Semiclassical propagation up to the Heisenberg time

Departamento de Física, Comisión Nacional de Energía Atómica - Av. del Libertador 8250, (C1429BNP) Buenos Aires, Argentina and Escuela de Ciencia y Tecnología, Universidad Nacional de General San Martín - Alem 3901, (B1653HIM) Villa Ballester, Argentina

Received: 7 June 2013
Accepted: 10 July 2013

Abstract

By using a quantum Hamiltonian system with classically chaotic dynamics, we demonstrate that it is possible to propagate waves, at a semiclassical level, for extremely long times of the order of the Heisenberg time. We achieve this unexpected result with a new formula that evaluates the autocorrelation function of a quantum state living in the neighborhood of a short periodic orbit, the so-called resonance, in terms of the set of homoclinic orbits; this set is given by the intersection of the stable and unstable manifolds of the periodic orbit. Here we study the manifolds of the shortest periodic orbit of the hyperbola billiard (a chaotic Hamiltonian system), finding a surprisingly simple tree structure. Then, we compute a complete set consisting of the first 18 146 homoclinic orbits, and by using this data we analyze the convergence of the new formula. Finally, we compare the quantum and semiclassical autocorrelation of resonances up to the Heisenberg time, obtaining a relative error O(ℏ) in correspondence with semiclassical predictions.