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.
Time-dependent
PT-symmetric quantum mechanics is featured by a varying inner-product metric and has stimulated a number of interesting studies beyond conventional quantum mechanics. In this paper, we explore geometric aspects of time-dependent
PT-symmetric quantum mechanics. We not only find a geometric phase factor emerging naturally from cyclic evolutions of
PT-symmetric systems, but also formulate a series of differential geometry concepts, including connection, curvature, parallel transport, metric tensor, and quantum geometric tensor. Our findings constitute a useful, perhaps indispensible, tool to tackle physical problems involving
PT-symmetric systems with time-varying system's parameters. To exemplify the application of our findings, we show that the unconventional geometrical phase [Phys. Rev. Lett. 91, 187902 (2003)], consisting of a geometric phase and a dynamical phase proportional to the geometric phase, can be expressed as a single geometric phase identified in this work.
We develop the broadest possible generalization of the well known connection between quantum-mechanical Bargmann invariants and geometric phases. The key concept is that of null phase curves in quantum-mechanical ray and Hilbert spaces. Examples of such curves are developed. Our generalization is shown to be essential for properly understanding geometric phase results in the cases of coherent states and of Gaussian states. Differential geometric aspects of null phase curves are also briefly explored.