Pulsed Weak-Value Tomography of Gauge-Invariant Electron Flow in an Aharonov--Bohm Interferometer: Protocol, Weak-Regime Certification, and Roadmap to Quantum Backflow

Abstract

I develop a fully operational proposal to reconstruct (tomographically) a gauge-invariant electron-flow field in a coherent Aharonov--Bohm (AB) interferometer using pulsed weak measurements and postselected weak values. The protocol targets a local kinetic-momentum density operator, ensuring gauge consistency in the presence of AB phases. A time-gated (pulsed) von Neumann interaction couples this local flow observable to a spin pointer, enabling extraction of the real part of the postselected weak value from conditional pointer shifts. From these data I reconstruct a drain-conditioned local velocity field and define experimentally testable flow lines as its integral curves. A central technical point is that time gating avoids a known failure mode of static interaction regions, where momentum sensitivity may be reduced by time-of-flight cancellation. I provide an internal certification strategy for the weak regime (linear signal scaling in coupling strength and quadratic visibility loss), a falsifiability program based on AB flux periodicity and drain dependence, and a detailed ``minimum viable experiment'' (MVE) that does not require spatial scanning. Finally, I outline an experimental roadmap from calibrated AB-ring flow tomography to a decisive nonclassical target in electronic quantum optics: electron quantum backflow, i.e., negative local probability current despite preparation with support restricted to positive kinetic momenta.