# Ultimate limits for quantum magnetometry via time-continuous measurements

## TL;DR

This paper demonstrates that continuous quantum measurements can achieve Heisenberg-limited precision in magnetic field estimation with atomic ensembles, surpassing standard quantum limits and maintaining robustness against detection inefficiencies.

## Contribution

It introduces a continuous measurement strategy that reaches Heisenberg-limited scaling in quantum magnetometry, proving its optimality and robustness.

## Key findings

- Continuous measurements enable Heisenberg-limited scaling of $1/J^2$.
- The strategy is robust against detection losses and finite measurement efficiency.
- The proposed method is proven to be optimal among possible strategies.

## Abstract

We address the estimation of the magnetic field B acting on an ensemble of atoms with total spin J subjected to collective transverse noise. By preparing an initial spin coherent state, for any measurement performed after the evolution, the mean-square error of the estimate is known to scale as $1/J$, i.e. no quantum enhancement is obtained. Here, we consider the possibility of continuously monitoring the atomic environment, and conclusively show that strategies based on time-continuous non-demolition measurements followed by a final strong measurement may achieve Heisenberg-limited scaling $1/{J^2}$ and also a monitoring-enhanced scaling in terms of the interrogation time. We also find that time-continuous schemes are robust against detection losses, as we prove that the quantum enhancement can be recovered also for finite measurement efficiency. Finally, we analytically prove the optimality of our strategy.