  We demonstrate that Berry phases may greatly affect the dynamics of
spin-orbit coupled Bose-Einstein condensates. The effective model Hamiltonian
under consideration is shown to be equivalent to the Exe Jahn-Teller model
first introduced in molecular physics. The corresponding conical intersection
is identified and the Berry phase acquired for a wave packet encircling the
intersection studied. It is found that this phase manifests itself in the
density profile of the condensate, making it a directly measurable quantity via
time-of-flight detection. Moreover, the non-Abelian gauge structure of the
system is addressed and we verify how it affects the dynamics.
