  We study the time evolution of ultra-cold atoms in an accelerated optical
lattice. For a Bose- Einstein condensate with a narrow quasi-momentum
distribution in a shallow optical lattice the decay of the survival probability
in the ground band has a step-like structure. In this regime we establish a
connection between the wave function renormalization parameter Z introduced in
[Phys. Rev. Lett. 86, 2699 (2001)] to characterize non-exponential decay and
the phenomenon of resonantly enhanced tunneling, where the decay rate is peaked
for particular values of the lattice depth and the accelerating force.
