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Journal of Modern Optics

Geometric Phase in Optics

Suresh Chandra Tiwari

Published May 1, 1992• 38 citations
Publisher

Abstract

A detailed critique on the classical and quantum descriptions of the geometrical phase in optics is presented. It is argued that both explanations are inadequate to understand the origin of the geometric phases. Significance of Byrne's analysis for describing the Pancharatnam phase in pointed out. A physical mechanism for the geometrical phase based on the angular momentum transfer is proposed, and a tentative approach utilising the gauge non-invariant angular momentum is suggested.

DOI
10.1080/09500349214551101

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A detailed critique on the classical and quantum descriptions of the geometrical phase in optics is presented. It is argued that both explanations are inadequate to understand the origin of the geometric phases. Significance of Byrne's analysis for describing the Pancharatnam phase in pointed out. A physical mechanism for the geometrical phase based on the angular momentum transfer is proposed, and a tentative approach utilising the gauge non-invariant angular momentum is suggested.
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Recently the physical mechanism for geometric phase in optics has been elucidated in terms of the angular momentum holonomy proposed in 1992. Aharonov and Kaufherr (PRL, 92, 070404, 2004) revisit the Aharonov-Bohm effect, and propose non-local exchange of a conserved, gauge invariant quantity that changes the modular momentum of the particle that is responsible for the AB phase shift. We suggest that the net angular momentum shifts proposed for GP may be analogous to the shift in the modulus momentum for the AB effect. At a single photon or electron level such non-trivial, geometric effects seem to hint at a new physics.
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.
Spin coherent states, proposed by several workers in the early 70s, are shown to contain the natural generalization to higher spins of the phase discovered earlier by Pancharatnam for polarized light. They also provide a simple setting in which the geometric phase of Berry and of Aharonov and Anandan can be exhibited and some of its further properties, noted by Samuel and Bhandari, can be illustrated. It is pointed out how many of the foregoing results might have been deduced in the 70s by proceeding in strict analogy with the earlier work of Pancharatnam. A similar point is made in connection with the geometric phase of SU(1,1) coherent states. In addition to its possible historical interest, this approach shows how the concept of the geometric phase can be introduced and developed in a largely kinematic fashion as an addendum to the standard quantum theory of angular momentum.