  Motivated by recent cold atom experiments in optical lattices, we consider a
lattice version of the Landau-Zener problem. Every single site is described by
a Landau-Zener problem, but due to particle tunnelling between neighboring
lattice sites this onsite single particle Landau-Zener dynamics couples to the
particle motion within the lattice. The lattice, apart from having a dephasing
effect on single site Landau-Zener transitions, also implies, in the presence
of a confining trap, an inter-site particle flow induced by the Landau-Zener
sweeping. This gives rise to an interplay between intra- and inter-site
dynamics. The adiabaticity constrain is therefor not simply given by the
standard one; the Hamiltonian rate of change relative to the gap of the onsite
problem. In experimentally realistic situations, the full system evolution is
well described by Franck-Condon physics, e.g. non-adiabatic excitations are
predominantly external ones characterized by large phononic vibrations in the
atomic cloud, while internal excitations are very weak as close to perfect
onsite transitions take place.
