Magnetic tensor gradiometry using Ramsey interferometry of spinor condensates

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 $50\mathrm{d}\mathrm{B}$. 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 $30,\mathrm{\mu G,cm^{-1}}$andasensorvolumeof$2\times10^{-5},$\mathrm{m}\mathrm{m}$^3$. We predict a spin-projection noise limited magnetic energy resolution of order $\mathrm{\hslash }$ for typical Zeeman coherence times of trapped condensates with this scheme, even with the low measurement duty cycle inherent to BEC experiments.