  Thouless's quantum adiabatic pumping is of fundamental interest to
condensed-matter physics. It originally considered a zero-temperature
equilibrium state uniformly occupying all the bands below a Fermi surface. In
the light of recent direct simulations of Thouless's concept in cold-atom
systems, this work investigates the dynamics of quantum adiabatic pumping
subject to dephasing, for rather general initial states with nonuniform
populations and possibly interband coherence. Using a theory based on
pure-dephasing Lindblad evolution, we find that the pumping is contributed by
two parts of different nature, a dephasing-modified geometric part weighted by
initial Bloch state populations, and an interband-coherence-induced part
compromised by dephasing, both of them being independent of the pumping time
scale. The overall pumping reflects an interplay of the band topology, initial
state populations, initial state coherence, and dephasing. Theoretical results
are carefully checked in a Chern insulator model coupled to a pure-dephasing
environment, providing a useful starting point to understand and coherently
control quantum adiabatic pumping in general situations.
