Statistical Modeling of Wave Chaotic Transport and Tunneling
Statistical Modeling of Wave Chaotic Transport and Tunneling
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Date
2013
Authors
Lee, Ming-Jer
Advisor
Ott, Edward
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Abstract
This thesis treats two general problem areas in the field of wave chaos.
The first problem area that we address concerns short wavelength tunneling
from a classically confined region in which the classical orbits are chaotic. We de-
velop a quantitative theory for the statistics of energy level splittings for symmetric
chaotic wells separated by a tunneling barrier. Our theory is based on the ran-
dom plane wave hypothesis. While the fluctuation statistics are very different for
chaotic and non-chaotic well dynamics, we show that the mean splittings of differ-
ently shaped wells, including integrable and chaotic wells, are the same if their well
areas and barrier parameters are the same. We also consider the case of tunneling
from a single well into a region with outgoing quantum waves.
Our second problem area concerns the statistical properties of the impedance
matrix (related to the scattering matrix) describing the input/output properties of
waves in cavities in which ray trajectories that are regular and chaotic coexist (i.e.,
`mixed' systems). The impedance can be written as a summation over eigenmodes
where the eigenmodes can typically be classified as either regular or chaotic. By
appropriate characterizations of regular and chaotic contributions, we obtain statis-
tical predictions for the impedance. We then test these predictions by comparison
with numerical calculations for a specific cavity shape, obtaining good agreement.