Tunneling in Complex Systems and Periodic Orbits

Marcus W. Beims, V. Kondratovich, J. B. Delos

We consider a system having decaying states with complex energies ${E}_{j}\ensuremath{-}i{\ensuremath{\Gamma}}_{j}/2$, and we define the ``width-weighted spectrum'' as $D\ensuremath{\Gamma}(E)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{j}^{}{\ensuremath{\Gamma}}_{j}\ensuremath{\delta}({E\ensuremath{-}E}_{j})$. We derive a semiclassical formula for this width-weighted spectrum, a formula which is analogous to the periodic-orbit representation of the density of states. The formula applies if classical motion is regular and if decay occurs by tunneling through a barrier. The semiclassical formula involves not periodic orbits or closed orbits, but action integrals associated with irreducible loops on the ``extremal torus,'' on which the particle climbs up the barrier and hangs at the top. Calculations confirm the validity of the formula.

Chaoming Song

The Gutzwiller trace formula establishes a profound connection between the quantum spectrum and classical periodic orbits. However, its application is limited by its reliance on the semiclassical saddle point approximation. In this work, we explore the full quantum version of the trace formula using the Lefschetz thimble method by incorporating complexified periodic orbits. Upon complexification, classical real periodic orbits are transformed into cycles on compact Riemann surfaces. Our key innovation lies in the simultaneous complexification of the periods of cycles, resulting in a fully quantum trace formula that accounts for all contributions classified by the homology classes of the associated Riemann surfaces. This formulation connects the quantum spectrum to contributions across all complex time directions, encompassing all relevant homology classes. Our approach naturally unifies and extends two established methodologies: periodic orbits in real time, as in Gutzwiller's original work, and quantum tunneling in imaginary time, as in the instanton method.

Joachim Ankerhold

A. S. Bhullar, R. Blümel, Patrick M. Koch

In the familiar concept of ray splitting, a ray incident on an interface splits into reflected and transmitted rays. Tunnelling adds something new to ray splitting: an incident Newtonian ray splits into a reflected Newtonian ray and a transmitted ray with complex classical action?a ghost orbit. Using novel periodic-orbit expansion techniques, we are able to compute explicitly and analytically energy eigenvalues of an energy-scaling one-dimensional step potential. Inclusion of all periodic orbits in our series expansions, in particular a class of orbits called pure ghosts with imaginary classical action, makes our formulae exact. We suggest an experiment that may be used to verify the importance of pure ghosts.

Max Meynig, Hal M. Haggard