Integrable and Chaotic Behaviour in the Paul Trap and the Hydrogen Atom in a Generalized van der Waals Potential

Abstract

For almost two decades the problem of Rydberg atoms interacting with external fields has proved to be a unique atomic laboratory for the study of chaos.^1–5 While some studies have specialized to interactions with explicitly time dependent microwave fields,1 others have dealt instead with static fields.^2–4 Many of the most outstanding cases of interest fall into the second category, and, further, their Hamiltonians turn out to be particular limits of the so-called generalized van der Waals Hamiltonian (GVDW)^6–7 (e.g., the notoriously chaotic quadratic Zeeman effect^2–5). Studies of atomic Rydberg states have clearly led to substantial progress being made in furthering an understanding of the implications of classical chaos in quantum systems. Quite recently, however, investigations of a different problem, the Paul trap,^8–9 suggest that this system might prove of comparable value in the study of chaotic dynamics in atomic systems. In this paper we show that the generalized van der Waals and trap Hamiltonians are special cases of a more general Hamiltonian and, remarkably, they share identical integrable limits. Despite their similitude, important differences also exist; the most significant of them being a double well structure in the effective potential energy surface for the Paul trap. This structure gives rise to a new type of two-ion crystal corresponding to a dynamical rather than a static equilibrium, leading to new resonances and modes of regular and chaotic behavior stemming from the dynamics of the associated separatrix layer.