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Macroscopic arrays of cold atoms trapped in optical cavities can reach thestrong atom-light collective coupling regime thanks to the simultaneousinteractions of the cavity mode with the atomic ensemble. In a recent work wereported a protocol that takes advantage of the strong and collectiveatom-light interactions in cavity QED systems for precise electric fieldsensing in the optical domain. We showed that it can provide between$10$-$20$~dB of metrological gain over the standard quantum limit in currentcavity QED experiments operating with long-lived alkaline-earth atoms. Here, wegive a more in depth discussion of the protocol using both exact analyticalcalculations and numerical simulations, and describe the precise conditionsunder which the predicted enhancement holds after thoroughly accounting forboth photon loss and spontaneous emission, natural decoherence mechanisms incurrent experiments. The analysis presented here not only serves to benchmarkthe protocol and its utility in cavity QED arrays but also sets the conditionsrequired for its applicability in other experimental platforms such as arraysof trapped ions.

Atom-light entanglement for precise field sensing in the optical domain