# Sideband cooling of small ion Coulomb crystals in a Penning trap

## TL;DR

This paper extends sideband cooling techniques to small ion Coulomb crystals in a Penning trap, achieving near-ground state cooling of multiple ions' axial modes despite complex sequences and outside the Lamb-Dicke regime.

## Contribution

It demonstrates efficient sideband cooling of small ion crystals in a Penning trap, including multi-ion planar configurations, with detailed measurements of cooling performance and heating rates.

## Key findings

- Achieved mean excitation numbers below 0.3 for two-ion modes.
- Successfully cooled multiple axial modes in small ion crystals.
- Measured low heating rates indicating stable ground state preparation.

## Abstract

We have recently demonstrated the laser cooling of a single $^{40}$Ca$^+$ ion to the motional ground state in a Penning trap using the resolved-sideband cooling technique on the electric quadrupole transition S$_{1/2} \leftrightarrow$ D$_{5/2}$. Here we report on the extension of this technique to small ion Coulomb crystals made of two or three $^{40}$Ca$^+$ ions. Efficient cooling of the axial motion is achieved outside the Lamb-Dicke regime on a two-ion string along the magnetic field axis as well as on two- and three-ion planar crystals. Complex sideband cooling sequences are required in order to cool both axial degrees of freedom simultaneously. We measure a mean excitation after cooling of $\bar n_\text{COM}=0.30(4)$ for the centre of mass mode and $\bar n_\text{B}=0.07(3)$ for the breathing mode of the two-ion string with corresponding heating rates of 11(2) s$^{-1}$ and 1(1) s$^{-1}$ at a trap frequency of 162 kHz. The ground state occupation of the axial modes is above 75% for the two-ion planar crystal and the associated heating rates 0.8(5) s$^{-1}$ at a trap frequency of 355 kHz.