After seeing this latest extremely good video from Veritasium, and looking back through my posts, I realized that while I've referenced it indirectly, I've never explicitly talked about the Aharonov-Bohm effect. The video is excellent, and that wikipedia page is pretty good, but maybe some people will find another angle on this to be helpful.
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| Still from this video. |
The ultrabrief version: The quantum interference of charged particles like electrons can be controllably altered by tuning a magnetic field in a region that the particles never pass through. This is weird and spooky because it's an entirely quantum mechanical effect - classical physics, where motion is governed by local forces, says that zero field = unaffected trajectories.
In quantum mechanics, we describe the spatial distribution of particles like electrons with a wavefunction, a complex-valued quantity that one can write as an amplitude and a phase \(\varphi\), where both depend on position \(\mathbf{r}\). The phase is important because waves can interfere. Crudely speaking, when the crests of one wave (say \(\varphi = 0\)) line up with the troughs of another wave (\(\varphi = \pi\)) at some location, the waves interfere destructively, so the total wave at that location is zero if the amplitudes of each contribution are identical. As quantum particles propagate through space, their phase "winds" with distance \(\mathbf{r}\) like \(\mathbf{k}\cdot \mathbf{r}\), where \(\hbar \mathbf{k} = \mathbf{p}\) is the momentum. Higher momentum = faster winding of phase = shorter wavelength. This propagation, phase winding, and interference is the physics behind the famous two-slit experiment. (In his great little popular book - read it if you haven't yet - Feynman described phase as a clockface attached to each particle.) One important note: The actual phase itself is arbitrary; it's phase differences that matter in interference experiments. If you added an arbitrary amount \(\varphi_{0}\) to every phase, no physically measurable observables would change.
Things get trickier if the particles that move around are charged. It was realized 150+ years ago that formal conservation of momentum gets tricky if we consider electric and magnetic fields. The canonical momentum that shows up in the Lagrange and Hamilton equations is \(\mathbf{p}_{c} = \mathbf{p}_{kin} + q \mathbf{A}\), where \(\mathbf{p}_{kin}\) is the kinetic momentum (the part that actually has to do with the classical velocity and which shows up in the kinetic energy), \(q\) is the charge of the particle, and \(\mathbf{A}(\mathbf{r}\)\) is the vector potential.
Background digression: The vector potential is very often a slippery concept for students. We get used to the idea of a scalar potential \(\phi(\mathbf{r})\), such that the electrostatic potential energy is \(q\phi\) and the electric field is given by \(\mathbf{E} = -\nabla \phi\) if there are no magnetic fields. Adding an arbitrary uniform offset to the scalar potential, \(\phi \rightarrow \phi + \phi_{0}\), doesn't change the electric field (and therefore forces on charged particles), because the zero that we define for energy is arbitrary (general relativity aside). For the vector potential, \(\mathbf{B} = \nabla \times \mathbf{A}\). This means we can add an arbitrary gradient of a scalar function to the vector potential, \(\mathbf{A} \rightarrow \mathbf{A}+ \nabla f(\mathbf{r})\), and the magnetic field won't change. Maxwell's equations mean that \(\mathbf{E} = -\nabla \phi - \partial \mathbf{A}/\partial t\). "Gauge freedom" means that there is more than one way to choose internally consistent definitions of \(\phi\) and \(\mathbf{A}\).
TL/DR main points: (1) The vector potential can be nonzero in places where \(\mathbf{B}\) (and hence the classical Lorentz force) is zero. (2) Because the canonical momentum becomes the operator \(-i \hbar \nabla\) in quantum mechanics and the kinetic momentum is what shows up in the kinetic energy, charged propagating particles pick up an extra phase winding given by \(\delta \varphi = (q/\hbar)\int \mathbf{A}\cdot d\mathbf{r}\) along a path.
This is the source of the creepiness of the Aharonov-Bohm effect. Think of two paths (see still taken from the Veritasium video), and threading magnetic flux just through the little region using a solenoid will tune the intensity detected on the screen on the far right. That field region can be made arbitrarily small and positioned anywhere inside the diamond formed by the paths, and the effect still works. Something not mentioned in the video: The shifting of the interference pattern is periodic in the flux through the solenoid, with a period of \(h/e\), where \(h\) is Planck's constant and \(e\) is the electronic charge.
Why should you care about this?
- As the video discusses, the A-B effect shows that the potentials are physically important quantities that affect motion, at least as much as the corresponding fields, and there are quantum consequences to this that are just absent in the classical world.
- The A-B effect (though not with the super skinny field confinement) has been seen experimentally in many mesoscopic physics experiments (e.g., here, or here) and can be used as a means of quantifying coherence at these scales (e.g., here and here).
- When dealing with emergent quasiparticles that might have unusual fractional charges (\(e^*\)), then A-B interferometers can have flux periodicities that are given by \(h/e^*\). (This can be subtle and tricky.)
- Interferometry to detect potential-based phase shifts is well established. Here's the paper mentioned in the video about a gravitational analog of the A-B effect. (Quibblers can argue that there is no field-free region in this case, so it's not strictly speaking the A-B analog.)
