Large, three-dimensional (3D) trapped-ion crystals offer improved sensitivity in quantum sensing protocols, and are expected to be implemented as platforms in near-future experiments. However, numerical techniques used to study the laser cooling of such crystals are inefficient as the number of ions, $N$, in the crystal increases. Here we develop a powerful numerical framework to simulate laser cooling of up to ${10}^5$ ions stored in a Penning trap. We apply this framework to characterize and optimize the cooling of ellipsoidal 3D crystals. We document pathways to enhanced cooling based on the addition of an axial component to the potential-energy-dominated $\mathit{E}\times \mathit{B}$ modes. Furthermore, we observe greatly enhanced cooling of the perpendicular kinetic energy to below 1 mK in prolate ion crystals, enabling a simplified cooling beam setup for such crystals. We propose specific values of trap and laser beam parameters which lead to optimal cooling in a variety of examples. This work illustrates the feasibility of preparing large 3D crystals for high-sensitivity quantum science protocols, motivating their use in future experiments.

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• Atom & ion cooling
• Nonneutral plasmas
• First-principles calculations in plasma physics
• Molecular dynamics
• Penning traps