  We study how well magnetic models can be implemented with ultracold bosonic
atoms of two different hyperfine states in an optical superlattice. The system
is captured by a two-species Bose-Hubbard model, but realizes in a certain
parameter regime actually the physics of a spin-1/2 Heisenberg magnet,
describing the second order hopping processes. Tuning of the superlattice
allows for controlling the effect of fast first order processes versus the
slower second order ones.
  Using the density-matrix renormalization-group method, we provide the
evolution of typical experimentally available observables. The validity of the
description via the Heisenberg model, depending on the parameters of the
Hubbard model, is studied numerically and analytically. The analysis is also
motivated by recent experiments [S. Foelling et al., Nature 448, 1029 (2007);
S. Trotzky et al., Sience 319, 295 (2008)] where coherent two-particle dynamics
with ultracold bosonic atoms in isolated double wells were realized. We provide
theoretical background for the next step, the observation of coherent
many-particle dynamics after coupling the double wells. Contrary to the case of
isolated double wells, relaxation of local observables can be observed. The
tunability between the Bose-Hubbard model and the Heisenberg model in this
setup could be used to study experimentally the differences in equilibration
processes for nonintegrable and Bethe ansatz integrable models. We show that
the relaxation in the Heisenberg model is connected to a phase averaging
effect, which is in contrast to the typical scattering driven thermalization in
nonintegrable models. We discuss the preparation of magnetic groundstates by
adiabatic tuning of the superlattice parameters.
