Several recent studies have investigated the dynamics of cold atoms in optical lattices subject to AC forcing; the theoretically predicted renormalization of the tunneling amplitudes has been verified experimentally. Recent observations include global motion of the atom cloud, such as giant "Super-Bloch Oscillations" (SBOs). We show that, in order to understand unexplained features of SBOs, in addition to the renormalization of the tunneling, a new and important phase correction must be included. For Fermionic systems with strong attractive interactions, one may engineer different types of collisions and recollisions between bound-pairs and unpaired atoms.

Magnetic domain patterns in a ferromagnetic Bose-Einstein condensate (BEC) show different properties depending on the quadratic Zeeman effect and dissipation. Another important factor that affects domain patterns and domain growth is superfluid flow of atoms. Domain growth in a ferromagnetic BEC with negative quadratic Zeeman energy is characterized by the same growth law as (classical) binary fluid in the inertial hydrodynamic regime. In the absence of the superfluid flow, the domain growth law for negative quadratic Zeeman energy is the same as that of scalar conserved fields such as binary alloys.

We study the spin dynamics of a spin-1 ferromagnetic Bose-Einstein condensate with magnetic dipole-dipole interaction (MDDI) based on the Gross-Pitaevskii and Bogoliubov theories. We find that various magnetic structures such as checkerboards and stripes emerge in the course of the dynamics due to the combined effects of spin-exchange interaction, MDDI, quadratic Zeeman and finite-size effects, and nonstationary initial conditions. However, the short-range magnetic order observed by the Berkeley group [Phys. Rev. Lett. 100, 170403 (2008)] is not fully reproduced in our calculations; the periodicity of the order differs by a factor of 3 and the checkerboard pattern eventually dissolves in the course of time. Possible reasons for the discrepancy are discussed.

Level statistics is discussed for XXZ spin chains with discrete symmetries for some values of the next-nearest-neighbor (NNN) coupling parameter. We show how the level statistics of the finite-size systems depends on the NNN coupling and the XXZ anisotropy, which should reflect competition among quantum chaos, integrability and finite-size effects. Here discrete symmetries play a central role in our analysis. Evaluating the level-spacing distribution, the spectral rigidity and the number variance, we confirm the correspondence between nonintegrability and Wigner behavior in the spectrum. We also show that non-Wigner behavior appears due to mixed symmetries and finite-size effects in some nonintegrable cases.