Integer and Fractionally Quantized Nonlinear Thouless Pumping

Thouless pumps are 1+1 dimensional reductions of Chern insulators for which one wavevector dimension is substituted with a periodic temporal modulation. As dimensionally reduced analogs of Chern insulators, Thouless pumps show topologically protected quantization of transport as long as the pump is adiabatic, the Fermi level is in the gap and interaction effects do not close the gap. Here, we demonstrate theoretically and observe experimentally, using a photonic system based on evanescently-coupled waveguides, that spatial solitons are transported by integer and fractionally quantized amounts per pumping cycle. These solitons form at high input power due to the optical Kerr effect. The dynamics of our system are well described by the focusing nonlinear Schrödinger equation (a.k.a. attractive Gross-Pitaevskii equation). We show analytically that low-power solitons track the motion of single-band Wannier functions and are thus pumped in a rigorously quantized fashion. For higher input power, solitons track the positions of maximally localized multi-band Wannier functions (equivalently, the multi-band Wilson loop eigenvalues), and thus exhibit fractional pumping. Finally, we numerically show that within the same model multiple plateaux of integer and fractionally quantized soliton transport appear as input power is varied.