Previous time-dependent Ginzburg–Landau simulations demonstrated that Abrikosov vortex dynamics in superconductors can be influenced by Aharonov–Bohm-type curl-less vector potentials even in simply connected geometries, without closed-loop trajectories enclosing magnetic flux [A. M. Gulian et al., Phys. Lett. A 382 (2018) 1058]. The origin of this effect has remained unexplained. Here we investigate further nonequilibrium effects in which the relation between a curl-less vector potential and an evolving quantum state produces observable consequences. During transient nonequilibrium state evolution, collective quantum systems may develop measurable responses to curl-less gauge fields despite a simply connected final equilibrium configuration with no persistent current. Using superconducting condensates, we formulate protocols based on the gauge-invariant coupling between quantum phase and the curl-less vector potential, analyze their dynamics using time-dependent Ginzburg–Landau theory and interpret rapid phase formation within the Kibble–Zurek picture. The predicted transient supercurrents and voltage signals provide a physical interpretation of nonequilibrium sensitivity to curl-less gauge fields and motivate experimental tests.
Previous time-dependent Ginzburg–Landau simulations demonstrated that Abrikosov vortex dynamics in superconductors can be influenced by Aharonov–Bohm-type curl-less vector potentials even in simply connected geometries, without closed-loop trajectories enclosing magnetic flux [A. M. Gulian et al., Phys. Lett. A 382 (2018) 1058]. The origin of this effect has remained unexplained. Here we investigate further nonequilibrium effects in which the relation between a curl-less vector potential and an evolving quantum state produces observable consequences. During transient nonequilibrium state evolution, collective quantum systems may develop measurable responses to curl-less gauge fields despite a simply connected final equilibrium configuration with no persistent current. Using superconducting condensates, we formulate protocols based on the gauge-invariant coupling between quantum phase and the curl-less vector potential, analyze their dynamics using time-dependent Ginzburg–Landau theory and interpret rapid phase formation within the Kibble–Zurek picture. The predicted transient supercurrents and voltage signals provide a physical interpretation of nonequilibrium sensitivity to curl-less gauge fields and motivate experimental tests.
In quantum theory, electromagnetic gauge fields enter directly into the phase evolution of the wavefunction and can even influence quantum systems in regions where the associated electric and magnetic fields vanish. The Aharonov-Bohm effect demonstrates that such gauge fields produce observable consequences when a coherent quantum system encloses magnetic flux along a doubly connected path. This has led to the widespread view that curl-free gauge fields are undetectable in simply connected systems, giving rise to a form of topological blindness. Here we show that this conclusion is not fundamental. During nonequilibrium processes, collective quantum systems can develop transient responses to curl-free gauge fields without enclosing magnetic flux in a static geometry. Using superconducting condensates as a concrete example, we demonstrate that the evolving phase of the macroscopic wavefunction generates supercurrents and voltage pulses whose time integral is proportional to the open-path line integral of the vector potential. In contrast to the conventional Aharonov-Bohm effect, the resulting response is not restricted modulo the flux quantum and may greatly exceed the scale associated with static doubly connected geometries. The mechanism can be interpreted as a dynamical closure of the gauge contour in spacetime and is supported by gauge-invariant arguments, time-dependent Ginzburg-Landau theory modeling, and numerical simulations. These results establish a general principle for detecting curl-free gauge fields and suggest new approaches for probing hidden gauge structures in quantum matter and beyond.
The Aharonov-Bohm (AB) effect highlights the fundamental role of electromagnetic potentials in quantum mechanics, manifesting as a phase shift for a charged particle in field-free regions. While well-established for static magnetic fluxes, the effect's behavior under time-varying fluxes remains an open and debated question. Employing the WKB method, we derive the AB phase shift for a time-dependent magnetic vector potential, demonstrating that for circular paths in the quasistatic regime, it is proportional to the time-averaged enclosed magnetic flux, ΔφAB=T1∫0TeΦ(t)dt, with the total phase shift, including kinetic contributions, equaling eΦ(0). For non-circular paths, the phase shift depends on both the flux history and path geometry, revealing the effect's hybrid nature involving gauge potentials and induced electric fields. We verify the consistency of our gauge choice with Maxwell's equations and discuss the implications for local versus nonlocal interpretations of the AB effect. We also generalize the results to scenarios with nonzero external magnetic fields, where the enclosed flux is through the actual electron paths, and for circular paths of radius R, the AB phase shift is also proportional to the time average of the enclosed flux Φenc(R,t), with the total phase shift depending only on the initial enclosed flux eΦenc(R,0); for general non-circular paths, the external magnetic field affects trajectories and phase accumulation through the Lorentz force, leading to additional path dependence. These findings clarify the role of gauge-dependent potentials and induced fields in the generalized AB effect, offering new theoretical insights and potential applications in quantum technologies.
This technical note gives a self-contained exposition of the Aharonov–Bohm effect, aimed at advanced undergraduate students. Starting from the role of electromagnetic potentials in classical electrodynamics, it introduces gauge freedom, the quantum Hamiltonian with minimal coupling, and the gauge covariance of the Schrödinger equation. It then derives the Aharonov–Bohm phase for both the time-dependent scalar configuration and the magnetic configuration, showing how a charged particle acquires an observable phase shift while travelling only through regions where the electromagnetic field vanishes. The final part reviews the experimental evidence, from the early electron-interference experiments of Chambers and of Möllenstedt and Bayh to the 1986 toroidal-magnet experiment of Tonomura and collaborators. In that experiment, superconducting shielding confined the magnetic flux completely and closed the leakage-field loophole, while flux quantization produced the characteristic half-fringe shift. The note is expository and pedagogical: it does not present original research results and has not been peer reviewed. Originally developed from study notes prepared for the PHYS02 scientific talk at Lady Margaret Hall, Oxford, in July 2026; an earlier version of the notes has been publicly available on GitHub since August 2026. Corrections and comments are more than welcome at giancarminesparso@gmail.com.
In this paper, we study the Aharonov–Bohm (A–B) effect and Anderson–Higgs mechanism in Ginzburg–Landau model of superconductors from the perspective of the decomposition of U (1) gauge potential. By the Helmholtz theorem, we derive exactly the expression of the transverse gauge potential A⊥ in A–B experiment, which is gauge-invariant and physical. For the case of a bulk superconductor, we find that the gradient of the total phase field θ provides the longitudinal component A‖, which reflects the Anderson–Higgs mechanism. For the case of a superconductor ring, the gradient of the longitudinal phase field θ1 provides the longitudinal component A‖, while the transverse phase field θ2 produces new physical effects such as the flux quantization inside a superconducting ring.
The most popular interpretation of the Aharonov–Bohm (AB) effect is that the electromagnetic potential locally affects the complex phase of a charged particle’s wave function in the magnetic field free region. However, since the vector potential is a gauge-variant quantity, multiple researchers suspect that it is just a convenient tool for calculating the force field. This motivates them to explain the AB effect without using the vector potential, which inevitably leads to some sort of non-locality. This frustrating situation is shortly summarized by the statement by Aharonov et al. that the AB effect may be due to a local gauge potential or due to non-local gauge-invariant fields. In the present paper, we shall give several convincing arguments which support the viewpoint that the vector potential is not just a convenient mathematical tool with little physical entity. Despite its gauge arbitrariness, the vector potential certainly contains a gauge-invariant piece, which solely explains the observed AB phase shift. Importantly, this component has a property such that it is basically unique and cannot be eliminated by any regular gauge transformations. To complete the discussion, we also discuss the role of remaining gauge arbitrariness still contained in the entire vector potential.
We present a rigorous no-go theorem that excludes \emph{all} gauge-invariant dynamical explanations of the generalized, time-dependent Aharonov--Bohm (AB) effect. In this generalization, the AB phase is determined by the time-averaged magnetic flux enclosed by the particle's path, ΔϕAB=ℏe⋅T1∫0TΦ(t)dt. Using a carefully designed protocol in which a constant flux Φ persists only for a finite interval Δt after wavepacket splitting and is switched to zero before detection, we demonstrate that during the phase-generating period all gauge-invariant observables of the electron remain identical to those in a zero-flux experiment. Relativistic causality further prohibits backreaction from the source. Consequently, no gauge-invariant quantity---whether local or nonlocal, field-based or source-mediated---can account for the observed phase, which depends solely on flux history. We also critically examine Healey's holonomy interpretation and show that it, too, fails to explain the time-dependent AB effect, as its static formalism lacks the dynamical structure needed to capture temporal averaging. This result establishes the inescapable physical reality of gauge potentials in quantum theory and clarifies the foundational status of the AB effect as a genuine manifestation of potentials rather than fields.