Cold atoms provide a flexible platform for synthesizing and characterizing topological materials in which geometric phases play a central role. However, cold atoms are inherently prone to thermal noise, which can overwhelm the topological response and interfere with their promised applications. On the other hand, the geometrical phase also determines the energy spectrum of particles subjected to static forces based on the polarization relationship between the Wannier-Stark ladder and the geometrical Sack phase. Exploiting this relationship, we develop a method to extract geometric phases from the energy spectrum of room-temperature superradiative lattices, which are momentum-space lattices of timed Dicke states. In such a momentum-space lattice, the thermal motion of atoms, rather than being a noise source, provides an effective force leading to the spectroscopic signature of the Zak phase. We directly measure the Zack phase from the anticrossing between the Wannier-Stark ladders in the Doppler broadened absorption spectrum of the superradiative grating. Our approach paves the way for measuring topological invariants and developing applications at room temperature atoms.
