Breakdown of homoclinic orbits to $L_1$ of the hydrogen atom in a circularly polarized microwave field

TL;DR

This paper investigates the chaotic dynamics near the $L_1$ equilibrium point of a Rydberg electron in a circularly polarized microwave field, combining analytical and numerical methods to compute invariant manifold distances.

Contribution

It introduces a semi-analytical method for computing manifold distances and a new class of Hamiltonians called Toy CP systems for comparison with existing theories.

Findings

Manifold distances are exponentially small in the perturbation parameter K.

The semi-analytical method effectively combines normal form, Melnikov, and averaging techniques.

Toy CP systems facilitate comparison between numerical results and theoretical predictions.

Abstract

We consider the Rydberg electron in a circularly polarized microwave field, whose dynamics is described by a 2 d.o.f. Hamiltonian, which is a perturbation of size $K>0$ of the standard rotating Kepler problem. In a rotating frame, the largest chaotic region of this system lies around a center-saddle equilibrium point $L_1$ and its associated invariant manifolds. We compute the distance between stable and unstable manifolds of $L_1$ by means of a semi-analytical method, which consists of combining normal form, Melnikov, and averaging methods with numerical methods. Also, we introduce a new family of Hamiltonians, which we call Toy CP systems, to be able to compare our numerical results with the existing theoretical results in the literature. It should be noted that the distance between these stable and unstable manifolds is exponentially small in the perturbation parameter $K$ (in analogy with the $L_3$ libration point of the R3BP).