Semiclassical propagation up to the Heisenberg time (Vol. 44 No. 6)

Semiclassical propagation of waves is a fruitful approach to understand and evaluate a wide set of physical processes. This is performed by associating quantum states with Lagrangian manifolds in phase space, and the propagation is accomplished by the evolution of manifolds. However, long time propagation in Hamiltonian systems with chaotic dynamics is a longstanding unsolved problem; the reason being that Lagrangian manifolds evolve into very complex objects.

Recently, we have shown that by using the stable and unstable manifolds of periodic orbits, the propagation is simplified enormously. For this reason, in this paper we study in detail the manifolds of a periodic orbit of the hyperbola billiard, finding that they are organized by a simple tree structure. Then, we compute a complete set of homoclinic orbits (resulting from the intersection of the manifolds), which is required to evaluate the autocorrelation function of a quantum state constructed in the neighborhood of the periodic orbit (resonance). Finally, we compare the quantum and semiclassical autocorrelation up to the Heisenberg time, finding a relative error of the order of the Planck constant.