# $\bar{\textrm{H}}^{+}$ Sympathetic Cooling Simulations with a Variable Time Step

## TL;DR

This paper introduces a new variable time step criterion for the velocity-Verlet algorithm, enabling accurate and efficient simulations of Coulomb interactions, demonstrated by cooling $ar{ extrm{H}}^{+}$ ions with a Coulomb crystal.

## Contribution

A novel variable time step criterion for the velocity-Verlet algorithm is proposed, improving simulation accuracy and efficiency for charged particle dynamics.

## Key findings

- Successfully simulated sympathetic cooling of $ar{ extrm{H}}^{+}$ ions in less than 10 ms.
- Validated the new criterion through physical arguments and numerical results.
- Demonstrated potential application for the GBAR project.

## Abstract

In this paper we present a new variable time step criterion for the velocity-Verlet algorithm allowing to correctly simulate the dynamics of charged particles exchanging energy via Coulomb collisions while minimising simulation time. We present physical arguments supporting the use of the criterion along with numerical results proving its validity. We numerically show that $\bar{\textrm{H}}^{+}$ ions with 18 meV initial energy can be captured and sympathetically cooled by a Coulomb crystal of $\textrm{Be}^{+}$ and $\textrm{HD}^{+}$ in less than 10 ms, an important result for the GBAR project.