Rapid production of many-body entanglement in spin-1 atoms via cavity output photon counting

We propose a simple and efficient method for generating metrologically useful quantum entanglement in an ensemble of spin-1 atoms that interacts with a high-finesse optical cavity mode. It requires straightforward preparation of $N$ atoms in the $m_F=0$ sublevel, tailoring of the atom-field interaction to give an effective Tavis-Cummings model for the collective spin-1 ensemble, and a photon counting measurement on the cavity output field. The photon number provides a projective measurement of the collective spin length $S$, which, for the chosen initial state, is heavily weighted around values $S\simeq\sqrt{N}$, for which the corresponding spin states are strongly entangled and exhibit Heisenberg scaling of the metrological sensitivity with $N$, as quantified by the quantum Fisher information.