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We demonstrate the fabrication of ultra-low-loss, all-fiber Fabry-P\'erotcavities containing a nanofiber section, optimized for cavity quantumelectrodynamics. By continuously monitoring the finesse and fiber radius duringfabrication of a nanofiber between two fiber Bragg gratings, we are able toprecisely evaluate taper transmission as a function of radius. The resultingcavities have an internal round-trip loss of only 0.31% at a nanofiber waistradius of 207 nm, with a total finesse of 1380, and a maximum expected internalcooperativity of $\sim$ 1050 for a cesium atom on the nanofiber surface. Ourability to fabricate such high-finesse nanofiber cavities may open the door forthe realization of high-fidelity scalable quantum networks.

Ultra-low-loss nanofiber Fabry-P\'erot cavities optimized for cavity quantum electrodynamics