  Theoretical studies of coherent atom transport have as yet mainly been
restricted to one-dimensional model systems with harmonic trapping potentials.
Here we investigate this important phenomenon -- a prerequisite for a variety
of quantum-technology applications based on cold neutral atoms -- under much
more complex physical circumstances. More specificially yet, we study fast
atomic transport in a moving {\em double-well optical lattice}, whose
three-dimensional (anharmonic) potential is nonseparable in the $x-y$ plane. We
first propose specific configurations of acousto-optic modulators that give
rise to the moving-lattice effect in an arbitrary direction in this plane. We
then determine moving-lattice trajectories that enable single-atom transport
using two classes of quantum-control methods: shortcuts to adiabaticity (STA),
here utilized in the form of inverse engineering based on a
quadratic-in-momentum dynamical invariant of Lewis-Riesenfeld type, and their
recently proposed modification termed enhanced STA (eSTA). Subsequently, we
quantify the resulting single-atom dynamics by numerically solving the relevant
time-dependent Schr\"{o}dinger equations and compare the efficiency of STA- and
eSTA-based transport by evaluating the respective fidelities. We show that --
except for the regime of shallow lattices -- the eSTA method consistently
outperforms its STA counterpart. This study has direct implications for
neutral-atom quantum computing based on collisional entangling two-qubit gates
and quantum sensing of constant homogeneous forces via guided-atom
interferometry.
