The MEG - Replication Project

Magnetic Aharonov–Bohm effect

The magnetic Aharonov–Bohm effect can be seen as a result of the requirement that quantum physics be invariant with respect to the gauge choice for the vector potential A. This implies that a particle with electric charge q travelling along some path P in a region with zero magnetic field ( $\mathbf{B} = 0 = \nabla \times \mathbf{A}$ ) must acquire a phase $\varphi$ , given in SI units by

$\varphi = \frac{q}{\hbar} \int_P \mathbf{A} \cdot d\mathbf{x},$

with a phase difference $\Delta\varphi$ between any two paths with the same endpoints therefore determined by the magnetic flux Φ through the area between the paths (via Stokes' theorem and $\nabla \times \mathbf{A} = \mathbf{B}$ ), and given by:

$\Delta\varphi = \frac{q\Phi}{\hbar}.$

This phase difference can be observed by placing a solenoid between the slits of a double-slit experiment (or equivalent). An ideal solenoid encloses a magnetic field B, but does not produce any magnetic field outside of its cylinder, and thus the charged particle (e.g. an electron) passing outside experiences no classical effect. However, there is a (curl-free) vector potential outside the solenoid with an enclosed flux, and so the relative phase of particles passing through one slit or the other is altered by whether the solenoid current is turned on or off. This corresponds to an observable shift of the interference fringes on the observation plane.

The same phase effect is responsible for the quantized-flux requirement in superconducting loops. This quantization is because the superconducting wave function must be single valued: its phase difference Δφ around a closed loop must be an integer multiple of 2π (with the charge q=2e for the electron Cooper pairs), and thus the flux Φ must be a multiple of h/2e. The superconducting flux quantum was actually predicted prior to Aharonov and Bohm, by London (1948)using a phenomenological model.

The magnetic Aharonov–Bohm effect is also closely related to Dirac's argument that the existence of a magnetic monopole necessarily implies that both electric and magnetic charges are quantized. A magnetic monopole implies a mathematical singularity in the vector potential, which can be expressed as an infinitely long Dirac string of infinitesimal diameter that contains the equivalent of all of the 4πg flux from a monopole "charge" g. Thus, assuming the absence of an infinite-range scattering effect by this arbitrary choice of singularity, the requirement of single-valued wave functions (as above) necessitates charge-quantization: $2qg/c\hbar$ must be an integer (in cgs units) for any electric charge q and magnetic charge g.

The magnetic Aharonov–Bohm effect was experimentally confirmed by Osakabe et al. (1986), following much earlier work summarized in Olariu and Popèscu (1984). Its scope and application continues to expand. Webb et al. (1985) demonstrated Aharonov–Bohm oscillations in ordinary, non-superconducting metallic rings; for a discussion, see Schwarzschild (1986) and Imry & Webb (1989). Bachtold et al. (1999) detected the effect in carbon nanotubes; for a discussion, see Kong et al. (2004).

Electric Aharonov–Bohm effect

Just as the phase of the wave function depends upon the magnetic vector potential, it also depends upon the scalar electric potential. By constructing a situation in which the electrostatic potential varies for two paths of a particle, through regions of zero electric field, an observable Aharonov–Bohm interference phenomenon from the phase shift has been predicted; again, the absence of an electric field means that, classically, there would be no effect.

From the Schrödinger equation, the phase of an eigenfunction with energy E goes as $\exp(-iEt/\hbar)$ . The energy, however, will depend upon the electrostatic potential V for a particle with charge q. In particular, for a region with constant potential V (zero field), the electric potential energy qV is simply added to E, resulting in a phase shift:

$\Delta\phi = -\frac{qVt}{\hbar} ,$

where t is the time spent in the potential.

The initial theoretical proposal for this effect suggested an experiment where charges pass through conducting cylinders along two paths, which shield the particles from external electric fields in the regions where they travel, but still allow a varying potential to be applied by charging the cylinders. This proved difficult to realize, however. Instead, a different experiment was proposed involving a ring geometry interrupted by tunnel barriers, with a bias voltage V relating the potentials of the two halves of the ring. This situation results in an Aharonov–Bohm phase shift as above, and was observed experimentally in 1998.