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arXiv:2301.04359 · Journal of the Optical Society of America A

Deciphering Pancharatnam’s discovery of geometric phase: retrospective

Luis A. Garza-Soto, Nathan A. Hagen, Dorilian Lopez-Mago

Published Mar 22, 2023• 17 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1364/josaa.485485

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The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects

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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.
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
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.
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.