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

In the Aharonov-Bohm (AB) effect, a superposed charge acquires a detectable phase by enclosing an infinite solenoid, in a region where the solenoid’s electric and magnetic fields are zero. Its generation seems therefore explainable only by the local action of gauge-dependent potentials, not of gauge-independent fields. This was recently challenged by Vaidman, who explained the phase by the solenoid’s current interacting with the electron’s field (at the solenoid). Still, his model has a residual nonlocality: it does not explain how the phase, generated at the solenoid, is detectable on the charge. In this Letter, we solve this nonlocality explicitly by quantizing the field. We show that the AB phase is mediated locally by the entanglement between the charge and the photons, like all electromagnetic phases. We also predict a gauge-invariant value for the phase difference at each point along the charge’s path. We propose a realistic experiment to measure this phase difference locally, by partial quantum state tomography on the charge, without closing the interference loop.

Figure 1

(a) A Mach-Zehnder-like setup for the AB effect. C is the charge, S the solenoid. (b) Quantum network for the AB phase generation along the charge’s path, via a local gate $U=\exp (i{H}_{\mathrm{AB}})$ defined at each point ${\mathbf{r}}_{\mathbf{c}}(\mathbf{t})$, involving the photon field.

Figure 2

Left: the reference electron A is superposed across the left (L) and right (R) modes; the solenoid is off. Center: the solenoid gradually reaches the desired current value; electron A acquires a fixed relative phase. Right: Electron B is superposed across L and R, acquiring the AB phase. Joint measurements of A and B, local to the left and right modes, reconstruct the partial AB phase.