Effect of defects on phonons and the effective spin-spin interactions of an ultracold Penning-trap quantum simulator

We generalize the analysis of the normal modes for a rotating ionic Coulomb crystal in a Penning trap to allow for inhomogeneities in the system. Our formal developments are completely general, but we choose to examine a crystal of Be+ ions with BeH+ defects to compare with current experimental efforts. We examine the classical phonon modes (both transverse and planar) and we determine the effective spin-spin interactions when the system is driven by an axial spin-dependent optical dipole force. We examine situations with up to approximately 15% defects. We find that most properties are not strongly influenced by the defects, indicating that the presence of a small number of defects will not significantly affect experiments.