TWICE in 20th-century physics, the notion of unpredictability has shaken
scientists鈥� view of the Universe. The first time was the development of quantum
mechanics, the theory that describes the behaviour of matter on an atomic scale.
The second came with the classical phenomenon of chaos. In both areas
unpredictable features changed scientists鈥� understanding of matter in ways that
were totally unforeseen.
How ironic then, that these two fields, which have something so fundamental
in common, should end up as antagonists when combined. For by rights, chaos
should not exist at all in quantum systems鈥攖he laws of quantum mechanics
actually forbid it. Yet recent experiments seem to show the footprints of
quantum chaos in remarkable swirling patterns of atomic disorder. These
intriguing patterns could illuminate one of the darkest corners of modern
physics: the twilight zone where the quantum and classical worlds meet.
Quantum theory is one of the most successful theories in modern science.
Developed in the 1920s, it accounts for a vast range of phenomena from the
nature of chemical bonds to the behaviour of subatomic particles, making
predictions that have been tested to unprecedented levels of accuracy. But at
its core, there are troublesome features: Prominent among them is
Heisenberg鈥檚 uncertainty principle鈥攊f you know the speed of a quantum
particle, for instance, you can never know its exact location. The notion that
some aspects of nature are simply unknowable has caused sleepless nights for
more than a few physicists.
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Chaos is a younger discipline. Although some of its conceptual elements had
already been appreciated by Leibnitz in the 17th century and Poincar茅 in
the 19th century, chaos theory did not become fashionable until the 1980s when
scientists began to realise that the phenomenon is widespread in the natural
world. It arises when a system is unusually sensitive to its initial conditions
so that a small perturbation of the system changes its subsequent behaviour in a
way that grows exponentially with time. Chaos has been observed in, among other
things, pendulums, the growth of populations, planetary dynamics and weather
systems. Probably the most famous example of chaos is the so-called 鈥渂utterfly
effect鈥�, in which, in theory, the tiny air disturbance from the flapping of a
butterfly鈥檚 wings can ultimately lead to a dramatic storm.
Of course, although both these theories place fundamental limits on what we
can know about the world, the unpredictabilities in quantum theory and chaos are
different in kind. But the particular problem with quantum chaos is that in
quantum mechanics small perturbations generally only lead to small perturbations
in subsequent states. Without the exponential divergence in evolutionary paths,
it is difficult to see how there can be any chaos.
This behaviour of quantum systems is often attributed to a special property
of the quantum equations: their linearity. An everyday example of linearity can
be seen in a rubber band. When it is stretched a little the extension is
proportional to the force. Nonlinearity steps in when you pull too far and the
band reaches its limit of elasticity. Stretch even further and it snaps. Because
nonlinearity is known to be a crucial ingredient in chaotic systems, it is often
said that quantum mechanics cannot be chaotic because it is linear.
But according to Michael Berry, a leading theorist in the study of quantum
chaos at the University of Bristol, this issue of linearity is a red herring.
鈥淭his is one of the biggest misconceptions in the business,鈥� he says. His
critique rests on the fact that it is possible to recast nonlinear classical
equations in a linear form and linear quantum equations in nonlinear form.
Berry鈥檚 preferred explanation for the difference between what happens in
classical and quantum systems as they edge towards chaos is that quantum
uncertainty imposes a fundamental limit on the sharpness of the dynamics. The
amount of uncertainty in a quantum system is quantified in Heisenberg鈥檚
uncertainty principle by a fixed value known as Planck鈥檚 constant. 鈥淚n classical
mechanics, objects can move along infinitely many trajectories,鈥� says Berry.
鈥淭his makes it easy to set up complicated dynamics in which an object will never
retrace its path鈥攖he sort of behaviour that leads to chaos. But in quantum
mechanics, Planck鈥檚 constant blurs out the fine detail, smoothing away the
肠丑补辞蝉.鈥�
This raises some interesting questions. What happens if you scale down a
classically chaotic system to atomic size? Do you still get chaos or does
quantum regularity suddenly prevail? Or does something entirely new happen? And
why is it that macroscopic systems can be chaotic given that everything is
ultimately built out of atoms and therefore quantum in nature? These questions
have been the subject of intense debate for more than a decade. But now a number
of experimental approaches have begun to offer answers.
Scrambled spectra
One of the earliest clues came from investigations of atomic absorption
spectra. If an atom absorbs a photon of light, it is possible for one of its
electrons to be kicked into a higher energy state. Normally, an atom鈥檚 energy
levels are spaced at mathematically regular intervals, accounted for by an
empirical formula given by the 19th-century physicist Johannes Rydberg. If an
atom absorbs photons with different energies, electrons are kicked into
different levels, and the result is a nice tidy absorption spectrum whose
details are characteristic of the chemical element involved. But when the atom
is subjected to a magnetic field, the line structure of the spectrum becomes
distorted. When the field is sufficiently intense, the spectrum becomes so
scrambled it looks pretty much random at higher energies.
The phenomenon is easier to understand in classical rather than quantum
mechanical terms. Viewed classically, atomic electrons move in orbits around the
nucleus rather like planets round the Sun. A magnetic field, though, introduces
an additional force which causes the electrons to swerve from their normal
trajectories. It鈥檚 rather like a stray star encroaching upon the Solar System.
If it got sufficiently close, the gravitational pull would at some point become
comparable to the pull between the Earth and our Sun. At this moment, the Earth
would find itself in a tug-of-war between the Sun and the interloping star. Such
a system would very probably be unstable, with the Earth switching erratically
between orbits around the Sun and the other star. The result would be a chaotic
orbit.
In the case of excited atoms, for small fields and lower energy states, the
magnetic swerving is small compared with the electrostatic pull towards the
nucleus and the electron continues to follow a stable orbit. But for strong
fields and highly excited states (where the electron is, on average, much
farther away from the nucleus) the swerving force becomes comparable to the
inward pull of the nucleus. In these circumstances, according to classical
predictions, the motion ought to be chaotic.
The effect was first studied back in 1969 by two astronomers, Garton and
Tomkins at Imperial College, London, who wanted to find out how the spectra of
stars would be affected by their powerful magnetic fields. Their experiments on
barium atoms produced one of the first surprises because their resulting
spectrum still displayed considerable regularity. A group at the University of
Bielefield in Germany repeated the experiments in the 1980s using higher
resolution equipment. Although the randomness was more apparent in their
spectra, it was still clear that quantum mechanics was in some strange way
superimposing its own order on the chaos.
Quantum billiards
More recently, signs of quantum suppression of chaos have come from another
experimental approach to quantum chaos: quantum billiards. On a conventional
rectangular table, it is quite common for a player to pot a ball by bouncing the
cue ball off the cushion first. In the hands of a skilled player, such shots are
often quite repeatable. But if you were to try the same shot on a rounded,
stadium-shaped table, the results are far less predictable: the slightest change
in starting position alters the ball鈥檚 trajectory drastically. So what you get
if you play stadium billiards is chaos.
In 1992, at Boston鈥檚 Northeastern University, Srinivas Sridhar and colleagues
substituted microwaves for billiard balls and a shallow stadium-shaped copper
cavity for the table. Sridhar鈥檚 team then observed how the microwaves settled
down inside the cavity. Although their apparatus is not of atomic proportions (a
cavity typically measures several millimetres across), the experiment exploits a
precise mathematical similarity between the wave equations of quantum mechanics
and the equations of the electromagnetic waves in this two-dimensional
situation. If microwaves behaved like billiard balls, you would not expect to
see any regular patterns. The experiments, however, reveal structures known as
鈥渟cars鈥� that suggest the waves concentrate along particular paths.
But where do these paths come from? One answer is provided by theoretical
work carried out back in the 1970s by Martin Gutzwiller of the IBM Thomas J.
Watson Research Center in Yorktown Heights near New York. He produced a key
formula that showed how classical chaos might relate to quantum chaos.
Basically, this indicates that the quantum regularities are related to a very
limited range of classical orbits. These orbits are ones that are periodic in
the classical system. If, for example, you placed a ball on the stadium table
and hit it along exactly the right path, you could get it to retrace its path
after only a few bounces off the cushions. However, because the system is
chaotic, these paths are unstable. You only need a minuscule error and the ball
will move off course within a few bounces. So classically you would not expect
to see these orbits stand out. But thanks to the uncertainty in quantum
mechanics, which 鈥渇uzzes鈥� the trajectories of the balls, tiny errors become less
significant and the periodic orbits are reinforced in some strange way so that
they predominate.
Sridhar鈥檚 millimetre-sized stadium was a good analogy for quantum behaviour,
but would the same effects occur in a truly quantum-sized system? This question
was answered recently by Laurence Eaves from the University of Nottingham, and
his colleagues at Nottingham and at Tokyo University. Eaves conducted his game
of quantum billiards inside an elaborate semiconductor 鈥渟andwich鈥�. He used
electrons for balls, and for cushions, he used a combination of quantum barriers
and magnetic fields. The quantum barriers are formed by the outer layers of the
sandwich, which gives the electrons a couple of straight edges to bounce back
and forth between. The other edges of the table are created by the restraining
effect of the magnetic field, which curves the electron motion in a complicated
way. As in Sridhar鈥檚 stadium cavity, the resulting dynamics ought to be
chaotic.
Number crunching
To do the experiments, Eaves needed ultraintense magnetic fields, so he took
his device to the High Magnetic Field Laboratory at the University of Tokyo,
which is equipped with some of the most powerful sources of pulsed magnetic
fields in the world. Meanwhile his colleagues in Nottingham, Paul Wilkinson,
Mark Fromhold and Fred Sheard, squared up to a heroic series of calculations,
deducing from purely quantum mechanical principles what the results should look
like.
In a spectacular paper that made the cover of Nature last month, the
team produced the first definitive evidence for quantum scarring, and precisely
confirmed the quantum mechanical predictions. Sure enough, the current flowing
through the device was predominantly carried by electrons moving along certain
鈥渟carred鈥� paths. Quantum regularity was lingering in the chaos rather like the
fading smile of the Cheshire Cat in Alice鈥檚 Adventures in Wonderland
.
In case these ideas seem academic it is worth noting that quantum chaos could
play an important role in the design of future semiconductor devices. At the
moment, transistor devices on silicon chips are still large enough for the
electrons to move through them diffusively like molecules in a gas. But as chip
manufacturers squeeze ever more logic gates onto silicon, says Eaves, in the
next 15 years transistors may become so small that electrons will instead flow
through them more like quantum billiard balls. 鈥淎t this point, we may well need
the principles of quantum chaos to understand how these devices will work,鈥� he
says.
But where does that leave the problem of how quantum mechanics turns into the
classical world on larger scales? One way of looking at the problem is to
investigate how a quantum chaos system actually evolves with time. Last
December, Mark Raizen and his colleagues at the University of Texas at Austin
managed to do just that, using an experimental version of a system called a
quantum kicked rotor. The idea is to couple two oscillating systems to produce
chaos. Imagine pushing a child鈥檚 swing. If you time your pushes in rhythm with
the swing, then it simply rises higher and higher. If you push at a different
frequency, the swing will sometimes be given a boost and sometimes slowed down.
If this is done too vigorously, the oscillations become chaotic.
In Raizen鈥檚 quantum version, ultracold sodium atoms were subjected to a
special kind of pulsed laser light. The laser beam was bounced between mirrors
to set up a short-lived standing wave鈥攁 periodic lattice of light that
remains motionless in space rather like the acoustic nodes on a violin string.
Depending on their precise location in the standing waves, the sodium atoms are
pushed around by the electromagnetic fields in the lattice. According to
classical calculations, the result is that the atoms should be kicked
chaotically along an increasingly energetic random walk. Raizen鈥檚 results
confirmed a long-standing prediction of the quantum theoretical descriptions of
these systems. The atoms did indeed move in a chaotic way to begin with. But
after around 100 microseconds (which corresponds to around 50 kicks) the
build-up in energy reached a plateau.
Break time
In other words, quantum mechanics does suppress the chaos but only after a
certain amount of time known as the 鈥渜uantum break time鈥�. This turns out to be
the crucial feature that distinguishes between quantum and classical predictions
of chaotic systems. Before the break time, quantum systems are able to mimic the
behaviour of classical systems by looking essentially random. But after the
break time, the system simply retraces its path. It is no longer random, but
stuckin a repeating loop albeit of considerable complexity.
But if this is right, how can classical systems exhibit chaos? Macroscopic
objects such as pendulums and planets are, after all, made out of atoms and are
therefore, ultimately, quantum systems. It turns out that classical systems are
in fact behaving exactly like quantum systems. The only difference is that for
classical systems, the quantum break times of macroscopic systems are
extraordinarily long鈥攆ar longer than the age of the Universe. If we could
study a classical system for longer than its quantum break time, we would see
that the behaviour was not really chaotic but quasi-periodic instead. Thus,
quantum and classical realities can be reconciled, with the classical world
naturally embedded in a larger quantum reality. Or, as physicist Dan Kleppner of
the Massachusetts Institute of Technology puts it, 鈥淎nything classical mechanics
can do, quantum mechanics can do better鈥�.
Since much of the experimental work on quantum chaos has agreed with
theoretical predictions, it could be tempting to say 鈥淪o what?鈥�. We already knew
that quantum theory was right. Well, research on quantum chaos does hold out the
promise of some remarkable discoveries. Berry is excited by what appears to be a
deep connection between the problem of finding the energy levels of a quantum
system that is classically chaotic and one of the biggest unsolved mysteries in
mathematics: the Riemann hypothesis. This concerns the distribution of prime
numbers. If you choose a number n and ask how many prime numbers there
are less than n it turns out that the answer closely approximates the
formula: n/log n. The formula is not exact, though: sometimes
it is a little high and sometimes it is a little low. Riemann looked at these
deviations and saw that they contained periodicities. Berry likens these to
musical harmonies: 鈥淭he question is what are the harmonies in the music of the
primes? Amazingly, these harmonies or magic numbers behave exactly like the
energy levels in quantum systems that classically would be chaotic.鈥�
Deep connection
This correspondence emerges from statistical correlations between the spacing
of the Riemann numbers and the spacing of the energy levels. Berry and his
collaborator Jon Keating used them to show how techniques in number theory can
be applied to problems in quantum chaos and vice versa. In itself such a
connection is very unusual. Although sometimes described as the queen of
mathematics, number theory is often thought of as pretty useless, so this deep
connection with physics is quite astonishing.
Berry is also convinced that there must be a particular chaotic system which
when quantised would have energy levels that exactly duplicate the Riemann
numbers. 鈥淔inding this system could be the discovery of the century,鈥� he says.
鈥淚t would become a model system for describing chaotic systems in the same way
that the simple harmonic oscillator is used as a model for all kinds of
complicated oscillators. It could play a fundamental role in describing all
kinds of 肠丑补辞蝉.鈥�
The search for this model system could become the Holy Grail of quantum chaos
research. Until it is found, we cannot be sure of its properties, but Berry
believes the system is likely to be rather simple, and expects it to lead to
totally new physics. It is a tantalising thought. Out there is a physical
structure waiting to be discovered. If we find it, the remarkable experiments
that we have recently witnessed in this discipline would be crowned by an
experimental apparatus that could do more than anything to unlock the secrets
of quantum chaos.