The Role of Monitoring Time and Detectors Efficiencies in Time-Continuous Quantum Magnetometry

We consider the estimation of a weak magnetic field B acting on a continuously monitored ensemble of atoms subjected to collective transverse noise. If N atoms are prepared in a coherent spin state and are not continuously monitored, the estimation precision scales with the total number of atoms according to the standard quantum limit δB2∼1/NδB2∼1/N. Remarkably, time-continuous monitoring of light that is coupled with the atomic ensemble, allows to achieve a Heisenberg limited precision δB2∼1/N2δB2∼1/N2. However this is typically obtained only for a large enough number of atoms N and with an asymptotic constant factor depending on the parameters characterizing the experiment. In this proceeding, after reviewing the analytical derivation of the effective quantum Fisher information that quantifies the ultimate precision achievable, we specifically address the role played by monitoring time and detectors measurement efficiency in obtaining a Heisenberg limited scaling. In particular we analyze the dependence on these experimentally relevant parameters of the asymptotic constant factor characterizing the effective quantum Fisher information, and, more importantly, the minimum value of atoms needed to observe the desired quantum enhancement.