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Department of Physics and Astronomy Quantum Chaos Research

Ph.D. Theses

Violations of ergodicity as observed in quantum chaotic systems and Coulomb blockade peak heights

by Cerruti, Nicholas Robert, Ph.D., Washington State University, 2000 , 222 pages [Thesis (7.3 MB)]

Abstract (Summary) Quantum systems that are classically chaotic were conjectured to follow the ergodic hypothesis stating that time averaging is equal to energy averaging. This dissertation investigates systematic deviations from this hypothesis in quantum chaotic systems and Coulomb blockade peak heights. The first part of this dissertation contains a semiclassical analysis of the response of eigenvalues to a perturbation in quantum chaotic systems. The variance of the response is related to a classical diffusion coefficient. Also, we developed a new measure that sensitively probes phase space localization properties of the eigenstates based upon a correlation between the eigenvalues and the eigenfunctions. In the ergodic model, the correlation is predicted to be zero, i.e. no localization of the eigenfunctions. However, we find large deviations from ergodic theory based on classical orbits of the system. The second part of this dissertation deals with quantum dots and an effect known as Coulomb blockade. Current can flow only if two different charge states of a quantum dot are tuned to have the same energy; this produces a peak in the conductance of the dot whose magnitude is directly related to the magnitude of the wave function near the contacts of the dot. Since dots are generally irregular in shape, the dynamics of the electrons are chaotic, and the characteristics of Coulomb blockade peaks reflect those of wave functions in chaotic systems. We developed a semiclassical theory of Coulomb blockade peak heights and showed that the dynamics in the dot lead to a large modulation of the peak heights. The corrections to the standard statistical theory, which assumes ergodicity, of peak height distributions, power spectra, and correlation functions are non-universal and can be expressed in terms of the classical periodic orbits of the dot that are well coupled to the leads. The resulting correlation function oscillates as a function of peak number in a way defined by such orbits; in addition, the correlation of adjacent conductance peaks is enhanced.