Effect of Bloch-Band Dispersion on the Quantized Transport in a Topological Thouless Pump

We study the spreading of an initially localized wave packet of a particle hopping on a one-dimensional superlattice during a cycle of a topological Thouless pump. Two contributions to the dispersion of the adiabatic pumping process are identified: a dynamical part and a geometrical part. The magnitude of the dynamical contribution to the spreading depends on the dispersion of the adiabatic transfer state and the cycle time. Unlike the dynamical one, the geometrical contribution does not depend on the duration of the adiabatic process and can be made much smaller than the lattice spacing. We show that as the adiabaticity is enhanced by prolonging the period of the pumping process, the uncertainty in coordinate space is increased linearly with the adiabaticity parameter. We propose a mechanism to smoothen the energy surface of the adiabatic transfer state to reduce the spreading of the spatial distribution of the transported particle. This diminishes or even eliminates (up to the geometric contribution) the dispersion of the coordinate during the transport process.