  We study Landau-Zener interband transitions for a non-symmetric optical
lattice in the presence of an external force. We show that gain and losses of
the light beam, as well as the relative occupation probabilities of the bands
involved in the transitions can be accurately managed upon tuning the amplitude
of the non-Hermitian component of the lattice. Exact expressions for the
transition and non-transition probabilities for a non-symmetric system obtained
within a two-mode approximation are provided. These equations successfully
account for the main features of the transitions in the optical lattice. We
also interpret the non-conventional Bloch oscillations at criticality studied
in Phys. Rev. Lett. 103, 123601 (2009) as a series of a Landau-Zener
transitions.
