Fig. 1: Relation between Zak phases and Wannier-Stark ladders in superradiance lattices.

a The Zak phase as an accumulated geometric phase of a particle adiabatically driven across a whole Brillouin zone. b The Wannier function in each unit cell. The zeroth unit cell is highlighted with yellow color. The positions of the Wannier centers are \(2n + \theta /\pi\). c Schematic WSLs shown by the projected density of states with different forces \(F = \delta\). The three dotted lines connect the corresponding WSLs in different forces. Quantum transport package Nanoskim⁵⁴ is used in the calculation. d Schematic configuration of the light fields. Inset: the coupling between the light fields and the atomic levels in the reference frame of the atom with the Rabi frequency \({{\Omega }}_i\) (\(i = 1,2,p\)), the coupling field detuning in the lab reference frame \({{\Delta }}_c\), and the Doppler shift δ. The shaded area indicates the envelope of the standing wave coupling field and the balls indicate atoms with velocity v. e The momentum-space superradiance lattice with tight-binding Hamiltonian \(H_s\) (upper) and a linear potential \(H_f\) (lower). \(\tilde d_j\left( {d = a,b} \right)\) are the timed Dicke states