Bose-Einstein Condensates in Time-Dependent Light Potentials: Adiabatic   and Nonadiabatic Behavior of Nonlinear Wave Equations
  The criteria for validity of adiabaticity for nonlinear wave equations are considered within the context of atomic matter-waves tunneling from macroscopically populated optical standing-wave traps loaded from a Bose-Einstein condensate. We show that even when the optical standing wave is slowly turned on and the condensate behaves adiabatically during this turn-on, once the tunneling-time between wells in the optical lattice becomes longer than the nonlinear time-scale, adiabaticity breaks down and a significant spatially varying phase develops across the condensate wave function from well to well. This phase drastically affects the contrast of the fringe pattern in Josephson-effect interference experiments, and the condensate coherence properties in general. 
Y.B. Band and Marek Trippenbach