Geometric phase in coupled neutron interference loops.

A geometric phase factor is derived for a split-beam experiment as an example of cyclic evolutions. Poincaré sphere descriptions for the split-beam experiment show its geometric property. We observe this geometric phase with a two-loop neutron interferometer, where a reference beam can be added to the beam from one interference loop. The combination of phase shifters and partial absorbers permitted the compensation of the dynamical phase and measurement of the geometric phase. All the experimental results show complete agreement with our theoretical treatment. We discuss a situation where a geometric phase exists even when the dynamical phase becomes zero.