A generalized Pancharatnam geometric phase formula for three level systems

We describe a generalisation of the well known Pancharatnam geometric phase formula for two level systems, to evolution of a three-level system along a geodesic triangle in state space. This is achieved by using a recently developed generalisation of the Poincare sphere method, to represent pure states of a three-level quantum system in a convenient geometrical manner. The construction depends on the properties of the group $SU(3)\/$ and its generators in the defining representation, and uses geometrical objects and operations in an eight dimensional real Euclidean space. Implications for an n-level system are also discussed.