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Journal of Modern Optics

The Geometric Phase

P. Hariharan, Maitreyee Roy, Peter A. Robinson, J. W. O'Byrne

Published May 1, 1993• 10 citations
Publisher

Abstract

The change in the phase of a beam of light produced by a cyclic change in its state of polarization is an example of the geometric phase that can be verified by interferometric measurements. This paper presents a quantum-field theoretical analysis of the geometric phase interferometer in the limit of a small photon number, as well as some experimental results that confirm that the optical effects due to the geometric phase persist down to the single-photon level.

DOI
10.1080/09500349314550881

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The change in the phase of a beam of light produced by a cyclic change in its state of polarization is an example of the geometric phase that can be verified by interferometric measurements. This paper presents a quantum-field theoretical analysis of the geometric phase interferometer in the limit of a small photon number, as well as some experimental results that confirm that the optical effects due to the geometric phase persist down to the single-photon level.
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.
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.
The geometric phase, originating from the cyclic evolution of a state, such as polarization on the Poincaré sphere, is typically measured through interferometric approaches that often include unwanted contributions from the dynamic phase. Here, we present a non-interferometric technique based on quantum correlation of pair photons to measure the geometric phase of a classical beam. The transfer of geometric phase of the classical pump beam arising from the cyclic evolution of its polarization state on the Poincaré sphere onto the polarization-entangled pair photons generated via spontaneous parametric down-conversion in a Sagnac interferometer enables easy control over the quantum state. Characterization of the generated quantum states reveals that the geometric phase of the pump beam controls the coincidence counts, entanglement visibility, Bell's parameter, quantum state tomography, and fidelity in close agreement with theoretical predictions. We observe sinusoidal modulation of the Bell's parameter and state fidelity with changes in the geometric phase, resulting in transitions between orthogonal Bell states and Bell-like maximally entangled states. Our results establish the geometric phase of the classical pump as a tunable parameter for quantum state control, offering a compact, passive platform for phase manipulation in quantum photonic systems, enabling geometric phase-based quantum gates, and compensating unwanted phase acquired by the quantum state on propagation.
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.