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.
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