Unambiguous measurement in an unshielded microscale magnetometer with sensitivity below 1 $\mathrm{pT}/\sqrt{\mathrm{Hz}}$

Cold-atom magnetometers exploit a dense ensemble of quanta with long coherence times to realize leading sensitivity on the micrometer scale. Configured as a Ramsey interferometer, a cold-atom sensor can approach atom shot-noise limited precision but suffers from fringe ambiguity, producing gross errors when the field falls outside a narrow predefined range. We describe how Hilbert-demodulated optical magnetometry can be realized on cold-atom sensors to provide field measurements both precise and unambiguous. Continuous reconstruction of the Larmor phase allows us to determine the dc magnetic field unambiguously in an unshielded environment, as well as measure ac variation of the field, in a single shot. The ac measurement allows us to characterize, and then neutralize, line-synchronous magnetic interference, extending reconstruction times. Using $1.6\times {10}^6{}^{87}\mathrm{Rb}$ atoms in a volume of $(68\text{μ}m{)}^3$, we measure a test field to be $86.0121261\left(4\right)\text{μ}T$ in a single shot, achieving dc sensitivity of $380\mathrm{fT}$ in a duration of $1000\mathrm{ms}$. Our results demonstrate that Hilbert-demodulated optical readout yields metrologically significant sensitivity without the fringe ambiguity inherent to Ramsey interferometry.