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There are diverse set approaches for vector magnetic field measurements
involving condensed matter and atomic physics systems. However, they all suffer
from various limitations, so the most widely used high-sensitivity vector
magnetometers are fluxgates using soft magnetically-saturable materials. Here
we describe a vector magnetometer by applying an external rotating magnetic
field to a scalar atomic magnetometer. Such an approach provides simultaneous
measurements of the total magnetic field and two polar angles relative to the
plane of magnetic field rotation. Crucially, it avoids several metrological
difficulties associated with vector magnetometers and gradiometers. We describe
in detail the fundamental, systematic, and practical limits of such vector
magnetometers. We use a field rotation rate faster than the spin relaxation
rate. We show that it eliminates a class of systematic effects associated with
heading errors in alkali-metal scalar magnetometers. We investigate several
other systematic effects, such as Berry's phase frequency shift and the effects
of eddy currents in nearby conductors. We also derive fundamental limits on the
sensitivity of such sensors and show that the vector sensitivity can approach
the sensitivity of scalar atomic magnetometers.
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