THE GREEK philosopher Zeno of Elea is famous for demonstrating that
motion is logically impossible. He argued that a runner who wishes to reach
a goal must first run from the starting point to the midpoint of the journey,
then run from this point to the midpoint of the remaining distance, and
so on, for ever. The runner will never reach the goal, said Zeno, because
it is impossible to traverse an infinite number of intervals.

Of course, Zeno refuted his own logical deduction each time he went
for a walk. Nevertheless, the paradox stimulated considerable philosophical
discussion, and led eventually to a mathematical understanding of the process
of infinite subdivision of a finite but continuous entity.

In quantum mechanics, the theory of the microscopic world, there is
a paradox which is similar in spirit to Zeno’s paradox. In 1977, B. Misra
and George Sudarshan of the University of Texas showed theoretically that
the decay of an unstable particle – for example, a radioactive nucleus –
is suppressed by the act of observation. The more times it is observed,
the greater is the suppression. When it is observed continuously, the decay
simply does not happen.

This has a quite staggering implication: a radioactive nucleus that
is watched constantly remains intact for ever, despite the fact that it
is intrinsically unstable. It is this phenomenon of ‘a watched pot never
boiling’ that Misra and Sudershan call the quantum Zeno effect.

For the effect to occur, one technical condition must be satisfied.
For a short interval of time after an unstable particle has been created,
the probability of it decaying should increase with the square of its age.
In practice, the condition is usually satisfied.

The interval of time is known as the Zeno time. If measurements of the
particle are made within one Zeno time of each other, a phenomenon known
as ‘the collapse of the wave function’ ensures that the decay is suppressed.

The wave function in quantum theory is a mathematical entity that contains
information about the dynamic behaviour of a particle. The Schrodinger equation
determines how the wave function evolves in time. While the wave function
is evolving, it contains within it all the possibilities of the particle’s
future; but the moment the particle is observed, it falls into one particular
state. Physicists say that the wave function ‘collapses’.

The quantum Zeno effect occurs in the following way. Initially, the
wave function of an unstable particle is concentrated around the undecayed
state. As time passes, however, the wave function spreads out into the decayed
state. But each time a measurement is made, the wave function snaps back,
or ‘collapses’, into the undecayed state.

The quantum Zeno effect would seem to imply that an unstable particle
in a bubble chamber will never decay, because the track it leaves behind
signifies that it is being observed continuously. But physicists see unstable
particles routinely when they inspect tracks made in bubble chambers.

In fact, there is no paradox. When physicists look more closely – on
the atomic scale – they find that the track is not continuous, but broken
up. This means that the observation of the unstable particle is not continuous
at all, but intermittent. Although the gap between consecutive observations
is small, it is longer than the Zeno time, which is exceedingly small.

The quantum Zeno effect remains wonderfully counterintuitive and defies
belief. Recently, however, David Wineland and his colleagues at the National
Institute of Standards and Technology, Boulder, Colorado, have performed
an experiment that leaves no room for doubt. The experiment carried out
by Wineland is a slight variant of an experiment proposed originally by
Richard Cook of the US Air Force Academy, Colorado Springs.

The researchers studied the behaviour of ions of beryllium. They confined
about 5000 ions in an apparatus known as an ion trap and applied a radio
frequency, or RF, field. They chose the frequency and strength of the RF
field carefully so that it would stimulate the ions to make a transition
from their lowest energy state to a state at higher energy. In the field,
the ions jumped back and forth between the two energy states once every
256 milliseconds.

Next, the physicists illuminated the beryllium ions with short pulses
of light, each lasting 2.4 milliseconds. These were the ‘measurement’ pulses.
If a particular ion happened to be in the low energy state, a photon in
the pulse would rapidly excite it to the high energy state. The beryllium
ion would then immediately remit the photon in a random direction, a process
known as scattering.

The light was scattered only by the ions which were in the low energy
state, not by those that had been stimulated by the RF field to make a transition
to the high energy state. This meant that the intensity of scattered light
indicated how many ions were still in the low energy state, having failed
to make the transition.

Wineland and his colleagues applied trains of pulses, with each train
lasting 256 milliseconds. Within each train, the pulses were equally spaced.
The most pulses they used was 64. At the end of each interval of 256 milliseconds,
they measured the scattered light intensity and hence the probability of
finding the ions in the high energy state.

Wineland and his colleagues found that the probability of finding ions
in the high energy state was 1 when they applied no measurement pulse during
the interval. But the probability decreased rapidly as they increased the
number of measurement pulses. This agreed well with theory. When the researchers
applied 64 measurement pulses, they found that almost no ions went from
the low energy to the high energy state. The transition was almost completely
suppressed.

In view of these results, there is an obvious question: will it be possible
to make our radioactive waste safe by monitoring each and every atom? The
answer, unfortunately, seems to be, no. It will be a long time before someone
invents an instrument which can monitor radioactive decays on timescales
shorter than the exceedingly short Zeno time. It may even be impossible
to devise such an instrument.

Wineland and his colleagues will be publishing the results of their
experiment in the journal Physical Review A.

Such a ‘virtual’ body of positively and negatively charged particles
makes the vacuum state resemble a state in which space is filled with ‘real’
particles – an equal mixture of positively and negatively charged ones.
But this is more than just a resemblance.

Both the vacuum and matter respond to electrical forces in a similar
way. For instance, when an electric field is applied to a so-called dielectric
material, the field shifts the positive and negative charges away from each
other, polarising the material. This is precisely what happens in a vacuum,
too. An electric field makes it polarised.

The phenomenon of vacuum polarisation has been known since the 1930s.
The first physicists to observe it, indirectly, were Willis Lamb and Robert
Retherford. In 1947, they measured the so-called ‘Lamb shift’ of the spectral
lines of atomic hydrogen, which is a direct result of the polarisation of
the vacuum.

Now, Y. J. Ding and A. E. Kaplan of Johns Hopkins University, Baltimore,
suggest that physicists, armed with today’s experimental techniques, should
be able to observe vacuum polarisation more directly. They propose setting
up an experiment to demonstrate in a vacuum a subtle optical phenomenon,
known as second-harmonic generation (Physical Review Letters, vol 63, p
2725).

The phenomenon is predicted by quantum electrodynamics and is a direct
consequence of the polarisation of the vacuum. Second-harmonic generation
occurs when photons scatter off each other in a strong magnetic field. Their
frequency is doubled, or raised to the level of their ‘second harmonic’.

Something similar occurs in materials whose optical properties are nonlinear
– that is, they respond to an electric field in a non-linear way. In these
materials, as in any other, the oscillating electric field in a light beam
polarises the medium, to form what is known as an oscillating electric dipole.

If the medium is nonlinear, the induced polarisation gives rise to a
static dipole, with no observable effect, and a dipole that oscillates at
twice the frequency of the original beam. This emits second-harmonic radiation.

Such radiation was first observed in 1961. Second-harmonic generation
in nonlinear materials is now an important technique for producing coherent
beams of light at frequencies that are otherwise unattainable.

All you need to demonstate second-harmonic generation in a vacuum, say
Ding and Kaplan, is a laboratory experiment in which an intense laser is
shined on a region of vacuum in a strong magnetic field. Already, they say,
suitable high-powered lasers are available, and there are techniques for
generating strong magnetic fields.

Vacuum polarisation can cause a pair of photons in the incident laser
beam to merge into a single photon, with double the laser frequency. This
means that experimenters should observe two beams emerging from the region
of the vacuum: one at the laser frequency, and one – much weaker – at twice
the laser frequency.

Although second-harmonic generation in a vacuum would seem to have no
technological applications, verifying it experimentally will be of great
importance in consolidating our understanding of the vacuum state.

Ding and Kaplan also point out that second-harmonic generation in a
vacuum may already be occurring in the vicinity of astronomical objects
which have strong magnetic fields.

‘It really is an entirely new technology which we believe will change
the way in which computers handle images,’ says Michael Barnsley, one of
its creators.

Lengthy digital codes are needed to define pictures and other images
for use by a computer. A typical picture with a modest resolution of 256
X 256 pixels, or picture elements, after coding for the colour or greyness
of each pixel, takes up to 64 kilobytes of space on a disc. Some method
for compressing the size of these codes is essential to simplify and to
reduce the costs of handling them.

Barnsley, with Alan Sloan of Iterated Systems in Georgia, has solved
this problem by using fractal geometry. They regard a picture as an assembly
of several fractals. Each fractal can be generated by a simple and compact
mathematical rule. These rules are devised using a technique based on a
mathematical result in fractal geometry, called the collage theorem, which
was proved by Barnsley and his colleagues several years ago.

Instead of storing the complete digital record of a picture, the Video
Modem stores the rules for generating the component fractals and then reconstructs
the pictures from these rules whenever it is required. The rules, being
compact, require much less storage than the complete pixel-by-pixel record.
Once the fractal of a picture is defined, the picture can be reconstructed
to any resolution required.

The Video Modem consists of two computer boards, an encoder and a decoder,
and its associated software.

For encoding a sequence of video pictures, a personal computer connected
to a video player relays the sequence, one frame at a time, to the encoder.
The encoder transforms the picture into fractals and encodes them mathematically;
it either stores the fractal codes on a disc or transmits them. Each frame
is encoded in two or three seconds. The fractal code of a typical frame
of a film requires about 1 kilobyte of storage, compressing the amount of
data by a factor of 64. This ratio is smaller, about 20, for a high-quality
still frame.

The decoder can accept fractal codes either from a disc or from a modem
and reconstruct the image on a graphics device.

The Video Modem can transmit at a rate of 30 frames of colour pictures
per second at a resolution of 256 X 256 pixels, with 8 bits for colour or
greyscale, fast enough to project a motion picture on a television screen.
Furthermore, these pictures can be reconstructed at a higher resolution
than the original.

* * *

Geometry of fractals

A fractal set is a geometrically complex collection of points in a mathematical
space in which the notion of near or far has a precise meaning. It is, in
fact, very much like a pointillist picture on a piece of rubber. With every
fractal, there is associated a specific way of distorting the underlying
space – stretching the piece of rubber – which does not affect the fractal
but distorts all other sets of points that may be defined in this space.
This distortion can be encoded very compactly by a mathematical rule which
also serves to encode the fractal itself.

In order to explain the results reported by Steven Jones and his colleagues,
theorists must reproduce a fusion rate of nearly 10-23 events per second
for every pair of deuterons. However, Leggett and Baym show, after making
some very reasonable assumptions, that the fusion rate in the metal lattice
cannot exceed 3 X 10-47 per second per deuteron pair (Physical Review Letters,
vol 63, p 191; Nature, vol 340, p 45).

Physicists have wondered whether there is any way that the complex environment
inside the lattice of titanium or palladium can lower the Coulomb barrier
between deuterons. The barrier exists because of the repulsive force between
like electric charges. In their study, Leggett and Baym estimate the maximum
amount that the Coulomb barrier can be lowered. Their estimate is both mathematically
rigorous and completely independent of any detailed features of the host
metal.

To determine the maximum fusion rate in a metal, it is only necessary
to calculate the probability that deuterons will tunnel through this minimal
barrier.

In normal circumstances, the probability of the pair of deuterons in
a deuterium molecule getting close enough to fuse is exceedingly small.
The rate of fusion is only 3 X 10-64 events per second for every molecule.

In calculating what is the maximum that the Coulomb barrier can be lowered,
Leggett and Baym assume thermodynamic equilibrium. They use four measurable
parameters in their calculations. These are the binding energies of electrons
in both hydrogen and helium atoms, and the affinity of the metal lattice
for an atom (that is, the energy released when an atom is inserted in the
crystal and allowed to settle into the lowest energy site) of hydrogen and
of helium.

Physicists know the values of the first three parameters. The fourth
one is slightly more of a problem because no one has measured the affinity
of palladium or titanium for helium. The quantity must be small, however,
because helium desorbs readily from these metals at room temperature.

Leggett and Baym’s analysis covers all except a few unlikely possibilities.
It is just conceivable, they say, that fusion could be enhanced if palladium
and titanium have an extraordinarily large affinity for helium. Leggett
and Baym also say that it is just conceivable that the fusion rate could
be enhanced if 1000 or more atoms act in concert through some unknown long-range
mechanism.

The only remaining realistic possibility is that fusion happens when
the deuteron-metal system is in a transient state, that is it is not in
thermodynamic equilibrium. But Leggett and Baym have an answer for this,
too. ‘Our argument,’ they say, ‘can also be used to put severe constraints
on the efficacy of non-equilibrium mechanisms, once these are specified
in detail.’

EXPERIMENTAL physicists often wish to increase the frequency of a laser pulse so that each photon is more energetic and packs a stronger punch. But all the methods currently available for altering the frequency are limited in some way. Now John Dawson and his colleagues from the University of California, Los Angeles, and the Los Alamos National Laboratory in New Mexico have proposed what they call a ‘photon accelerator’ (Physical Review Letters, vol 62, p 2600).

The scheme involves continuously increasing the frequency of laser light. It extends an earlier idea for designing compact particle accelerators which could one day replace the gigantic machines now in use in high-energy physics.

Dawson and his colleagues propose transferring energy from one pulse of light to another through the medium of a plasma. A plasma is a highly ionised gas consisting of negatively charged electrons and positively charged atomic ions. Overall, the plasma is electrically neutral.

The researchers note that when a pulse of laser light travels through a plasma, it sets the electrons moving and creates alternating layers of positive and negative charge in its wake. The electrons in a negatively charged layer are pulled by the neighbouring positive charge and move towards it, but overshoot due to their inertia, creating oscillations in the charge. The pattern of oscillation of positively and negatively charged regions set up in the wake of the light pulse is known as a Langmuir wave. This plasma wave travels at very nearly the speed of light and saps energy from the light pulse.

If a second laser pulse is introduced into the plasma, it will create its own Langmuir wave. However, Dawson and his colleagues say that if its introduction is timed carefully, the crests of the second wave will coincide with the troughs of the first, and vice versa. This means that there will be no net plasma oscillation in the wake of the trailing laser pulse. The plasma wave generated by the leading pulse will extend only as far as the trailing pulse. It will serve as a conduit for transferring energy from the leading pulse to the trailing pulse. As a result, the shape of the trailing pulse will change and its frequency will increase.

Another way of looking at the method is to think of each laser pulse as a packet of photons, each possessing an effective mass and moving with the velocity of the laser pulse. This is possible because energy and mass are equivalent, as Einstein pointed out. The photons experience a force acting on them when they are in the presence of a gradient in the density of the plasma. This is because wherever there is such a gradient, there is an electric force parallel to it. The plasma wave, because it is a density gradient and moves with the photons, can continuously increase their energy.

Dawson and his colleagues have estimated that their scheme can give a fivefold increase in the frequency of a photon a 1-metre stretch of a moderately dense plasma. Smaller increases could be obtained by tailoring the length of the plasma column.

THE SHAPE of a molecule is one of its most fundamental properties but, for many polyatomic molecules and molecular ions which are held together loosely, it is very difficult to measure. Now, a team of scientists from the Weizmann Institute of Science in Israel and the Argonne National Laboratory in the US has perfected a technique which maps directly the motion of the atoms which make up a molecule. The new technique, called Coulomb explosion imaging, reveals indirectly the shape of a molecule (Science, vol 244, p 426), and it works equally well for tightly-bound and loosely-bound molecules.

To carry out Coulomb explosion imaging, the researchers accelerate a beam of neutral molecules to a very high speed, nearly one-fiftieth the speed of light, and smash it into a foil of solid material which is only about 30 atoms thick. Because neutral molecules are difficult to accelerate, the researchers must first add an extra electron to each molecule. This they remove later with a 10-nanosecond pulse from a dye laser.

The nuclei in the molecule go through the foil unscathed and their relative arrangement is not changed by the collision (very rarely, they may hit a nucleus in the foil). Most of the molecular electrons, however, and all the electrons which the molecule together, are scattered away by the electrons in the foil.

The process is over in one-hundredth of the time required by the nuclei to execute one vibrational oscillation. As a result, the attractive force which keeps the nuclei together in the molecule disappears in mid-oscillation, and, suddenly, the nuclei are subject to the purely repulsive forces of like charges. The molecule flies apart, very much like an exploding shell – hence the term Coulomb explosion.

The nuclear fragments are carried forward to a detector where the speed of each fragment and the direction from which it has come are recorded. It is a simple matter then to calculate the original arrangement of the nuclei.

At any given time, however, different molecules in the beam are in different stages of their vibrational motion. The integrated measurement of the relative positions of the nuclei in the entire collection of molecules in the beam gives the probability distribution of nuclear position. This is a direct measurement of the most fundamental quantum mechanical quantity – the absolute magnitude squared of the nuclear wave function.