A group theoretic treatment of the geometric phase

J. Anandan, Leo Stodolsky

A group-theoretic treatment is given of the new phase found by Berry in the adiabatic evolution of a quantum-mechanical system in a finite-dimensional Hilbert space. It is shown how the Berry phases for the various eigenstates of the Hamiltonian are obtained from a set of angles associated with a group element. For the special case of a two-level system there is just one such angle which corresponds to the holonomy transformation associated with parallel transport around a closed curve on a sphere.

M. J. Rave, J. E. Lawson

J. Anandan, Yakir Aharonov

The geometric phase in quantum mechanics is formulated for charged particles in a gauge-invariant, geometric manner. It is then extended to an evolution resulting from a sequence of measurements as in the work of Pancharatnam and Aharonov and Vardi. Its close connection to the Feynman formulation of quantum mechanics is pointed out. The geometric angles, which are generalizations of the classical, adiabatic angles introduced by Hannay and the quantum, adiabatic angles introduced by Anandan and Stodolsky in their group-theoretic treatment of Berry's phase, are studied in quantum and classical physics. The geometric phase for a quantum spin in a magnetic field due to a second particle is obtained using the quantum reference frame defined by the latter. The question of whether the geometric phase and angles are local or nonlocal and their relationship to the electromagnetic and gravitational phases are also discussed.

Armin Uhlmann

Yuji Hasegawa, Michael Zawisky, H. Rauch +1 more

A geometric phase factor is derived for a split-beam experiment as an example of cyclic evolutions. Poincar\'e 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. \textcopyright{} 1996 The American Physical Society.

Arun Kumar Pati