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Physical Review A

Berry phases for partial cycles

Thomas F. Jordan

Published Aug 1, 1988• 65 citations
Publisher

Abstract

A simple definition of Berry phases based on the Pancharatnam phase convention is stated in the language of ordinary quantum mechanics. It is shown to be equivalent to previous definitions for a cycle of transformations built continuously from infinitesimal transformations. It also applies to discrete transformations and partial cycles. The phase for a partial cycle can be the same as for the complete cycle. This suggests possibilities for simpler experiments. In an example considered here, each part of a split beam of light is put through two half-wave plates. The light is not brought back to the original polarization state, but interference produced by the Berry phase can be observed because the different parts of the beam are brought to the same polarization state.

DOI
10.1103/physreva.38.1590

Related papers

The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects

Francisco De

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

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We extend the concepts of Pancharatnam’s connection and phase difference of optical beams to cover random, stationary, partially polarized quasimonochromatic light. The geometric and dynamic structures of the phase are assessed within the framework of classical statistical polarization theory when the beam undergoes a cyclic, unitary polarization-state evolution. The existence of the two phase types is confirmed experimentally by conducting an interferometric measurement where the phase changes are found from the variations in the position of an intensity fringe pattern. The formalism also leads to a definition of the geometric phase for partially polarized beams, which is consistent with the results on mixed (qubit) quantum states.

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According to Pancharatnam [S. Pancharatnam, in Geometric Phases in Physics, edited by A. Shapere and F. Wilczek (World Scientific, Singapore, 1989)], two light beams of differing elliptical polarization can be said to be ``in phase'' if their interference is maximally constructive. In the same spirit, the relative phase between two beams a and b is the amount by which the phase of beam b needs to be changed to put it in phase with beam a. Berry has established a simple relation between Pancharatnam's concept and a phase relation between two quantum-mechanical states: If two nonorthogonal states \ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{a}}$〉 and \ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{b}}$〉 are phased in such a way that \ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{a}}$〉+\ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{b}}$〉 has maximum norm, i.e., such that 〈${\mathrm{\ensuremath{\psi}}}_{\mathit{a}}$\ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{b}}$〉 is real and positive, they can be said to be in phase or parallel, and in general the relative phase between them is the phase angle of 〈${\mathrm{\ensuremath{\psi}}}_{\mathit{a}}$\ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{b}}$〉. This concept plays a role in the theory of the Abelian geometric phase. In the non-Abelian case, one has two n-dimensional subspaces instead of two single states, and instead of choosing the phase of each state one has to choose an orthonormal basis (including phase) in each subspace. We show that the concept of ``block diagonalization'' introduced by Cederbaum, Shirmer, and Meyer [J. Phys. A 22, 2427 (1989)] in the context of a molecular problem provides a natural generalization of the Pancharatnam-Berry phase relation.

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Measurement of the Pancharatnam phase for a light beam

Thomas H. Chyba, R. Simon, L. J. Wang +1 more

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.

Optics Letters•Jul 1, 1988•127 citations
A simple definition of Berry phases based on the Pancharatnam phase convention is stated in the language of ordinary quantum mechanics. It is shown to be equivalent to previous definitions for a cycle of transformations built continuously from infinitesimal transformations. It also applies to discrete transformations and partial cycles. The phase for a partial cycle can be the same as for the complete cycle. This suggests possibilities for simpler experiments. In an example considered here, each part of a split beam of light is put through two half-wave plates. The light is not brought back to the original polarization state, but interference produced by the Berry phase can be observed because the different parts of the beam are brought to the same polarization state.
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
We extend the concepts of Pancharatnam’s connection and phase difference of optical beams to cover random, stationary, partially polarized quasimonochromatic light. The geometric and dynamic structures of the phase are assessed within the framework of classical statistical polarization theory when the beam undergoes a cyclic, unitary polarization-state evolution. The existence of the two phase types is confirmed experimentally by conducting an interferometric measurement where the phase changes are found from the variations in the position of an intensity fringe pattern. The formalism also leads to a definition of the geometric phase for partially polarized beams, which is consistent with the results on mixed (qubit) quantum states.
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.
According to Pancharatnam [S. Pancharatnam, in Geometric Phases in Physics, edited by A. Shapere and F. Wilczek (World Scientific, Singapore, 1989)], two light beams of differing elliptical polarization can be said to be ``in phase'' if their interference is maximally constructive. In the same spirit, the relative phase between two beams a and b is the amount by which the phase of beam b needs to be changed to put it in phase with beam a. Berry has established a simple relation between Pancharatnam's concept and a phase relation between two quantum-mechanical states: If two nonorthogonal states \ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{a}}$〉 and \ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{b}}$〉 are phased in such a way that \ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{a}}$〉+\ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{b}}$〉 has maximum norm, i.e., such that 〈${\mathrm{\ensuremath{\psi}}}_{\mathit{a}}$\ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{b}}$〉 is real and positive, they can be said to be in phase or parallel, and in general the relative phase between them is the phase angle of 〈${\mathrm{\ensuremath{\psi}}}_{\mathit{a}}$\ensuremath{\Vert}${\mathrm{\ensuremath{\psi}}}_{\mathit{b}}$〉. This concept plays a role in the theory of the Abelian geometric phase. In the non-Abelian case, one has two n-dimensional subspaces instead of two single states, and instead of choosing the phase of each state one has to choose an orthonormal basis (including phase) in each subspace. We show that the concept of ``block diagonalization'' introduced by Cederbaum, Shirmer, and Meyer [J. Phys. A 22, 2427 (1989)] in the context of a molecular problem provides a natural generalization of the Pancharatnam-Berry phase relation.