A Matrix Element for Chaotic Tunnelling Rates and Scarring Intensities

Abstract

It is shown that tunnelling splittings in ergodic double wells and resonant widths in ergodic metastable wells can be approximated as easily-calculated matrix elements involving the wavefunction in the neighbourhood of a certain real orbit. This orbit is a continuation of the complex orbit which crosses the barrier with minimum imaginary action. The matrix element is computed by integrating across the orbit in a surface of section representation, and uses only the wavefunction in the allowed region and the stability properties of the orbit. When the real orbit is periodic, the matrix element is a natural measure of the degree of scarring of the wavefunction. This scarring measure is canonically invariant and independent of the choice of surface of section, within semiclassical error. The result can alternatively be interpretated as the autocorrelation function of the state with respect to a transfer operator which quantises a certain complex surface of section mapping. The formula provides an efficient numerical method to compute tunnelling rates while avoiding the need for the exceedingly precise diagonalisation endemic to numerical tunnelling calculations.

Submitted to Annals of Physics. This work has been submitted to Academic Press for possible publication

References in corpus (2)

• Semiclassical theory of magnetotransport through a chaotic quantum well
• Semiclassical description of resonant tunneling

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• Semiclassical transmission across transition states
• Short-Time Effects on Eigenstate Structure in Sinai Billiards and Related Systems
• Scarring Effects on Tunneling in Chaotic Double-Well Potentials
• Scarring and the statistics of tunnelling
• Uniform approximation of barrier penetration in phase space
• A realistic example of chaotic tunneling: The hydrogen atom in parallel static electric and magnetic fields
• Investigating quantum transport with an initial value representation of the semiclassical propagator
• Riemann surfaces of complex classical trajectories and tunnelling splitting in one-dimensional systems
• Regularization of Tunneling Rates with Quantum Chaos
• Resonance- and Chaos-Assisted Tunneling
• Estimating the spectral density of unstable scars