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Where Two Worlds Meet |
Butterflies can cause hurricanes, according to the classical theory of chaos. But what happens when chaos encounters the quantum world?
Julian Brown
Twice in 20th-century physics, the notion of
unpredictability has shaken scientists' 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' 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 -the 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's
uncertainty principle -if 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.
Chaos is a younger discipline . Although some of its conceptual
elements had already been appreciated
by
Leibnitz in the 17th century,and
Poincaré 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
"butterfly effect", in which, in theory, the tiny
air disturbance from the flapping of a butterfly's 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.
"Quantum regularity lingers in the chaos like the fading smile of the Chesire Cat"
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. "This is one of the biggest misconceptions
in the business," 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's 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's
uncertainty principle by a fixed value known as
Planck's constant. In classical mechanics,
objects can move along infinitely many trajectories," says Berry. This makes
it easy to set up complicated dynamics in which an object will never retrace
its path-the sort of behaviour that leads to chaos. But in quantum mechanics,
Planck's constant blurs out the fine detail, smoothing away the chaos."
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's 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's 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 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's trajectory drastically.
So what you get if you play stadium billiards is chaos. In 1992, at Boston's
Northeastern University, Srinivas Sridhar and colleagues substituted microwaves
for billiard balls and a shallow stadium-shaped copper cavity for the table.
Sridhar's 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 "scars"
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 "fuzzes" 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's 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 "sandwich" . 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 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's stadium cavity,the resulting dynamics ought to be chaotic.
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Number Crunching
To do the experiments,Eaves needed ultraintense magnetic fields, so he took
his device to the High Magnetic Field Laboratory at 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,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 "scarred"
paths. Quantum regularity was lingering in the chaos rather like the fading
smile of the Cheshire Cat in
Alice's Adventures in Wonderland.
"Finding this system could be the discovery of the century"
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 . "At this
point, we may well need the principles of quantum chaos to understand how
these devices will work," 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's 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's quantum version, ultra cold 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 -a 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's 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 "quantum break time". 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 stuck in 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-far 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, " anything classical mechanics can do, quantum mechanics can do
better".
"What
are the harmonies in the music of the
primes?"
Since much of the experimental work on quantum chaos has agreed
with theoretical predictions , it could be tempting to say "So what?". 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 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 : "The 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."
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. "Finding this
system could be the discovery of the century," he says. " It 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 chaos."
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.
| A WEIRD NEW WORLD Throughout the history of physics,
large-scale phenomena have been explained by smaller scale ones. Gases, for
example, can be regarded as swarms of hyperactive atoms. Atoms can be described
as tiny electrons orbiting dense nuclei. These, in their turn, can be broken
down into protons, then quarks and so on. But perhaps we shouldn't expect
the laws of the microworld to explain the world on the largest scale. It's
verging on heresy, but Thomas Banks of Rutgers University and the University
of California in Santa Cruz believes that we simply can't build everything
from the bottom up - that some large-scale aspects of the cosmos may be just
as fundamental as the laws that govern particles. In fact, he goes so far
as to suggest that we could be living in a top-down Universe. [New Scientist 10 Aug 2002] |