The asymmetric rotating saddle potential as a mechanical analog to the RF Paul trap

Under specific conditions, a rotating saddle potential can confine the motion of a particle on its surface. This time-varying hyperbolic potential shares key characteristics with the RF electric quadrupole ion trap (RF Paul trap), making it a valuable mechanical analog. Previous work has primarily focused on symmetric saddles, characterized by equal curvatures along the trapping and anti-trapping directions. However, most applications of RF Paul traps-such as atomic clocks, quantum computing, and quantum simulations-require asymmetry in the quadrupole potential to break the degeneracy of motional modes, which is essential for processes like laser cooling and other quantum manipulations. In this paper, we investigate the motion of trapped particles in asymmetric rotating saddles. We demonstrate that even minor asymmetries, including those arising from manufacturing imperfections, can significantly affect particle trajectories and stability. Our analysis includes both theoretical modeling and experimental measurements. We derive the equations of motion for asymmetric saddles and solve them to explore stability and precession effects. Additionally, we present lifetime measurements of particles in saddles with varying degrees of asymmetry to map key features of the a-q stability diagram, including counterintuitive demonstrations of stability for saddles with negative asymmetry. This study underscores the importance of incorporating asymmetry into mechanical models of ion traps to better reflect real-world implementations. Although motivated primarily by RF Paul traps, these asymmetry-related results are also relevant to emerging gravitational analogs, such as rotating saddle potentials in certain binary black hole systems.