Aharanov–Bohm effect: what happens if I turn off the magnetic field?
I've recently been trying to really understand what's going on in the Aharanov–Bohm effect, and I've crystallized my questions into the following variant of the typical Aharanov–Bohm setup.
Begin with the typical <PERSON>–Bohm setup. Consider a charge-$q$ particle moving around in an annular region. If we inject a flux $\Phi$ into the region and the particle makes a single rotation around the region, it picks up a relative phase $e^{2\pi i \Phi/\Phi_0}$, where $\Phi_0 = h/q$ is the flux quantum. This relative phase can be experimentally detected.
Now consider the following variant: First, move the particle an angle $0 < \theta < 2\pi$ around the annulus. Then, holding the particle as stationary as possible consistent with the Heisenberg uncertainty principle, slowly turn off the magnetic field in the center of the region. Then continue to rotate the particle the remaining angle $2\pi - \theta$ to its starting point. What is the relative phase in this case?
This process has three stages: (1) the initial rotation of angle $\theta$, (2) turning the magnetic field off, and (3) the final rotation of $2\pi - \theta$. 474
I expect step (2) will result in changes to the phase, because I can convince myself of different answers to the question if I consider just parts (1) and (3) in different gauges:
* First, consider the conventional gauge for Aharanov–Bohm where $A(r,\varphi,z) = \frac{\Phi}{2\pi r} \hat{\varphi}$ in the annular region when the flux is on. I pick up a relative phase $e^{\theta i \Phi/\Phi_0}$ during (1). When the magnetic field is off, $A\equiv 0$ and I pick up no additional phase during (3). Thus, the total phase picked up during (1) and (3) is $e^{\theta i \Phi/\Phi_0}$.
* Second, choose a gauge in which $A \equiv 0$ along the arc of angle $\theta$ while the magnetic field is turned on and $A\equiv 0$ everywhere after the field is turned off. This is possible because integrals along $A$ are path-independent in a simply connected domain containing the arc of angle $\theta$ during part (1). In this gauge, the phase picked up during steps (1) and (3) is trivial $=1$.
Based on these two contradictory answers, I expect that something happens during (2) which corrects for the gauge-dependent answers I get by just treating (1) and (3). This leads me to ask: What relative phase does a charged particle pick up in an <PERSON>–Bohm experiment where the magnetic field is turned off mid-experiment (i.e., all three steps (1), (2), and (3))? 474