Spin Grid States for Quantum Metrology in Atomic Clocks Limited by Spontaneous Emission

Entanglement can enhance precision in phase estimation and in frequency estimation with atomic clocks, but it remains a central challenge to identify useful states and measurements under realistic noise processes. Here, we study quantum metrology with an ensemble of atoms subject to spontaneous emission, which limits frequency estimation by constraining the useful interrogation time. We identify spin grid states (SGSs) as the relevant near-optimal probes beyond a crossover at an ensemble size of 51, where GHZ-like states cease to be optimal. SGSs display a periodic grid on the Bloch sphere and can be generated with two one-axis-twisting (OAT) operations separated by a collective rotation. By optimizing the quantum Fisher information over permutationally symmetric states, we find that SGSs perform close to the globally optimal probes for intermediate and large ensembles, which turn out to be spin GKP states. We further present a sequential readout strategy that nearly saturates the corresponding quantum Fisher information. This strategy uses an OAT echo to map spontaneous-emission events onto collective rotations, making the corresponding jump sectors distinguishable and allowing the measurement basis to be conditioned on the number of decay events. Together, these results show that SGSs provide a practical route to quantum-enhanced phase and frequency metrology in atomic ensembles limited by spontaneous emission.