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arXiv:2610.01540

Measuring Geometric Phase based on Indefinite Causal Order in a Sagnac Interferometer

Lucas Marques Fagundes, Finlay Campbell, Richard Aguiar Maduro, Renné Medeiros de Araújo, Stephen Mark Barnett, Croke, Sarah, Sonja Franke‐Arnold

Published Oct 1, 2026• 0 citations
arXivHTMLPDF

Abstract

Sagnac interferometers are important tools for precision measurements. Cancellation of common-path noise ensures a high degree of stability against external perturbations, while their sensitivity to rotations provides the basis of laser gyroscopes. Here we highlight another, less explored feature of Sagnac interferometers: the fact that optical elements are traversed in reverse order for clockwise and counterclockwise propagating light components. This provides the opportunity to explore a classical realization of indefinite causal order, where the order of events within a sequence is not fixed. We explore this concept to identify the geometric phase associated with two non-commuting polarization operations with a single measurement. This idea may have applications for the rapid determination of polarization manipulations or optical activity within chiral media, but foremost it provides a geometric illustration of indefinite causal order.

DOI
10.48550/arxiv.2610.01540

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Sagnac interferometers are important tools for precision measurements. Cancellation of common-path noise ensures a high degree of stability against external perturbations, while their sensitivity to rotations provides the basis of laser gyroscopes. Here we highlight another, less explored feature of Sagnac interferometers: the fact that optical elements are traversed in reverse order for clockwise and counterclockwise propagating light components. This provides the opportunity to explore a classical realization of indefinite causal order, where the order of events within a sequence is not fixed. We explore this concept to identify the geometric phase associated with two non-commuting polarization operations with a single measurement. This idea may have applications for the rapid determination of polarization manipulations or optical activity within chiral media, but foremost it provides a geometric illustration of indefinite causal order.
It has been shown that measurements involving indefinite causal order can be superior to those in which a sequence of operations occurs in a specified order. In optics, such measurements are realized naturally in a Sagnac interferometer. We show that such an arrangement can measure the solid angle (on the Poincaré sphere) enclosed by a sequence of unitary transformations of the polarization. This is the Pancharatnam-Berry phase. Extension from the classical or single-photon treatment to a fully quantized treatment allows the analysis of the interferometer for arbitrary quantum states of light.This article is part of the theme issue 'The quantum theory of light'.
A novel scheme to realize stable single-photon interference and router-control is demonstrated in this report. Phase difference between clockwise and counter-clockwise single-photons was controlled by time-division phase modulation in an optic fiber Sagnac[1] loop. Due to the annular geometric configuration of the Sagnac loop, clockwise and counter-clockwise single-photons traveled the same optical path and had the same slowly-varied phase drifts, which were automatically compensated. Furthermore, clockwise and counter-clockwise single-photons encounter the same polarization mode dispersion, which was also automatically compensated at the exit port of the Sagnac interferometer.
A rotating interferometer with paths that enclose a physical area exhibits a phase shift proportional to this area and to the rotation rate of the frame. Understanding the origin of this so-called Sagnac effect has played a key role in the establishment of the theory of relativity and has pushed for the development of precision optical interferometers. The fundamental importance of the Sagnac effect motivated the realization of experiments to test its validity for waves beyond optical, but precision measurements remained a challenge. Here, we report the accurate test of the Sagnac effect for matter waves, by using a Cesium atom interferometer featuring a geometrical area of 11 cm 2 and two sensitive axes of measurements. We measure the phase shift induced by Earth’s rotation and find agreement with the theoretical prediction at an accuracy level of 25 parts per million. Beyond the importance for fundamental physics, our work opens practical applications in seismology and geodesy.
A novel scheme to realize stable single-photon interference and router-control is demonstrated in this paper. Phase difference between clockwise and counter-clockwise single-photons is controlled by time-division phase modulation in an optic fiber Sagnac loop. Due to the annular geometric configuration of the Sagnac loop, clockwise and counter-clockwise single-photons travel the same optical path and have the same slowly-varying phase drifts, which are automatically compensated. Furthermore, clockwise and counter-clockwise single-photons encounter the same polarization mode dispersion, which is also automatically compensated at the exit port of the Sagnac interferometer. Long-distance single-mode optic fiber at 1550 nm is used in our experimental realization. Fringe visibility of single-photon interference higher than 98% and fidelity of single-photon router-control higher than 90% have been obtained in a Sagnac fiber loop of 5 km. We also realized a stable single-photon interference in Sagnac fiber loops as long as 27 and 52 km. The corresponding fringe visibilities were higher than 94% and 84%, respectively.
Quantum information processing with geometric features of quantum states may provide promising noise-resilient schemes for quantum metrology. In this work, we theoretically explore phase-space geometric Sagnac interferometers with trapped atomic clocks for rotation sensing, which could be intrinsically robust to certain decoherence noises and reach high precision. With the wave guide provided by sweeping ring traps, we give criteria under which the well-known Sagnac phase is a pure or unconventional geometric phase with respect to the phase space. Furthermore, corresponding schemes for geometric Sagnac interferometers with designed sweeping angular velocity and interrogation time are presented, and the experimental feasibility is also discussed. Such geometric Sagnac interferometers are capable of saturating the ultimate precision limit given by the quantum Cram\'er-Rao bound.
Subject of study . A phase modulation device for encoding quantum states based on an electro-optical phase modulator asymmetrically located in a Sagnac interferometer is studied. Aim of study. An easily implemented and repeatable method for measuring the path difference to a phase modulator inside a Sagnac interferometer in quantum key distribution systems is developed. Method. A phase difference between the clockwise and counterclockwise continuous propagating light is introduced in the beamsplitter of a Sagnac interferometer when an electrical pulse is transmitted to a phase modulator. Moreover, the asymmetric location of the phase modulator means that this single electrical pulse introduces twice the optical phase difference. Main results. A technique for measuring the difference in propagation time for light traveling in opposite directions to reach the phase modulator in a Sagnac interferometer was proposed and experimentally tested; this technique can be used for simultaneous modulation of phase and intensity in quantum key distribution systems. Practical significance. Our results regarding this highly accurate and easily implementable technique for measuring the difference in propagation time for light traveling in opposite directions to reach the asymmetrically positioned phase modulator in a Sagnac interferometer are relevant for the production of optoelectronic modules used in quantum key distribution systems, where this time difference directly affects the system quality and overall performance.