Active Frequency Measurement on Superradiant Strontium Clock Transitions

We develop a stochastic mean-field theory to describe active frequency measurements of pulsed superradiant emission, studied in recent experiments with 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 the intriguing dynamics of atomic ensembles with multiple transition frequencies, and it reproduces the superradiant beats signal, noisy power spectra, and frequency uncertainty in remarkable agreement with the experiments. Moreover, by reducing the number of atoms, elongating the superradiant pulses and shortening the experimental duty cycle, we predict a short-term frequency uncertainty $9\times10^{-16} \sqrt{\tau/s}$, which makes active frequency measurements with superradiant transitions comparable with the record performance of current frequency standards [M. Schioppo, et al., Nat. Photonics, 11, 48 (2017)]. Our theory combines cavity-quantum electrodynamics and quantum measurement theory, and it can be readily applied to explore conditional quantum dynamics and describe frequency measurements for other processes such as steady-state superradiance and superradiant Raman lasing.