# Metrologically useful states of spin-1 Bose condensates with macroscopic magnetization

## TL;DR

This paper demonstrates that spin-1 Bose condensates with macroscopic magnetization can achieve Heisenberg-limited precision in quantum metrology, with scaling of quantum Fisher information confirmed across various interaction types and conditions.

## Contribution

It provides a theoretical analysis showing Heisenberg scaling of quantum Fisher information in spin-1 Bose condensates with macroscopic magnetization, applicable to different interactions and temperature regimes.

## Key findings

- Heisenberg scaling of quantum Fisher information demonstrated
- Scaling applies to both antiferromagnetic and ferromagnetic interactions
- Precision gain explained by phase space quasi-distribution features

## Abstract

We study theoretically the usefulness of spin-1 Bose condensates with macroscopic magnetization in a homogeneous magnetic field for quantum metrology. We demonstrate Heisenberg scaling of the quantum Fisher information for states in thermal equilibrium. The scaling applies to both antiferromagnetic and ferromagnetic interactions. The effect preserves as long as fluctuations of magnetization are sufficiently small. Scaling of the quantum Fisher information with the total particle number is derived within the mean-field approach in the zero temperature limit and exactly in the high magnetic field limit for any temperature. The precision gain is intuitively explained owing to subtle features of the quasi-distribution function in phase space.