  Previously, we have shown that the transition probability of the Landau-Zener
problem in periodic lattice systems becomes large by taking into account the
nonlinearity of the energy spectra, compared with the probability by the
conventional Landau-Zener formula. The enhancement comes from the nonlinearity
peculiar to the periodic lattice system, and this effect from the lattice on
transition action cannot be neglected in the transition process.
  In the present paper, we first give a brief review of the previous work, and
construct the transfer matrix of the Landau-Zener problem by the semiclassical
description for lattice systems. Next, we study a ladder lattice system and
show that the transition action obtains a phase due to the nonlinearity. Then,
we consider the double-passage problem of the ladder system within the
semiclassical description. We find the oscillation of the probability by the
transition phase by the lattice effect. This phase comes from the semiclassical
analysis unlike the Stokes phase, and we show that the oscillation is mainly
contributed by the transition phase by the lattice effect, when the
hybridization of the ladder is strong.
