We demonstrate a precise magnetic microscope based on direct imaging of the Larmor precession of a
87Rb spinor Bose-Einstein condensate. This magnetometer attains a field sensitivity of
8.3 pT/Hz1/2 over a measurement area of $120\text{ }\text{ }\ensuremath{\mu}{\mathrm{m}}^{2}$, an improvement over the low-frequency field sensitivity of modern SQUID magnetometers. The achieved phase sensitivity is close to the atom shot-noise limit, estimated as
0.15 pT/Hz1/2 for a unity duty cycle measurement, suggesting the possibilities of spatially resolved spin-squeezed magnetometry. This magnetometer marks a significant application of degenerate atomic gases to metrology.
We provide a theoretical treatment of the quantum backaction of Larmor frequency measurements on a spinor Bose-Einstein condensate by an off-resonant light field. Two main results are presented; the first is a ``quantum jump'' operator description that reflects the abrupt change in the spin state of the atoms when a single photon is counted at a photodiode. The second is the derivation of a conditional stochastic master equation relating the evolution of the condensate density matrix to the measurement record. We provide a few examples of the application of this formalism and comment on its application to metrology.