Zak phase reconstruction enabled by thermal motion of atoms

Ultracold atoms provide a flexible platform for simulating topological phases. However, they are intrinsically prone to thermal noise, which can overwhelm topological response and hamper promised applications. On the other hand, topological invariants can be reconstructed from the energy spectra of particles subjected to a static force, based on the polarization relation between Wannier-Stark ladders and Zak phases. Such spectroscopic methods circumvent the obstacle to measuring topological invariants in thermal atoms. Here we reconstruct Zak phases from the anti-crossing between Wannier-Stark ladders in room-temperature superradiance lattices. Without dynamical manipulation routinely needed in ultracold atoms, we evaluate Zak phases directly from the absorption spectra. The thermal atomic motion, an essential source of noise in cold atoms, plays a key role in extracting Zak phases. Our approach paves the way of measuring topological invariants and developing their applications in room-temperature atoms.