AOH :: HOLONOMY.TXT|
Phase shifts track a quantum system in "state space."
by Gregory Greenwell
To understand the latest surprise served up by the quantum world take
a pencil, lay it on the north pole of a globe and point it in the
direction of any line of longitude that radiates from the pole. Move
the pencil down along the line to the equator and, keeping it
perpendicular to the equator, move it along the equator to another
longitude. Move the pencil back to the north pole along the new
longitude and you will find that although the pencil has returned to
its starting spot, it no longer points along the original line of
This is an example of a purely geometric effect, known as holonomy,
resulting from the fact that the pencil was forced to trace out a
circuit on the surface of a sphere while remaining parallel to the
meridians. The holonomy caused by such "parallel transport" around a
circuit on a curved surface is not limited to tangible objects such as
globes and pencils. The results of three different experiments
recently reported in the same issue of Physical Review Letters show
that geometric holonomy can exist even for abstract constructs in the
microscopic realm of quantum physics.
In the quantum realm the state of a physical system is best regarded
as a wave whose characteristics are determined by parameters, or
physical quantities, that may affect the system. In the case of a
photon such a parameter might be, say, its polarization (the direction
of its associated electric field) or its intrinsic spin. Another
important component of quantum waves is their phase: the positions of
each wave's crests and troughs in relation to one another.
As was first pointed out in 1983 by Michael V. Berry of the University
of Bristol, as a result of geometric holonomy a quantum system can
exhibit different phases in its initial and final wave representations
even though the system begins and ends with the same parameter values.
The key to understanding how this can happen is to visualize all
possible states of the system as points on the surface of a sphere in
"state space." Because the initial and final states of the system are
represented by the same point on the sphere, the intermediate states
lie on a closed curve on the sphere.
The phase of the wave representing the system can then be envisioned
as undergoing parallel transport (like the pencil on the globe) as the
system goes from state to state around the curve, completing a
circuit. As a consequence, when the system returns to its starting
point it no longer has its original phase. In fact, the magnitude of
the change in phase reveals the general form of the curve, because it
is proporational to the area on the state-space sphere enclosed by the
Rajendra Bhandari and Joseph Samuel of the Raman Research Institute in
India succeeded in measuring such a phase shift in an interference
pattern produced by a laser beam that was split and recombined. The
shift was caused by varying the polarization state of the photons is
one of the split beams while ensuring that the beam's photons began
and ended in the same polarization state as those in the other split
beam. A team headed by Raymond Y. Chiao of the University of
California at Berkeley and Howard Nathel of the Lawrence Livermore
National Laboratory tried altering the direction of the photons'
momentum rather than their polarization. The team observed the
predicted phase shifts even though the photons traced out a circuit in
a different state space.
The third experiment, also done by a group at Berkeley, detected a
quantum holonomy not in laser photons but in the radio signals emitted
by the spins of atomic nuclei. Applying a variation of
nuclear-magnetic-resonance interferometry, Dieter Suter, Karl T.
Mueller and Alexander Pines worked with molecules consisting of atoms
whose nuclear spins are aligned. They subjected the atoms to magnetic
pulses that caused the nuclei to change their spin state and then
return to their original state. As expected, the radio signals later
emitted by the nuclei were phase-shifted.
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