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Optics Letters

Measurement of the Pancharatnam phase for a light beam

Thomas H. Chyba, R. Simon, L. J. Wang, L. Mandeļ

Published Jul 1, 1988• 127 citations
Publisher

Abstract

By using a Michelson interferometer arrangement, we have measured the phase shift (the Pancharatnam phase) experienced by a light beam whose state of polarization is made to follow, by successive unitary transformations, a closed circuit on the Poincaré sphere. We confirm that the phase shift equals half of the solid angle subtended by the closed circuit at the center of the sphere. This phase is closely related to the topological Berry phase.

DOI
10.1364/ol.13.000562

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By using a Michelson interferometer arrangement, we have measured the phase shift (the Pancharatnam phase) experienced by a light beam whose state of polarization is made to follow, by successive unitary transformations, a closed circuit on the Poincaré sphere. We confirm that the phase shift equals half of the solid angle subtended by the closed circuit at the center of the sphere. This phase is closely related to the topological Berry phase.
We report a direct experimental observation of Pancharatnam's phase, which is closely related to Berry's topological phase. The experiment involves measurement of the phase change in one beam of a laser interferometer as the polarization state of light is taken along a closed circuit on the Poincar\'e sphere. The phase change is found to be equal to half the solid angle subtended by the circuit at the center of the Poincar\'e sphere. Apart from providing a striking demonstration of the topological phase, the experiment demonstrates that unitary time evolution of a system is not essential for the appearance of the topological phase.
We study two types of topological phases, i.e., Aharonov-Bohm phases, by means of optical interferometry. The first arises from a cycle of redirections of the photon spin and is closely related to the Aharonov-Anandan phase. The second arises from a cycle of polarization changes on the Poincare sphere and was discovered by Pancharatnam. Both of these phases, and their combined effects, have been investigated by a nonplanar Mach-Zehnder interferometer. The signs of the fringe shifts arising from these topological effects have been determined and agree with theory.
We report on theoretical calculations and experimental observations of Pancharatnam's phase originating from arbitrary SU(2) transformations applied to polarization states of light. We have implemented polarimetric and interferometric methods, which allow us to cover the full Poincar\'e sphere. As a distinctive feature, our interferometric array is robust against mechanical and thermal disturbances, showing that the polarimetric method is not inherently superior over the interferometric one, as previously assumed. Our strategy effectively amounts to feeding an interferometer with two copropagating beams that are orthogonally polarized with respect to each other. It can be applied to different types of standard arrays, such as a Michelson, a Sagnac, or a Mach-Zehnder interferometer. We exhibit the versatility of our arrangement by performing measurements of Pancharatnam's phases and fringe visibilities that closely fit the theoretical predictions. Our approach can be easily extended to deal with mixed states and to study decoherence effects.
It has been shown that measurements involving indefinite causal order can be superior to those in which a sequence of operations occurs in a specified order. In optics, such measurements are realized naturally in a Sagnac interferometer. We show that such an arrangement can measure the solid angle (on the Poincaré sphere) enclosed by a sequence of unitary transformations of the polarization. This is the Pancharatnam-Berry phase. Extension from the classical or single-photon treatment to a fully quantized treatment allows the analysis of the interferometer for arbitrary quantum states of light.This article is part of the theme issue 'The quantum theory of light'.
In 1984 Berry addressed a quantum system undergoing a unitary and cyclic evolution under the action of a time-dependent Hamiltonian (M. V. Berry, 1984). The process was supposed to be adiabatic, meaning that the time scale of the system’s evolution was much shorter than the time scale of the changing Hamiltonian. Until Berry’s study, it was assumed that for a cyclic Hamiltonian the quantum state would acquire only so-called dynamical phases, deprived of physical meaning. Such phases could be eliminated by redefining the quantum state through a “gauge” transformation of the form |ψ〉 → eiα |ψ〉. However, Berry discovered that besides the dynamical, there was an additional phase that could not be “gauged away” and whose origin was geometric or topological. It depended on the path that |ψ〉 describes in the parameter space spanned by those parameters to which the Hamiltonian owed its time dependence. Berry’s discovery was the starting point for a great amount of investigations that brought to light topological aspects of both quantum and classical systems. Berry’s phase was soon recognized as a special case of more general phases that showed up when dealing with topological aspects of different systems. For example, the Aharonov-Bohm phase could be understood as a geometric phase. The rotation angle acquired by a parallel-transported vector after completing a closed loop in a gravitationally curved space-time region, is also a geometric, Berry-like phase. Another example is the precession of the plane of oscillation of a Foucault pendulum. Berry’s original formulation was directly applicable to the case of a spin-1/2 system evolving under the action of a slowly varying magnetic field that undergoes cyclic changes. A spin-1/2 system is a special case of a two-level system. Another instances are two-level atoms and polarized light, so that also in these cases we should expect to find geometric phases. In fact, the first experimental test of Berry’s phase was done using polarized, classical light (A. Tomita, 1986). Pancharatnam (S. Pancharatnam, 1956) anticipated Berry’s phase when he proposed, back in 1956, how to decide whether two polarization states are “in phase”. Pancharatnam’s prescription is an operational one, based upon observing whether the intensity of the interferogram formed by two polarized beams has maximal intensity. In that case, the two polarized beams are said to be “in phase”. Such a definition is analogous to the definition of distant parallelism in differential geometry. Polarized states can be subjected to different transformations which could be cyclic or not, adiabatic or not, unitary or not. The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects
Photon pairs produced by type-I and type-II parametric down-conversion are passed through a Michelson interferometer. Two quarter-wave plates in one arm of the interferometer allow variation of the polarization state of the photons. We investigate experimentally the geometric phase or Pancharatnam phase acquired by single photons and photon pairs dependent on the solid angle that is subtended by the circuit that represents the varying state of polarization on the Poincar\'e sphere. It is found that the geometric phase acquired by the pair depends on the initial polarization state of the two photons. If both photons are in equal states of linear polarization, we observe a doubling of the geometric phase compared to single photons; in the case of orthogonal states of linear polarization, the geometric phase is completely canceled. Our results show the role of Pancharatnam's phase in nonclassical two-photon interference phenomena and the interplay between the geometric phase and the dynamical phase in these phenomena.