Interacting Mathieu equation, synchronization dynamics and collision-induced velocity exchange in trapped ions

TL;DR

This paper investigates the dynamics of trapped ions modeled by the Mathieu equation, revealing synchronization and velocity exchange phenomena, and discusses their implications for quantum simulation and mathematical physics.

Contribution

It introduces a detailed analysis of many-body Mathieu equations in trapped ions, highlighting synchronization and collision dynamics, and explores effects of anharmonic potentials.

Findings

Velocity exchange resembles Newton's cradle dynamics.

Synchronization occurs independently of initial conditions.

Anharmonic potentials cause desynchronization during collisions.

Abstract

Recently, large-scale trapped ion systems have been realized in experiments for quantum simulation and quantum computation. They are the simplest systems for dynamical stability and parametric resonance. In this model, the Mathieu equation plays the most fundamental role for us to understand the stability and instability of a single ion. In this work, we investigate the dynamics of trapped ions with the Coulomb interaction based on the Hamiltonian equation. We show that the many-body interaction will not influence the phase diagram for instability. Then, the dynamics of this model in the large damping limit will also be analytically calculated using few trapped ions. Furthermore, we find that in the presence of modulation, synchronization dynamics can be observed, showing an exchange of velocities between distant ions on the left side and on the right side of the trap. These dynamics resemble to that of the exchange of velocities in Newton's cradle for the collision of balls at the same time. These dynamics are independent of their initial conditions and the number of ions. As a unique feature of the interacting Mathieu equation, we hope this behavior, which leads to a quasi-periodic solution, can be measured in current experimental systems. Finally, we have also discussed the effect of anharmonic trapping potential, showing the desynchronization during the collision process. It is hopped that the dynamics in this many-body Mathieu equation with damping may find applications in quantum simulations. This model may also find interesting applications in dynamics systems as a pure mathematical problem, which may be beyond the results in the Floquet theorem.