## 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 ()
must acquire a phase
,
given in
SI units by

with a phase difference 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 ), and given by:

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*=2*e* for the
electron Cooper pairs),
and thus the flux Φ must
be a multiple of *h*/2*e*.
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
4π*g* 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:
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
.
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:

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.