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In Collaboration with, Dong-Ho Wu, Naval Research Laboratory, Washington, DC 20375, US Our recent research has shown that tunneling in symmetric, closed billiard systems separated by a barrier is very different for integrable vs. completely chaotic systems. Numerical calculations have shown that fluctuations in tunneling rates for many integrable billiard shapes can easily range over several orders of magnitude and generally increase with increasing eigenstate energy. On the contrary the fluctuations for chaotic shapes are one to two orders of magnitude smaller than for integrable shapes and on average the fluctuations decrease with increasing energy in the semiclassical regime. We have developed a random-plane-wave theory that appears to capture correctly the average and variance (fluctuation amplitude) of the tunneling rates in chaotic billiards. Such strong dependence of tunneling rates on potential well shapes suggest that shape can be used in quantum dot devices as a means to control or modify tunneling rates in such devices. However, there remain many unanswered questions about shape dependence in tunneling which we think a semiclassical approach could shed some light on. These questions are, (1) How is tunneling rate shape dependence manifested in open systems?, (2) How can one treat more realistic (non-hard wall) potentials?, (3) What happens in mixed (integrable and chaotic) systems?, (4) Are the observations in the semiclassical regime of the calculations and approximate random plane wave theory of increasing fluctuations in integrable systems and decreasing fluctuations in chaotic systems with increasing energy generic?, and (5) How, if at all, can semiclassical techniques such as Initial Value Representations and Ray Splitting approaches be turned into efficient and accurate numerical methods? I hope to explore these and other questions with the participants of the Advanced Study Group for Dynamical Tunneling. |
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