Waves of Chaos

Srinivas Sridhar

Billiard tables, vibrating drums, microwaves in cavities, and quantum mechanics: these seemingly unconnected things come together in the research on quantum chaos that my students and I carry out.

The field of quantum chaos merges two of the major intellectual developments in physics during this century: quantum mechanics, and chaos in classical mechanics. Classical mechanics is concerned with the motion of objects in our everyday world and is well described by Newton’s laws. Quantum mechanics describes the unseen actions of electrons and other elementary particles.

Both quantum mechanics and chaos defy our common sense and intuition. In chaos, a particle seemingly disobeys the commonly observed repeatability to which we are accustomed. In quantum mechanics, a particle does not even behave like a particle; instead it acquires the characteristics of a wave. This latter notion has occupied some of the most brilliant minds of the twentieth century and still offers many fascinating puzzles. The interface between quantum and classical properties is a key area of modern research in physics.

The paradigm of classical chaos is best illustrated in terms of a billiard table. If you strike a ball with your cue stick, it follows straight-line trajectories between hits at the table edges. Another ball struck nearly, but not exactly, the same way will follow closely along the initial trajectory. The standard rectangular billiard table is not very sensitive to initial conditions (in this case, the details of the initial strike). Consider instead a billiard table with a cylindrical post in the middle. Now, however, two balls struck only slightly differently will travel along completely different paths on the table. Such a table is called a Sinai billiard table after the famous Soviet mathematician Y. G. Sinai, who first considered the mechanics of a particle on such a billiard table and showed that it obeyed the technical definition of chaos, namely, a sensitivity to initial conditions (in this case, the exact direction of the initial strike). Indeed, it would be almost impossible to play on such a table, since it would be intolerant of the slightest error in the initial strike. The difference between playing a regular table and a Sinai billiard table is not unlike facing a regular pitcher and a knuckleballer!

How does this manifest itself in the tiny world of electrons and other quantum particles? Are the properties of an electron different between a regular and a Sinai billiard table of tiny, atomic dimensions? In other words, does classical chaos affect quantum mechanics? One of the first scientists to raise this issue was Einstein, although it took another half a century before further answers emerged.

Since quantum properties are essentially wavelike, I use waves of a different kind, microwaves, to study the same phenomena. Our experiments are done in microwave cavities, which are simply metallic boxes similar to a large microwave oven, but very thin in the vertical dimension. Under these conditions the equations that describe the quantum properties of an electron and the description of the microwaves in the cavity are exactly the same. A key advance we made in 1991 was to devise a means to observe the wave patterns inside the microwave cavity. When we studied a Sinai billiard cavity, we found that waves tend to line up along certain preferred directions called periodic orbits. These orbits are trajectories along which a real billiard ball would bounce back and forth forever if placed exactly on the table — any error would send the ball careening all over the table. The waves, however, seem to compensate for any errors and tend to stay along these orbits. This phenomenon, called a scar,was predicted theoretically in 1984, but our experiment, reported in Physical Review Letters in 1991, was the first to directly observe these scars.

As often occurs in my research, a key technical advance — the method to observe wavefunctions (the technical name for the wave patterns inside the cavity) — led to the elucidation of an important physical phenomenon. The initial experiments were “bootlegged” on other projects, and were done in my spare time during weekends and evenings so as not to take away from my other research program on superconductivity and from my teaching duties. Soon after, I obtained seed funding from the Research and Scholarship Development Fund and the Center for Electromagnetics Research, both at Northeastern, and the success of our work then led to funding for the project from the National Science Foundation and the Office of Naval Research.

A powerful principle in physics is that of universality, which recognizes the underlying commonality of apparently diverse phenomena. Thus wave phenomena, from matter waves to light waves to seismic waves, have common features, the most familiar of which are interference and diffraction. Our experiments demonstrate that there is another set of properties that only depend on the shape of the container in which they bounce around; they are universal in that they are common to waves and also to quantum properties that are wavelike. As mentioned earlier, the shape determines whether the motion of a billiard ball or light ray is regular or chaotic. For those shapes where the classical motion is chaotic, the corresponding wave spectra have certain universal properties that are the same regardless of the nature of the wave. Furthermore, these properties are different from those where the shape is regular. This distinction between regular and chaotic applies even to nuclear, atomic, and chemical physics. Thus, a profound underlying similarity between, say, a benzene molecule and microwaves in a Sinai billiard cavity has emerged from these studies. The elucidation of fundamental laws is, of course, the basic endeavor of the physicist.

A quantitative description, not just a record of an observation, is the essential goal of a physicist attempting to understand nature. Thus, theoretical calculations play an important role in combination with our experiments. The interplay of chaos and disorder is now understood in terms of what are called supersymmetry theories, which originally emerged in another field of physics, that of elementary particles.

Chaos is just one limit of the influence of the classical motion on wave properties. The next stage is that of disorder, which leads to another phenomenon common to waves, localization. This means that under certain conditions waves can interfere constructively, that is, add up, to become large intensities in certain regions. This is likely to happen when the medium or region in which the waves are bouncing around is “disordered.” We have been able to study these in specially made microwave cavities in which we introduce disorder, using copper tiles much like pebbles on a beach. When the wave functions are measured, we find that they are forced to be concentrated in certain areas of the cavity (see Figure 1). The conditions are magically right — the frequency and size of the region — for the waves to get concentrated in these regions. As the microwave frequency is changed, the regions of concentration may suddenly appear elsewhere. This phenomenon of localization is, once again, common to particle electrons in a solid, as well as to light and sound waves, and an understanding of this phenomenon is useful in diverse areas, from electronic devices to light scattering in random media.

In 1994 we had a very enjoyable interlude when our experiments turned out to be relevant to an important mathematical problem. More than thirty years ago, a famous mathematician, Mark Kac, asked the ques-tion, “Can you hear the shape of a drum?” What he meant is this: suppose you had perfect hearing and could identify every tone of a vibrating drum; would you then be able to determine its shape? In principle you could, provided there are no drums that have different shapes but have the same spectrum, that is, are isospectral. This question fascinated mathematicians, and in 1992, Carolyn Gordon, Scott Wolpert, and Dennis Webb showed that there were real (two- dimensional) drums with different shapes but the same spectrum.

When I casually mentioned to Murray Schwartz of the mathematics department at Northeastern that we could do experiments on these special shapes, he immediately put me in touch with other mathematicians who showed me simpler shapes to work with. We constructed these shapes and observed that, indeed, every vibrating frequency in one was exactly equal in the other. Our experiment, which was reported in Physical Review Letters, captured the imagination of mathematicians, physicists, and reporters and was profiled in international science newsmagazines such as Physics World and La Recherche.

On one level, our work examines fundamental intellectual issues, such as the connection between classical and quantum mechanics, as well as a new way of describing the behavior of electromagnetic waves. On another level, the work can have impact on a variety of practical situations. Electronic devices are now reaching sizes where the quantum chaotic properties we study may play an important role in performance. The general principles derived here apply to microwave scattering from various objects and could lead to algorithms to reduce “clutter” and correct for errors in the transmission of light or microwaves through rain and snow. I envisage a class of microwave devices that exploit, rather than avoid, the chaotic properties we elucidate. It is particularly satisfying to be involved in research that ranges from the purely intellectual to the very practical.

Srinivas Sridhar is a professor of physics. He received his B.Sc. from Calcutta University in India, an M.Sc. from Madurai University in India, an M.S. from Ohio State University, and his Ph.D. from the California Institute of Technology. His research is funded by the National Science Foundation and the Air Force Office of Scientific Research.