Parametric Amplification of Spin-Motion Coupling in Three Dimensional Trapped-Ion Crystals*

Three-dimensional crystals offer a route to scale up trapped ion systems for quantum sensing and simulation applications. At the same time, engineering coherent spin-spin interactions in large crystals poses technical challenges associated with decoherence and prolonged timescales to generate high levels of entanglement. Here, we explore the possibility to magnify the effective spin-spin interactions in 3D crystals via parametric amplification. We derive a general Hamiltonian for the parametric amplification of spin-motion coupling by taking the ion crystal geometry and normal mode properties into account. Our modeling is applicable to crystals of any dimension in both RF Paul traps and Penning traps. Unlike in lower-dimensional crystals, we find that the ability to faithfully (uniformly) amplify the spin-spin interactions in 3D crystals depends on the physical implementation of the spin-motion coupling. We consider the light-shift (LS) gate and the so-called phase-insensitive and phase-sensitive Molmer-Sorensen (MS) gates, and find that only the latter can be faithfully amplified in general 3D crystals. We discuss some special cases where non-uniform amplification can be advantageous. We also reconsider the impact of counter-rotating terms on PA and find that they are not as detrimental as previous studies suggest. Our work is of relevance to current and near-term trapped ion experiments in both RF Paul traps and Penning traps where 3D crystals of hundreds to thousands of trapped ions can be realized for quantum information applications.