  The generation of many-body entangled states in atomic samples should be
fast, as this process always involves a subtle interplay between desired
quantum effects and unwanted decoherence. Here we identify a controllable and
scalable catalyst that allows metrologically useful entangled states to be
generated at a high rate. This is achieved by immersing a collection of bosonic
atoms, trapped in a double-well potential, in an optical cavity. In the
dispersive regime, cavity photons collectively couple pairs of atoms in their
ground state to a molecular state, effectively generating, photon-number
dependent atom-atom interactions. These effective interactions entangle atoms
at a rate that strongly scales with both the number of photons and the number
of atoms. As a consequence, the characteristic time scale of entanglement
formation can be much shorter than for bare atom-atom interactions, effectively
eliminating the decoherence due to photon losses. Here, the control of the
entanglement generation rate does not require the use of Feshbach resonances,
where magnetic field fluctuations can contribute to decoherence. Our protocol
may find applications in future quantum sensors or other systems where
controllable and scalable many-body entanglement is desired.
