  We have realized a magnetic tensor gradiometer by interferometrically
measuring the relative phase between two spatially separated Bose-Einstein
condensates (BECs). We perform simultaneous Ramsey interferometry of the
proximate $^{87}$Rb spin-1 condensates in freefall and infer their relative
Larmor phase -- and thus the differential magnetic field strength -- with a
common-mode phase noise suppression exceeding $50\,\mathrm{dB}$. By
appropriately biasing the magnetic field and separating the BECs along
orthogonal directions, we measure the magnetic field gradient tensor of ambient
and applied magnetic fields with a nominal precision of $30\,\mathrm{\mu
G\,cm^{-1}}$ and a sensor volume of $2\times10^{-5}\,\mathrm{mm}^3$. We predict
a spin-projection noise limited magnetic energy resolution of order $\hbar$ for
typical Zeeman coherence times of trapped condensates with this scheme, even
with the low measurement duty cycle inherent to BEC experiments.

Quantum Physics;Quantum Gases;Atomic Physics