  Garrison and Wright showed that upon undergoing cyclic quantum evolution a
meta-stable state acquires both a geometric phase and a geometric decay
probability. This is described by a complex geometric ``phase'' associated with
the cyclic evolution of two states and is closely related to the two state
formalism developed by Aharonov et al.. Applications of the complex geometric
phase to the Born--Oppenheimer approximation and the Aharonov--Bohm effect are
considered. A simple experiment based on the optical properties of absorbing
birefringent crystals is proposed.
