Quantum Chaos is the study of the quantum mechanics of classically chaotic systems. The field is currently a very active area of research. Our main thrust is to explore, via electromagnetic experiments, the manifestations of classical chaos in wave mechanics.
Our research program has led to several noteworthy observations listed below:
- Experimental “proof” of a mathematical theorem on “Not Hearing the Shape of Drums”. Click here to see cover story in Science News.
- Direct experimental observation of scars in quantum eigenfunctions of microwave cavities. Phys. Rev. Lett. , 67, 785 (1991)
- Experimental observation of localized wavefunctions in disordered billiards, and deviations from the Porter-Thomas distribution due to localization in disordered billiards. These were the first experiments to be performed on disordered billiards. Phys. Rev. Lett., 75, 822 (1995)
- First experimental observation of quantum fingerprints of classical Ruelle-Pollicott resonances. Phys. Rev. Lett., 85, 2360 (2000), Nobel Symposium on “Quantum Chaos Y2K”
- Experimental studies of correlations of chaotic and disordered eigenfunctions and comparison with supersymmetry nonlinear sigma models. Phys. Rev. Lett., 85, 2360 (2000)
- Tunneling Proximity Resonances: Experimental observation using dielectric resonators Phys. Lett. A, 268 (4-6), 399 (2000)
- Precision tests of universal aspects of quantum spectra, observation of a “correlation hole”, and tests of applicability of Random Matrix Theories to microwave cavities. Phys. Rev. E. (Rapid Comm.), 49, R11 (1994)
- Observation of Porter-Thomas distribution and fluctuations in eigenfunctions of chaotic billiards. Phys. Rev. Lett., 75, 822 (1995)
- Observation of quantum resonances, universal properties, and comparison with semiclassical theories of the n-disk system. Phys. Rev. Lett., 82, 5233 (1999), Phys. Rev. E, 61, 3652 (2000)
It is evident that the experiments are able to explore a remarkable range of issues in Quantum Chaos, including tests of fundamental theories and different classical limits. The microwave experiments have yielded entirely new insights and perspectives concerning the quantum-classical correspondence. The reasons for this success are the ability to study well-defined geometries where the classical dynamics is clearly known, and the precision and flexibility of the experiments. The experiments have led to tests of key theoretical ideas while at the same time raising entirely new questions and motivating theorists in new directions.
Our work in this area is currently funded by National Science Foundation (Division of Atomic, Molecular and Optical Physics).
Not Hearing the Shape of Drums
While the theorem of isospectral domains was proved on abstract mathematical grounds, the actual resonance frequencies and wave functions of the geometries were not known. The microwave experiments were able to make an unique contribution by exploiting the the fact that below a certain frequency, the equation obeyed in thin microwave cavities is the same as the Helmholtz wave equation. The resonances were obtained by actually constructing the shapes using copper. The wavefunctions are obtained using a cavity perturbation technique. Therefore these experiments were not only able to physically realize drums which sounded alike, but also were able to show that there were no degeneracies in the first 54 resonances.