  We show that the adiabatic dynamics of a Bose-Einstein condensate (BEC) in a
double well potential can be described in terms of a dark variable resulting
from the combination of the population imbalance and the spatial atomic
coherence between the two wells. By means of this dark variable, we extend, to
the non-linear matter wave case, the recent proposal by Vitanov and Shore
[Phys. Rev. A 73, 053402 (2006)] on adiabatic passage techniques to coherently
control the population of two internal levels of an atom/molecule. We
investigate the conditions to adiabatically split or transport a BEC as well as
to prepare an adiabatic self trapping state by the optimal delayed temporal
variation of the tunneling rate via either the energy bias between the two
wells or the BEC non-linearity. The emergence of non-linear eigenstates and
unstable stationary solutions of the system as well as their role in the
breaking down of the adiabatic dynamics is investigated in detail.
