Conditional Spin-Squeezing of a Large Ensemble via the Vacuum Rabi Splitting

We demonstrate that the collective vacuum Rabi splitting can be used to perform quantum nondemolition (QND) measurements of the psuedo-spin projection $J_z$ for the two-level clock states of nearly $10^6~^{87}$Rb atoms confined in a low finesse $F =710(10)$ optical cavity. The QND measurement is used to prepare a conditionally spin-squeezed state. We infer a 3.4(6)~dB improvement in quantum phase estimation relative to the standard quantum limit for a coherent spin state composed of uncorrelated atoms. The measurement is enhanced using a large ensemble and may lead to more precise atomic sensors and tests of fundamental physics.