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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.

Butterfly in GlassBerry'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.

watercolour of butterflySridhar'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.

Chesire Cat

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.

Professor chasing butterflyBerry 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]


Julian Brown is a freelance science writer based in London





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