  We propose a two-dimensional (2D) version of Thouless pumping that can be
realized by using ultracold atoms in optical lattices. To be specific, we
consider a 2D square lattice tight-binding model with an obliquely introduced
superlattice. It is demonstrated that quantized particle transport occurs in
this system, and that the transport is expressed as a solution of a Diophantine
equation. This topological nature can be understood by mapping the Hamiltonian
to a three-dimensional (3D) cubic lattice model with a homogeneous magnetic
field. We also propose a continuum model with obliquely introduced superlattice
and obtain the amount of pumping by calculating the Berry curvature. For this
model, the same Diophantine equation can be derived from the plane-wave
approximation. Furthermore, we investigate the effect of a harmonic trap by
solving the time-dependent Schr\"odinger equation. Under a harmonic trap
potential, as often used in cold atom experiments, we show, by numerical
simulations, that nearly quantized pumping occurs when the phase of the
superlattice potential is driven at a moderate speed. Also, we find that two
regions appear, the Hofstadter region and the rectifying region, depending on
the modulation amplitude of the superlattice potential. In the rectifying
region with larger modulation amplitudes, we uncover that the pumping direction
is restricted to exactly the $x$-axis or the $y$-axis direction. This
difference in these two regions causes a crossover behavior, characterizing the
effect of the harmonic trap.

Quantum Gases;Mesoscale and Nanoscale Physics;Quantum Physics