We can also measure the geometric phases from the bulk energy bands to obtain the topological invariants.
On the other hand, the topological invariants are proportional to the geometric phases accumulated across a whole Brillouin zone. However, edges are not always available in atomic quantum simulators 7. They are usually measured by the response from gapless edge states based on the bulk-edge correspondence. Characterizing topological invariants is a central task in synthesizing and simulating topological phases of matter. These invariants change stepwisely only when the bulk goes through a topological phase transition, which involves band gap closing and reopening. Topological matter has promising applications in noise resilient devices and quantum information processing 1, 2, thanks to the robust topological response guaranteed by global geometric quantities of the Bloch bands, namely, the topological invariants 3, 4, 5, 6. Our approach paves the way of measuring topological invariants and developing their applications in room-temperature atoms. We measure Zak phases directly from the anti-crossings between Wannier-Stark ladders in the Doppler-broadened absorption spectra of superradiance lattices. In such momentum-space lattices the thermal motion of atoms, instead of being a source of noise, provides effective forces which lead to spectroscopic signatures of the Zak phases. By exploiting this relation, we develop a method to extract geometric phases from energy spectra of room-temperature superradiance lattices, which are momentum-space lattices of timed Dicke states. On the other hand, geometric phases also determine the energy spectra of particles subjected to a static force, based on the polarization relation between Wannier-Stark ladders and geometric Zak phases. However, cold atoms are intrinsically prone to thermal noise, which can overwhelm the topological response and hamper promised applications. Cold atoms provide a flexible platform for synthesizing and characterizing topological matter, where geometric phases play a central role.
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