## Neutron Interferometry: Lessons in Experimental Quantum Mechanics

*by Helmut Rauch and Samuel A. Werner*

448 pp. Clarendon Press, Oxford, 2000

Reviewed in *American Journal of Physics* by Mark P. Silverman

It is all too easy, when one reads standard textbooks of quantum
mechanics, to focus so intently on abstract state vectors in Hilbert
space or on mathematical techniques for solving the Schrödinger
equation, that one loses track of (or perhaps never encounters) the
fascinating experiments on real physical particles for which the
principles of quantum mechanics are required. *Neutron
Interferometry* helps motivate the theoretical side of quantum
mechanical instruction by disclosing a world of experimental detail,
centered on the neutron, that calls for and tests the principles of
quantum theory. The book is not a textbook, but I have recently used
it, together with my own book
on the quantum interference of electrons (both free and bound in
atoms), for instructive, thought-provoking examples - some for
mathematical analysis, others for qualitative discussion - in a
junior-senior level course of quantum mechanics.

The authors, who have, both independently and in collaboration, made pioneering contributions to neutron interferometry, begin with the analogy between neutron optics and light optics, and from there develop seminal concepts relating to coherence, diffraction, and interference. I find this approach congenial to my own way of teaching quantum mechanics, which, in brief, is to begin with the analogy to classical optics rather than with ties to classical mechanics. In this way, students may find that certain aspects of quantum mechanics are not as unintuitive as physics popularizers, or even quantum mechanics teachers, are wont to claim, if by "intuitive" one means the capacity to predict qualitatively the behavior of a system on the basis of past experience. With classical optics (instead of classical mechanics) as past experience, a variety of single-particle quantum phenomena, e.g. those involving step potentials, barriers, and wells, become reasonably intuitive.

Although quantum mechanics takes its name from the discreteness -
quantization - of energy, angular momentum, and other dynamical
observables, I believe a good case can be made (and I have made it
elsewhere) that what distinguishes quantum mechanics most from classical
mechanics is superposition and interference. If interference is to
occur, then the superposing waves (or states) must exhibit some degree
of coherence. Ironically, for all its fundamentality as the concept
underlying quantum interference, I have not found many quantum mechanics
textbooks in which the term coherence even appears in the index (apart,
perhaps, from the topic of coherent oscillator states), let alone in the
discussion of interference phenomena. Students all too frequently may
be left with an erroneous impression that it is the de Broglie
wavelength that sets the size scale for objects or apertures to give
rise to interference effects. By contrast, *Neutron
Interferometry* gives a thorough discussion of the important
coherence parameters (longitudinal coherence length, transverse
coherence length, coherence volume, coherence time, and so forth) that
enter into an analysis of quantum interference, as well as experimental
procedures for measuring these coherence parameters in the case of
neutron beams.

For readers looking for satisfyingly detailed descriptions of quantum
interference phenomena, *Neutron Interferometry* is a gold mine
of illustrative examples. As in my own book whose title asserts that,
contrary to Feynman's oft-quoted remark, there is more to the "mystery"
of quantum mechanics than two-slit interference, Rauch and Werner
outline the basic theory and experimental features of various
inequivalent categories of quantum interference phenomena involving spin
superposition, topological phase, gravitational and noninertial effects,
nonlinearity of the Schrödinger equation, particle-antiparticle
oscillations, quantum statistics, quantum entanglement, and much more.
Some of these examples are experiments that have already been done (in
fact, many years ago), and others are speculative experiments waiting
for appropriate advances in technology. Because book reviews are
expected to be reasonably brief, I will comment on only a few of the
numerous experiments that have interested me most and which represent
quantum interference phenomena conceptually different from the standard
example of two-slit interference that one encounters most often in
textbooks.

The Aharonov-Bohm (AB) effect is a quantum interference effect that
depends on spatial topology and can be manifested only by particles
endowed with electric charge. A split electron beam, for example, made
to pass in field-free space around (and not through) a region of space
within which is a confined magnetic flux, will, upon recombination,
exhibit a flux-dependent pattern of fringes. Thus, by a judicious
adjustment of the magnetic flux, one can produce an interference
*minimum* in the forward direction, even though the optical path
length difference of the two beam components is null. The electrons do
not experience a magnetic field locally, and therefore are not acted
upon by a classical Lorentz force. As neutral particles, neutrons do
not exhibit what is traditionally regarded as *the* AB effect.
However, neutrons have a magnetic moment and give rise to a companion
topological phenomenon known as the Aharonov-Casher (AC) effect. In the
latter, a split neutron beam is made to pass around a region of space
within which is a confined electric charge and, upon recombination,
gives rise to a charge-dependent interference pattern. The
experimental confirmation of this effect, which may be interpreted as an
example of spin-orbit coupling, was performed at the University of
Missouri Research Reactor in 1991. Rauch and Werner summarize the
theoretical interpretations and experimental features of the AC effect
and its variants very well.

All particles, quantum as well as classical, are subject to the
attractive force of gravity. In quantum mechanics, however, potential
differences in the absence of classical forces can give rise to quantum
interference effects (as just illustrated above in the case of a
topological phase). In their book, the authors describe the so-called
COW experiments (for Colella-Overhauser-Werner) in which a beam of
neutrons, coherently split into two components moving parallel, but
displaced vertically from one another, are recombined to yield an
interference pattern that depends on the gravitational potential
difference of the two beams. Here is an example where, ideally, the net
work done by gravity on the two beams is the same, as well as is the
optical path length difference of the two beams. There is a
gravitationally-induced quantum interference in the absence of a net
gravitational force. Quite by chance, I was lecturing on the COW
experiments to my quantum mechanics class at about the time (2002) when
the first experiments reporting the quantization of neutron energy
states in a gravitational field were reported in *Nature* - an
experiment that I hope will be included in the next edition of this
book.

The AB, AC, and COW experiments are examples of single-particle
self-interference. Among the entries in the chapter on "forthcoming and
speculative experiments" is the neutron analogue of the optical Hanbury
Brown-Twiss (HBT) experiments that demonstrated the correlated "wave
noise" in chaotic light. From a quantum perspective, such correlations
are known as photon bunching and represent a type of quantum
interference attributable to the bosonic nature of the photon.
Neutrons, however, like electrons, are fermions and are therefore
governed by Fermi-Dirac statistics. A neutron HBT experiment would show
a negative correlation or antibunching effect. In my own book I
analyzed a variety of HBT experiments on free electron beams and had
come to the conclusion that the degeneracy parameter of the most
coherent field-emission electron sources available was marginally large
enough for such experiments to be performed. (The degeneracy parameter
is a measure of the mean number of electrons per cell of phase space.)
The much lower (by orders of magnitude) degeneracy of known neutron
sources led me to conclude that a neutron HBT experiment was virtually
hopeless. Rauch and Werner point out, however, the very interesting
possibility of obtaining correlated neutrons from the deuteron
disintegration reactions D(n,p)2n and
D(* p^{-}, g*)2n, a
proposition similar to my proposal of obtaining correlated electrons
from the disintegration of the exotic ion

*(the muonic analogue of H-).*

**m**^{+}e^{-}e^{-}Throughout their book, the authors describe clearly and objectively the successful applications of quantum mechanics to neutron interferometry, eschewing philosophical digressions over such matters as the completeness or interpretation of the quantum mechanical formalism. In the final chapter, however, they give a comprehensive neutral summary of the principal positions that have emerged in answer to the epistemological questions: (a) What is the meaning of the wavefunction? (b) How is the measurement process described? (c) How can a classical world appear out of quantum mechanics? (d) How can non-locality be explained? That such questions remain after more than 75 years of extensive use and meticulous testing of quantum mechanics testifies to how odd the quantum world can be - a world humorously, and not inaptly, mirrored in the Charles Addams cartoon that decorates the cover of the book: the skier who in some mysterious way has left one ski track around each side of a tall pine tree.

### About the author

Mark P. Silverman is Jarvis Professor of Physics at Trinity College. He wrote of his investigations of light, electrons, nuclei, and atoms in his books *Waves and Grains: Reflections on Light and Learning* (Princeton, 1998), *Probing the Atom* (Princeton, 2000), and *A Universe of Atoms, An Atom in the Universe* (Springer, 2002). His latest book *Quantum Superposition* (Springer, 2008) elucidates principles underlying the strange, counterintuitive behaviour of quantum systems.