Broadening of the Drumhead Mode Spectrum due to In-Plane Thermal Fluctuations of Two-Dimensional Trapped Ion Crystals in a Penning Trap*

Two-dimensional nonneutral plasma crystals stored in Penning traps are a leading platform for quantum simulation and sensing experiments. For small amplitudes, the out-of-plane motion of such crystals, which is exploited for quantum information protocols, can be described by a discrete set of normal modes called the drumhead modes. However, experimental observations of crystals with Doppler cooled and even near ground-state cooled drumhead modes reveal an unresolved drumhead mode spectrum. In this work, we establish in-plane thermal fluctuations in ion positions as a major contributor to the broadening of the drumhead mode spectrum. In the process, we demonstrate how the confining magnetic field leads to unconventional properties for the in-plane normal modes. These properties, in turn, have implications for the sampling procedure required to choose the in-plane initial conditions for molecular dynamics simulations. For current operating conditions of the NIST Penning trap, our study suggests that the two dimensional crystals produced in this trap undergo in-plane potential energy fluctuations in the range of 10 mK. Our study therefore motivates the need for designing improved techniques to cool the in-plane degrees of freedom.