# Precision bounds for gradient magnetometry with atomic ensembles

## TL;DR

This paper establishes quantum precision bounds for gradient magnetic field estimation using atomic ensembles, considering various states and configurations, including localized particles and Bose-Einstein condensates.

## Contribution

It introduces a comprehensive method to calculate quantum Fisher information-based precision bounds for gradient magnetometry with diverse atomic states and spatial arrangements.

## Key findings

- Single observable measurement suffices for invariant states.
- Simultaneous measurement needed for sensitive states.
- Method applicable to localized particles and Bose-Einstein condensates.

## Abstract

We study gradient magnetometry with an ensemble of atoms with arbitrary spin. We calculate precision bounds for estimating the gradient of the magnetic field based on the quantum Fisher information. For quantum states that are invariant under homogeneous magnetic fields, we need to measure a single observable to estimate the gradient. On the other hand, for states that are sensitive to homogeneous fields, a simultaneous measurement is needed, as the homogeneous field must also be estimated. We prove that for the cases studied in this paper, such a measurement is feasible. We present a method to calculate precision bounds for gradient estimation with a chain of atoms or with two spatially separated atomic ensembles. We also consider a single atomic ensemble with an arbitrary density profile, where the atoms cannot be addressed individually, and which is a very relevant case for experiments. Our model can take into account even correlations between particle positions. While in most of the discussion we consider an ensemble of localized particles that are classical with respect to their spatial degree of freedom, we also discuss the case of gradient metrology with a single Bose-Einstein condensate.