Normal modes, rotational inertia, and thermal fluctuations of trapped ion crystals

Abstract

The normal modes of a trapped ion crystal are derived using an approach based on the Hermitian properties of the system's 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 fields are developed, for which the spectrum breaks into E × B 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.