# Ultimate quantum limits for noisy magnetometry via time-continuous measurements. (arXiv:1706.00485v2 [quant-ph] UPDATED)

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 quantum-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.