Normal Modes, Rotational Inertia, and Thermal Fluctuations of Trapped Ion Crystals

TL;DR

This paper develops a simplified, general method to analyze normal modes and thermal fluctuations of trapped ion crystals, including effects of magnetic fields and rotation, with applications to Penning traps.

Contribution

It introduces a Hermitian-based approach for classical systems, derives rotational inertia including magnetic effects, and explores unique rotational behaviors in symmetric traps.

Findings

Normal modes are derived using a Hermitian approach, simplifying analysis.

Thermal fluctuations are consistent with the Bohr-van-Leeuwen theorem when neutrally-stable modes are included.

A magnetic contribution to the rotational inertia dominates at high magnetic fields.

Abstract

The normal modes of a trapped ion crystal are derived using an approach based on the Hermitian properties of the systems dynamical matrix. This method is equivalent to the standard Bogoliubov method, but for classical systems it is arguably simpler and more general in that canonical coordinates are not necessary. The theory is developed for stable, unstable, and neutrally-stable systems. The method is then applied to ion crystals in a Penning trap. Reduced eigenvalue problems for the case of large applied magnetic field are developed, for which the spectrum breaks into ExB drift modes, axial modes, and cyclotron modes. Thermal fluctuation levels in these modes are analyzed and shown to be consistent with the Bohr-van-Leeuwen theorem, provided that neutrally-stable modes associated with crystal rotations are included in the analysis. An expression for the rotational inertia of the crystal is derived, and a magnetic contribution to this inertia, which dominates in large magnetic fields, is described. An unusual limit is discovered for the special case of spherically-symmetric confinement, in which the rotational inertia does not exist and changes in angular momentum leave the rotation frequency unaffected.