Quantized transport of solitons in nonlinear Thouless pumps: From Wannier drags to topological polarons

Recent progress in synthetic lattice systems has opened the door to novel explorations of topological matter. In particular, photonics devices and ultracold matter waves offer the unique possibility of studying the rich interplay between topological band structures and tunable nonlinearities. In this emerging field of nonlinear topological physics, a recent experiment [Nature 596, 63 (2021)] revealed the quantized motion of localized nonlinear excitations, known as solitons, upon driving a Thouless pump sequence; the reported results suggest that the quantized displacement of solitons is dictated by the Chern number of the band from which they emanate. In this work, we elucidate the origin of this nonlinear topological effect, by showing that the motion of solitons is established by the quantized displacement of Wannier functions. Furthermore, we consider the quantum-mechanical parent of this setting and relate the quantized motion of solitons to the transport of Bose polarons (dressed impurities). By demonstrating the quantized pumping of Bose polarons, our work introduces a novel instance of topological polaron, which could be observed in ultracold atomic mixtures.