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We develop a stochastic mean-field theory to describe active frequencymeasurements of pulsed superradiant emission, studied in recent experimentswith strontium-87 atoms trapped in an optical lattice inside an optical cavity[M. Norcia, et al., Phys. Rev. X 8, 21036 (2018)]. Our theory reveals theintriguing dynamics of atomic ensembles with multiple transition frequencies,and it reproduces the superradiant beats signal, noisy power spectra, andfrequency uncertainty in remarkable agreement with the experiments. Moreover,by reducing the number of atoms, elongating the superradiant pulses andshortening the experimental duty cycle, we predict a short-term frequencyuncertainty $9\times10^{-16} \sqrt{\tau/s}$, which makes active frequencymeasurements with superradiant transitions comparable with the recordperformance of current frequency standards [M. Schioppo, et al., Nat.Photonics, 11, 48 (2017)]. Our theory combines cavity-quantum electrodynamicsand quantum measurement theory, and it can be readily applied to exploreconditional quantum dynamics and describe frequency measurements for otherprocesses such as steady-state superradiance and superradiant Raman lasing.

Active Frequency Measurement on Superradiant Strontium Clock Transitions