We investigated Berry's phase for noncyclic evolution using the rotation of a polarization azimuth of linearly polarized light in a partially wound optical fiber over the surface of a cylinder. Using a rotation gauge around the rotation axis, the observed rotation of the polarization azimuth corresponds to the area of a spherical rectangle over the surface enclosed by the contour
C of actual evolution, a large circle on the equator, and a longitudinal line connecting them, whereas the rotation of the polar gauge encloses a spherical triangle connecting the zenith of the sphere. The observed values were converted to Berry's phase by transformation from the rotation gauge to the geodesic gauge. Consequently, we could confirm that Berry's phase for a noncyclic evolution is given by the geodesic rule proposed by Samuel and Bhandari.
A simple definition of Berry phases based on the Pancharatnam phase convention is stated in the language of ordinary quantum mechanics. It is shown to be equivalent to previous definitions for a cycle of transformations built continuously from infinitesimal transformations. It also applies to discrete transformations and partial cycles. The phase for a partial cycle can be the same as for the complete cycle. This suggests possibilities for simpler experiments. In an example considered here, each part of a split beam of light is put through two half-wave plates. The light is not brought back to the original polarization state, but interference produced by the Berry phase can be observed because the different parts of the beam are brought to the same polarization state.
The Berry phases for a spherically symmetric atom in a circularly polarized semiclassical radiation field are calculated using an operator decomposition scheme. The author then takes the two-level atomic limit and recovers the Berry phases already calculated by other means. His attention is then focused on the two-level atom. He shows that in the semiclassical problem the Rabi oscillations do not arise from cyclic wavefunctions. Hence any comparison of the phases of the initial and final states must use the Pancharatnam connection. In the quantum model he no longer gets perfect Rabi oscillations as there are partial collapses and revivals. These collapses and revivals are exploited in alternative quantum optical models, the Raman coupling model and intensity-dependent coupling model, for which they are exactly cyclic. Again he finds that he must invoke the Pancharatnam connection.