Translational dynamics of diatomic molecule in magnetic quadrupole trap

TL;DR

This paper investigates the classical translational dynamics of homonuclear diatomic molecules in a magnetic quadrupole trap, revealing complex behaviors including chaos and non-integrability, with analytical solutions for certain motions.

Contribution

It provides a detailed analysis of the classical translational motion of diatomic molecules in magnetic traps, demonstrating non-integrability and deriving analytical solutions for specific trajectories.

Findings

Molecular motion is confined to a bounded region that expands with energy.

Motion is non-integrable and can be chaotic at higher energies.

Analytical solutions are obtained for motion on symmetry axes and planes.

Abstract

We study the translational motions of homonuclear diatomic molecules prepared in their ${}^3\Sigma$ electronic states, deeply bound vibrational states, and rotational states of well-defined parity. The trapping potential arises due to the interaction of the total spin of electrons and orbital angular momentum of nuclei with the trap's quadrupole magnetic field. The translational motion of a molecule is treated classically. We examine the Hamilton equations that govern the center of mass dynamics numerically and analytically. Using data of a hydrogen molecule at the ground vibrational state, we present global dynamics using the Poincar\'e section method and various types of trajectories: periodic, quasi-periodic and chaotic. We prove that the Hamiltonian system governing this motion is non-integrable. The particle's orbits are confined to a bound region of space that grows with energy, but for small energies (< 1.8 K), the motion is restricted to a processing chamber (a few centimetres). Solutions of equations of motion occurring on the symmetry axis and the horizontal plane are expressed in terms of Jacobi elliptic functions.