Aharonov-Bohm Phase is Locally Generated Like All Other Quantum Phases

Abstract

Aharonov and Bohm noted that the wave-function of a charge acquires a detectable phase when superposed along two paths enclosing an infinite solenoid, even though that wave-function is nonzero exclusively where the solenoid's electric and magnetic fields are zero. This phase was long considered radically different from all other quantum phases, because it seems explainable only via local action of gauge-dependent potentials, not of the gauge-invariant electromagnetic fields.Recently Vaidman proposed a model explaining the phase only in terms of the electron's (magnetic) field at the solenoid. Later Kang proposed a Lagrangian model from which Vaidman's can be derived. However, their analysis is still non-local. For it does not explain how the phase, generated at the solenoid, is detectable on the charge when closing the interference loop. To explain this, one needs to quantise the EM field and consider how the interaction between the solenoid and the electron is mediated by photons. In this paper we propose a local model, where the field is quantised. This shows that the Aharonov-Bohm (AB) phase is generated by the same quantum local mechanism as all other electromagnetic phases. Specifically, it is mediated by the entanglement between the superposed charge and the photons. Surprisingly, the quantised model produces some experimentally different predictions from the semiclassical accounts of the AB phase, because it predicts that the phase at any point along the charge path is gauge-invariant and locally generated, and therefore in principle detectable. We propose a realistic experiment, within current technological reach, where the predicted phase difference along the path is measured locally, by performing (a partial) quantum state tomography on the charge without closing the interfering charge paths.

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

“…This suggests that a general interplay may exist between dynamical nonlocality and kinematical nonlocality (see also ref. [99]). The latter is a broad category that includes, for instance, Bell scenarios.…”

“…However, there is no final consensus on this issue and, thus, the origins of this phase are investigated until this day with works that include the idea of modular variables [61] and models that consider the source of magnetic field as part of the dynamics or quantize the magnetic field itself. [85,[90][91][92][93][94][95][96][97][98][99][100][101][102] Furthermore, the interplay between Berry and AB phases in various scenarios can also be of interest. For instance, the phases acquired by a quantum charge with a large spreading while it encircles one or multiple solenoids were studied in ref.…”

A Classical (Local) Account of The Aharonov-Bohm Effect

“…Then all we need to do is compute the electron field and the A field along the tube. The explanation as to how the AB phase is acquired is now identical to the quantum treatment [5] (see also [6]). For us, it suffices to note that when the electron field changes slowly (meaning that the amplitude change can be neglected and all that is left is the change in the phase of the electron wave), we can let φ = 0 and the second of the field equations then reduces to e 2 Āµ (x) Āµ (x) − m 2 = 0 .…”

“…Of course, we do not need to talk about electrons at all; the change in the electron field wavelength will suffice. The expression for the AB phase is gauge invarient (without having to close the loop [5,7,8]) because it relies on the difference between vector potentials. This is of course always the case, since in any wave theory (and quantum waves inherit this too [6]) only the relative phase has a physical meaning.…”

A Simple Field Theoretic Description of Single-Photon Nonlocality

“…Our main result is that the physically observable electric field of light in the single-photon state (1), measured simultaneously at two different locations behind the two ports of the beam splitter, is completely determined by the field associated with the single-photon state entering the device. The use of quantum field theory to explain optical interference phenomena is becoming increasingly popular nowadays [27][28][29][30][31].…”