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Physical Review Letters

Observation of topological phase by use of a laser interferometer

Rajendra Bhandari, Joseph Samuel

Published Mar 28, 1988• 234 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.60.1211

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Journal of Modern Optics•May 1, 1998•13 citations
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.
The wave function of a quantum system may acquire an additional phase factor (the geometric phase), when its parameters undergo a cyclic change. An optical manifestation of the geometric phase is the change in the phase of a light wave produced when its state of polarization is taken around a circuit on the Poincare sphere. This phase change can be verified by interferometric measurements. We describe an interferometer in which the two beams traverse the same path in opposite directions, so that their optical paths are always equal, and the phase difference between them is varied by operating on the geometric phase. Measurements made with this interferometer, at a light level at which the probability of more than one photon being present simultaneously in the interferometer was negligibly small, confirmed that the interference effects observed due to the geometric phase persist down to the single photon level.
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.
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
In crystal optics the concept of the geometric phases can be demonstrated to the naked eye with comparatively simple apparatus. Pancharatnam's phase is described by an area on the Poincaré sphere which is enclosed by geodesic lines (great circles) while the concept of the Berry phase does not restrict the nature of the contour. With a simple experiment we demonstrate that Pancharatnam's phase is accompanied by a second phase which compensates the pure area effect for areas enclosed by small circles. This compensating phase changes sign at time reversal and, thus, exhibits the features of a geometric phase.