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Journal of Modern Optics

A simple demonstration of the Pancharatnam phase as a geometric phase

P. Hariharan, Sushil A. Mujumdar, Hema Ramachandran

Published Jul 1, 1999• 9 citations
Publisher

Abstract

To demonstrate the Pancharatnam phase as a geometric (Berry's) phase, each polarization state must be obtained by projecting the previous state on it. We describe a simple interferometric arrangement for such a demonstration which only uses a single rotation linear analyser to introduce a continuously variable phase difference between the two beams.

DOI
10.1080/09500349908231346

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To demonstrate the Pancharatnam phase as a geometric (Berry's) phase, each polarization state must be obtained by projecting the previous state on it. We describe a simple interferometric arrangement for such a demonstration which only uses a single rotation linear analyser to introduce a continuously variable phase difference between the two beams.
The geometric phase was first anticipated 1956 by Pancharatnam, in the context of polarised light. Much later, in 1984, Berry made an analogue discovery, but this time in the context of quantum mechanics. It turns our that the geometric phase, or Berry's phase, or the Pancharatnam phase, is a very general phenomenon. It is a phase shift that can be acquired by a system, when it has undergone a adiabatic process, bringing it back to its original initial conditions. If the parameter space of the involved Hamiltonian is curved, the final condition may involve a phase change. This effect has been demonstrated in a range of different interferometry experiments. Although the Pancharatnam phase has been seen, this has always been in studies involving the first order correlation, g(1), this is in the intereferogramm of two or more wave amplitudes. Here, we demonstrate experimental proofs of the effect of the Pancharatnam phase in intensity interferometry, or in other words, where the second order correlation factor g(2) is the relevant description of the interference.
Young’s dual-pinhole interference experiment with arbitrary fully correlated and polarized vector light fields leads to a Pancharatnam–Berry geometric phase that is related to the associated dynamical phase. We demonstrate theoretically and experimentally how the dynamical phase across the interference pattern can be deciphered from the total phase, thereby leaving only the geometric phase. Our results constitute the first genuine interferometric phase measurements that yield the Pancharatnam–Berry geometric phase in Young’s two-beam interference setup.
We present a novel interferometric arrangement that makes it possible to measure with great versatility geometric phases produced in polarization states of classical light. Our arrangement is robust against thermal and mechanical disturbances and can be set up in a Mach-Zehnder, a Michelson or a Sagnac configuration. We present results concerning the geometric phase as an extension of previous measurements of the Pancharatnam, or total phase. The geometric phase is obtained by compensating the dynamical contribution to the total phase, so as to extract out of it a purely geometric phase. This can be achieved over trajectories on the Poincare sphere that are not necessarily restricted to be great circles (geodesics). We thus demonstrate the feasibility of our method for dynamical extraction of the geometric contribution to the total phase, a prerequisite for building geometric quantum gates. Although our results correspond to polarization states of classical light, the same methodology could be applied in the case of polarization states of single photons.
While Pancharatnam discovered the geometric phase in 1956, his work was not widely recognized until its endorsement by Berry in 1987, after which it received wide appreciation. However, because Pancharatnam's paper is unusually difficult to follow, his work has often been misinterpreted as referring to an evolution of states of polarization, just as Berry's work focused on a cycle of states, even though this consideration does not appear in Pancharatnam's work. We walk the reader through Pancharatnam's original derivation and show how Pancharatnam's approach connects to recent work in geometric phase. It is our hope to make this widely cited classic paper more accessible and better understood.
Abstract The geometric phase for classical electromagnetic light beams, in its original formulation as introduced by Pancharatnam, concerns fields experiencing cyclic, discrete in-phase polarization-state changes. A similar phase was later recognized by Berry to govern the behavior of adiabatic quantum systems, with consequent extensions to nonadiabatic and noncyclic evolutions of the quantum state. However, no optical counterpart for the noncyclic, adiabatic (continuous) evolution has been demonstrated. Here we employ a modified Young’s two-pinhole setup with controlled pinhole polarizations and intensities to generate on interference an arbitrary continuous spatial evolution of the polarization state, an optical analogue to the adiabatic case. The customized arrangement allows separating at any point the accumulated dynamical and geometric phases from the total phase, enabling a detailed study of the noncyclic Pancharatnam–Berry phase in a continuous transformation. Our theoretical and experimental results are in excellent agreement and consistent with the geodesic rule for noncyclic evolutions.