Ray-Splitting Correction to the Weyl Formula: Experiment versus Theory

Peter M. Koch and Corrie Vaa
Department of Physics and Astronomy, State University of New York, Stony Brook, NY 11794-3800 USA
Reinhold Blümel
Department of Physics, Wesleyan University, Middletown, CT 06459-0155 USA

The problem of blackbody radiation led physicists, notably Max Planck, to the quantum theory in 1900. Lecturing in Göttingen in 1910, physicist Hendrik Lorentz aroused the interest of mathematicians with a conjecture that the total number of resonances below frequency f in a three-dimensional electromagnetic cavity is proportional to its volume, independent of its shape, when f is large. Though his mathematician host David Hilbert is said to have predicted that the proof would not be done in his lifetime, Hermann Weyl began the next year to publish such proofs [1] for various resonant systems. Thus began the derivation of mathematical expressions, now called Weyl formulae, that estimate the number of resonances based only on dimensionality, geometry (e.g., volume, surface area, length, topology, connectivity), and what kind of wave-boundary conditions apply. Acoustics researchers first examined in 1939 theoretical correction terms needed when f is not asymptotically large and made the first measurement of such a correction [2] using quasi two-dimensional acoustical cavities (air-filled chambers). Emphasizing acoustics but making connections with theoretical problems in quantum chaos, Couchman et al. [3] considered the consequences of ray splitting, which occurs when a wave encountering a ray-splitting boundary is split into a reflected and transmitted wave. Experiments by Sirko et al. [4] confirmed the existence of the ray-splitting effect. Prange et al. [5] predicted that an abrupt change in the dielectric constant of media filling a microwave cavity would require a new ray-splitting correction to the Weyl formula. In this talk we report the first experimental confirmation [6] of the predicted ray-splitting correction [7]. We use a rectilinear microwave cavity whose height, a few centimeters, is much smaller than its length and width, each nearly a meter. Two rectilinear paraffin wax bars that fill the length and height but only part of the width are put inside the cavity. When the bars are pushed together, they supply only two ray-splitting air/wax boundaries. When the bars are moved apart, they supply four boundaries. The key to controlling systematic effects in our experiment and being able to reveal the small ray-splitting correction is to compare spectra of resonances for “bars apart” versus “bars together”. Moving the bars carefully does not change their dimensions, the cavity dimensions, or the dielectric constants of the air and wax, as long as we keep the temperature constant to half a degree Celsius and avoid other systematic effects. Without any missing or spurious added resonances, we obtain complete spectra with at least 152 resonances up to 1.95 GHz. The rectilinear geometry of the cavity and wax bars allows numerical simulations to be carried out that confirm the completeness of the experimental spectra. That all measured resonances correspond to “two-dimensional” cavity modes makes the electromagnetic cavity experiment equivalent mathematically to a two-dimensional quantal ray-splitting billiard [8]. Therefore, the experiment also demonstrates the existence of ray-splitting corrections to the Weyl formula for quantum levels that would exist in, e.g., quantum dots or other quantum microcavities having ray-splitting boundaries.

NSF grants (PHY-9732443 and -0099398 to Stony Brook; PHY-9984075 to Wesleyan) supported this work.

[1] H. Weyl, Gesammelte Abhandlungen (Springer, Berlin, 1968).
[2] R.H. Bolt, J. Acoust. Soc. Am. 11, 184 (1939).
[3] L. Couchman, E. Ott, and T.M. Antonsen, Jr., Phys. Rev. A 46, 6193 (1992).
[4] R.E. Prange, E. Ott, T.M. Antonsen, Jr., B. Georgeot, and R. Blümel, Phys. Rev. E 53, 207 (1996).
[5] L. Sirko, P.M. Koch, and R. Blümel, Phys. Rev. Lett. 78, 2940 (1997).
[6] C. Vaa, P.M. Koch, and R. Blümel, Phys. Rev. Lett. (in press, 2003).
[7] A. Kohler and R. Blümel, Ann. Phys. (NY), 267, 249 (1998).
[8] H.-J. Stöckmann, Quantum Chaos (Cambridge University Press, New York, 1999).