systems

We propose a realistic scheme for the full characterization of Floquet topological insulators realized in optical lattices. We construct a family of drives that connects the target Hamiltonian to a topologically trivial reference point for the periodically driven systems that we identify as the high-frequency regime. Our proposal relies on the detection of topological singularities in the Floquet spectrum with respect to the reference point as the drive parameters are varied within a specific drive family. Bypassing the difficulties of adiabatic preparation through a topological transition, we show how the topological charge of individual singularities can be measured using state tomography techniques. We demonstrate our scheme by using two concrete examples of periodically driven systems in two dimensions relevant to experiments. Our proposal paves the way to full experimental characterization of nonequilibrium topological phases in driven systems with the measurement of both regular and anomalous Floquet topological invariants.