# Investigations on Dynamical Stability in 3D Quadrupole Ion Traps

## TL;DR

This paper investigates the dynamical stability of ions in 3D quadrupole traps using classical and quantum models, introducing a new stability model based on the Hessian matrix and applying it to various trap configurations.

## Contribution

It introduces a novel model for dynamical stability analysis of ion traps using the Hessian matrix, applicable to classical and quantum many-body systems, and demonstrates its effectiveness in identifying equilibrium states.

## Key findings

- Weak coupling condition is impractical for detection.
- Collective modes show a peak at specific ion masses.
- The model effectively infers trap parameters and equilibrium configurations.

## Abstract

We firstly discuss classical stability for a dynamical system of two ions levitated in a 3D Radio-Frequency (RF) trap, assimilated with two coupled oscillators. We obtain the solutions of the coupled system of equations that characterizes the associated dynamics. In addition, we supply the modes of oscillation and demonstrate the weak coupling condition is inappropriate in practice, while for collective modes of motion (and strong coupling) only a peak of the mass can be detected. Phase portraits and power spectra are employed to illustrate how the trajectory executes quasiperiodic motion on the surface of torus, namely a Kolmogorov-Arnold-Moser (KAM) torus. In an attempt to better describe dynamical stability of the system, we introduce a model that characterizes dynamical stability and the critical points based on the Hessian matrix approach. The model is then applied to investigate quantum dynamics for many-body systems consisting of identical ions, levitated in 2D and 3D ion traps. Finally, the same model is applied to the case of a combined 3D Quadrupole Ion Trap (QIT) with axial symmetry, for which we obtain the associated Hamilton function. The ion distribution can be described by means of numerical modeling, based on the Hamilton function we assign to the system. The approach we introduce is effective to infer the parameters of distinct types of traps by applying a unitary and coherent method, and especially for identifying equilibrium configurations, of large interest for ion crystals or quantum logic.