# Entanglement-based dc magnetometry with separated ions

## TL;DR

This paper demonstrates a novel entanglement-based method for measuring inhomogeneous dc magnetic fields using separated trapped ions, achieving high spatial resolution and sensitivity by employing Bell states and Bayesian estimation.

## Contribution

It introduces a new technique utilizing entangled ions in a segmented trap for precise magnetic field difference measurements with high spatial resolution.

## Key findings

- Measured magnetic field differences up to 6.2 mm apart.
- Achieved measurement accuracy of around 300 fT.
- Sensitivity down to 12 pT/√Hz and spatial resolution of 10 nm.

## Abstract

We demonstrate sensing of inhomogeneous dc magnetic fields by employing entangled trapped ions, which are shuttled in a segmented Paul trap. As \textit{sensor states}, we use Bell states of the type $\left|\uparrow\downarrow\right>+\text{e}^{\text{i}\varphi}\left|\downarrow\uparrow\right>$ encoded in two $^{40}$Ca$^+$ ions stored at different locations. Due to the linear Zeeman effect, the relative phase $\varphi$ serves to measure the magnetic field difference between the constituent locations, while common-mode fluctuations are rejected. Consecutive measurements on sensor states encoded in the $\text{S}_{1/2}$ ground state and in the $\text{D}_{5/2}$ metastable state are used to separate an ac Zeeman shift from the linear dc Zeeman effect. We measure magnetic field differences over distances of up to $6.2~\text{mm}$, with accuracies of around 300~fT, sensitivities down to $12~\text{pT} / \sqrt{\text{Hz}}$, and spatial resolutions down to $10~\text{nm}$. For optimizing the information gain while maintaining a high dynamic range, we implement an algorithm for Bayesian frequency estimation.