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arXiv:2110.05824 · Advanced Quantum Technologies

Geometric Phases and The Sagnac Effect: Foundational Aspects and Sensing Applications

Ismael L. Paiva, Rain Lenny, Eliahu Cohen

Published Jan 15, 2022• 20 citations
arXivHTMLPDF

Abstract

Abstract Geometric phase is a key player in many areas of quantum science and technology. In this review article, several foundational aspects of quantum geometric phases and their relations to classical geometric phases are outlined. How the Aharonov–Bohm and Sagnac effects fit into this context is then discussed. Moreover, a concise overview of technological applications of the latter, with special emphasis on gravitational sensing, like in gyroscopes and gravitational wave detectors is presented.

DOI
10.1002/qute.202100121

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Abstract Geometric phase is a key player in many areas of quantum science and technology. In this review article, several foundational aspects of quantum geometric phases and their relations to classical geometric phases are outlined. How the Aharonov–Bohm and Sagnac effects fit into this context is then discussed. Moreover, a concise overview of technological applications of the latter, with special emphasis on gravitational sensing, like in gyroscopes and gravitational wave detectors is presented.
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Recently we showed that quantum systems in orbit around a massive body may be compared to ground clocks to test the gravitational Aharonov-Bohm effect. To be sensitive, orbits must have non-zero eccentricity, so geometric phase effects related to the time varying gravitation potential will be present. Since the quantum system will be in free-fall, by the equivalence principle, the quantum system is locally screened from the gravitational field. However, the time varying gravitational potential will change the energy levels of the quantum system and develop side bands, which are harmonics of the orbit frequency and is the signature of the scalar Aharonov-Bohm effect. Experiments that have the potential to measure such geometric phases include atomic clocks in space, such as the ACES mission, as well as other missions that propose optical clocks in space. Also, worth investigating is the possible use of data from Galileo clocks with non-zero eccentricity.
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Quantum eigenstates undergoing cyclic changes acquire a phase factor of geometric origin. This phase, known as the Berry phase, or the geometric phase, has found applications in a wide range of disciplines throughout physics, including atomic and molecular physics, condensed matter physics, optics, and classical dynamics. In this article, the basic theory of the geometric phase is presented along with a number of representative applications. The article begins with an account of the geometric phase for cyclic adiabatic evolutions. An elementary derivation is given along with a worked example for two-state systems. The implications of time-reversal are explained, as is the fundamental connection between the geometric phase and energy level degeneracies. We also discuss methods of experimental observation. A brief account is given of geometric magnetism; this is a Lorenz-like force of geometric origin which appears in the dynamics of slow systems coupled to fast ones. A number of theoretical developments of the geometric phase are presented. These include an informal discussion of fibre bundles, and generalizations of the geometric phase to degenerate eigenstates (the nonabelian case) and to nonadiabatic evolution. There follows an account of applications. Manifestations in classical physics include the Hannay angle and kinematic geometric phases. Applications in optics concern polarization dynamics, including the theory and observation of Pancharatnam's phase. Applications in molecular physics include the molecular Aharonov-Bohm effect and nuclear magnetic resonance studies. In condensed matter physics, we discuss the role of the geometric phase in the theory of the quantum Hall effect.
This work reports an effort to introduce a few advanced concepts in quantum physics to undergraduate students who have only a minimal familiarity with wave mechanics. The geometric phase is not introduced in most introductory courses on quantum mechanics. It is of pivotal significance for the understanding of topological phase transitions which are becoming increasingly important in a large number of physical phenomena, including those that involve platforms for robust quantum computing. This article would provide a gentle introduction to the geometric phase. Toward this goal, Feynman’s path integral approach to quantum mechanics and its application in the interpretation of the wondrous Aharonov–Bohm effect is introduced in a gentle manner. A computer simulation of this interesting phenomenology is presented to help gain an early insight into some advanced concepts.
With the work of Aharonov and Bohm in 1959, history saw a constant growth in the number of works that focused their efforts on researching the concept of geometric phases, in particular, the area of condensed matter physics and its applications made great contributions to the understanding of the cause and effects of such an abstraction.So profound was the research that by itself it only formed a completely new branch.Going through the conception of holonomic processes in quantum mechanics, the adiabatic theorem, the Berry phase, and its categorization as independent of time and the Aharonov-Bohm, it is possible to have an understanding of the occurrence and use of geometric phases.Moreover, we can reference contemporary applications, specifically in the quantum computing branch.As well, how these concepts can be applied in order to create systems with stability and fault-tolerant characteristics.