Bloch-Landau-Zener oscillations in a quasi-periodic potential

Bloch oscillations and Landau-Zener tunneling are ubiquitous phenomena which are sustained by a band-gap spectrum of a periodic Hamiltonian and can be observed in dynamics of a quantum particle or a wavepacket in a periodic potential under action of a linear force. Such physical setting remains meaningful for aperiodic potentials too, although band-gap structure does not exist anymore. Here we consider the dynamics of noninteracting atoms and Bose-Einstein condensates in a quasi-periodic one-dimensional optical lattice subjected to a weak linear force. Excited states with energies below the mobility edge, and thus localized in space, are considered. We show that the observed oscillatory behavior is enabled by tunneling between the initial state and a state (or several states) located nearby in the coordinate-energy space. The states involved in such Bloch-Landau-Zener oscillations are determined by the selection rule consisting of the condition of their spatial proximity and condition of quasi-resonances occurring at avoided crossings of the energy levels. The latter condition is formulated mathematically using the Gershgorin circle theorem. The effect of the inter-atomic interactions on the dynamics can also be predicted on the bases of the developed theory. The reported results can be observed in any physical system allowing for observation of the Bloch oscillations, upon introducing incommensurablity in the governing Hamiltonian.