  We report a theoretical description of the synthetic momentum-state lattices
with a 3D Gross-Pitaevskii equation (GPE), where both the external trap
potential and the mean-field spatial-density-dependent many-body interactions
are naturally included and exactly treated. The GPE models exhibit better
performance than the tight-binding model to depict the experimental
observations. Since the trap modifies the dispersion relation for free
particles and shapes the spatial density distribution that leads to
inhomogeneous interactions, decoherences (damping oscillation) appear even for
a short-time evolution. Our parametric calculations for the two-state
oscillation suggest that we should work with a relatively shallow trap in the
weakly interacting regime, especially when the long-term dynamics are
concerned. The impact of the mean-field interaction, i.e., the self-trapping
behavior, on the transport dynamics and the topological phase transition in a
finite multiple-state lattice chain is also specifically investigated. Such an
accurate treatment of the inhomogeneous interactions allows for further
investigations on the interplay with disorder, the pair correlation dynamics,
and the thermalization process in momentum space.
