DEPARTMENT OF MATHEMATICAL SCIENCES

APPLIED AND COMPUTATIONAL MATHEMATICS SEMINAR

**Speaker:** Lou Pecora, Naval Research Laboratory

**Title: ***
Regularization of Tunneling Rates in Two-dimensional Systems with Quantum Chaos
*

**Abstract:**
In quantum mechanics particles can tunnel through barriers. This phenomenon is used in transistors, photon detectors and quantum nano systems. Tunneling in one-dimensional systems is a trivial undergraduate problem. Tunneling in higher-dimensional systems is a research problem. Real systems are at least two-dimensional so we must understand tunneling in those more complex systems. We study quantum tunneling of particles in various shaped, two-dimensional, flat, double wells by calculating the energy splitting between symmetric and anti-symmetric state pairs. For shapes that have regular or nearly regular classical behavior (e.g. rectangular or circular wells) we find that tunneling rates for nearby energy states vary over wide ranges. Rates for energetically close quantum states can differ by several orders of magnitude. As we transition to well shapes that admit more classically chaotic behavior (e.g. the stadium, the Sinai billiard) the range of tunneling rates narrows, often by an order of magnitude or more. For well shapes in which the classical behavior appears to be fully chaotic (as determined from numerical bounce maps) the tunneling rates' range narrows to about a factor of 2 or so between the smallest and largest rates in a wide range of energies. This dramatic narrowing appears to come from destabilization of periodic orbits in the regular wells that produce the largest and smallest tunneling rates. It is in this sense that we say the quantum chaos regularizes the tunneling rates and presents a possible approach to controlling tunneling in devices. I will present a numerical method (the Boundary Element Method) which we used to calculate the quantum states and tunneling rates along with the above results. Furthermore, we have recently developed a theory based on a random plane wave approximation appears to qualitatively reproduce the changes in tunneling rate distributions seen in the numerical results.

This talk is based on joint work of
Louis M. Pecora, Hoshik Lee, and Dong-Ho Wu

Naval Research Laboratory, code 6362, Washington, DC, 20375, USA

Ed Ott, Tom Antonsen, and Ming-Jer Lee, U. Maryland, College Park, MD USA

**Time:** Friday, October 15, 2010, 1:30-2:30 p.m.

**Place:** Science and Tech I, Room 242

Department of Mathematical Sciences

George Mason University

4400 University Drive, MS 3F2

Fairfax, VA 22030-4444

http://math.gmu.edu/

Tel. 703-993-1460, Fax. 703-993-1491