Localization of ultracold atoms in Zeeman lattices with incommensurate spin-orbit coupling

We consider a particle governed by one-dimensional Hamiltonian in which artificial periodic spin-orbit coupling and Zeeman lattice have incommensurate periods. Using best rational approximations to such quasiperiodic Hamiltonian, the problem is reduced description of spinor states superlattice. In absence constant splitting, system acquires an additional symmetry, hinders localization. However, if lattices are deep enough, then localized can appear even for field with zero or small mean value. Spatial distribution modes nearly uniform directly related topological properties effective superlattice: center-of-mass coordinates determined Zak phases computed from superlattice band structure. The feature `memory' effect: each approximation holds information about energies spatial obtained under preceding, less accurate approximations. Dispersion low-energy initial wavepackets characterized law $\propto t^\beta$ $\beta$ varying between $1/2$ at stage $1$ longer, but still finite-time, evolution. dynamics wavepackets, exciting mainly modes, manifests quantum revivals.