  We study the dynamics of a quantum two state system driven through an avoided
crossing under the influence of a super Ohmic environment, i.e. a longitudinal
as well as a transversal one. The crossing time window, in which relaxation
influences the dynamics, is centered around the avoided crossing. We determine
the dynamics and the Landau-Zener probability employing the numerical exact
quasi-adiabatic path integral. At weak coupling we show that the numerically
less demanding nonequilibrium Bloch equations provide an accurate description.
The crossing time depends strongly not only on the system-bath coupling
strength but also on the bath spectral cut-off frequency in contrast to the
situation in an Ohmic bath. Our results enable to design quantitative protocols
which drive quantum systems out of the influence range of relaxation.
