# How to Directly Measure Floquet Topological Invariants in Optical Lattices

## TL;DR

This paper introduces a method to directly measure topological invariants called winding numbers in Floquet systems using cold atoms in optical lattices, enabling better characterization of driven quantum systems.

## Contribution

It proposes a practical scheme to measure Floquet topological invariants by connecting the system to a trivial reference through a family of drives and identifying band-touching singularities.

## Key findings

- Method for measuring winding numbers in Floquet systems
- Connection of Floquet invariants to band-touching singularities
- Proposed spectroscopic technique for time evolution operators

## Abstract

The classification of topological Floquet systems with time-periodic Hamiltonians transcends that of static systems. For example, spinless fermions in periodically driven two-dimensional lattices are not completely characterized by the Chern numbers of the quasienergy bands, but rather by a set of winding numbers associated with the quasienergy gaps. We propose a scheme for measuring these winding numbers in a system of fermionic cold atoms in a periodically driven optical lattice efficiently and directly. It is based on the construction of a one-parameter family of experimentally feasible drives, continuously connecting the Floquet system of interest to a trivial reference system. The winding numbers are then determined by the identification and the tomography of the band-touching singularities occurring on the way. As a byproduct, we also propose a method for probing spectral properties of time evolution operators via a time analog of crystallography.