Detection of a weak magnetic field via cavity enhanced Faraday rotation

We study the sensitive detection of a weak static magnetic field via Faraday rotation induced by an ensemble of spins in a bimodal degenerate microwave cavity. We determine the limit of the resolution for the sensitivity of the magnetometry achieved using either single-photon or multiphoton inputs. For the case of a microwave cavity containing an ensemble of Nitrogen-vacancy defects in diamond, we obtain a magnetometry sensitivity exceeding 0.5~\text{\nano\tesla}/\sqrt{\text{\hertz}}, utilizing a single photon probe field, while for a multiphoton input we achieve a sensitivity about 1 \text{\femto\tesla}/\sqrt{\text{\hertz}}, using a coherent probe microwave field with power of P_\text{in}=1~\text{\nano\watt}.Comment: 8 pages, 7 figure