Computing classical escape rates from periodic orbits in chaotic hydrogen

When placed in parallel magnetic and electric fields, the electron trajectories of a classical hydrogen atom are chaotic. The classical escape rate of such a system can be computed with classical trajectory Monte Carlo techniques, but these computations require enormous numbers of trajectories, provide little understanding of the dynamical mechanisms involved, and must be completely rerun for any change of system parameters, no matter how small. We demonstrate an alternative technique to classical trajectory Monte Carlo computations based on classical periodic orbit theory. In this technique, escape rates are computed from a relatively modest number (a few thousand) of periodic orbits of the system. One only needs the orbits' periods and stability eigenvalues. A major advantage of this approach is that one does not need to repeat the entire analysis from scratch as system parameters are varied; one can numerically continue the periodic orbits instead. We demonstrate the periodic orbit technique for the ionization of a hydrogen atom in applied parallel electric and magnetic fields. Using fundamental theories of phase space geometry, we also show how to generate nontrivial symbolic dynamics for acquiring periodic orbits in physical systems. A detailed analysis of heteroclinic tangles and how they relate to bifurcations in periodic orbits is also presented.