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

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