Dissipative generation of spin squeezing in the resolved vacuum Rabi splitting limit

Harnessing dissipation in the presence of strong symmetries has recently emerged as a promising route for generating entanglement in atomic clocks. However, previous proposals relied on regimes where cavity photons can be adiabatically eliminated, significantly limiting their applicability to experimentally relevant cavity-QED regimes that lie in or near the resolved vacuum Rabi splitting regime. Here we show that symmetry-protected dissipative spin squeezing can be realized even when cavity photons actively participate in the dynamics, extending the experimental relevance of the protocol. We study a three-level ensemble of ⁸⁷Sr atoms coupled to an optical cavity in the resolved vacuum Rabi splitting regime and demonstrate that, with smooth ramps of the drive amplitude and detunings, the driven-dissipative dynamics enters a stable low-photon regime in which nonadiabatic cavity excitations and sector-resolving photon leakage can be controlled. Within this low-photon operating window, sector-resolving photon leakage is suppressed and the sector-dependent geometric phase realizes effective one-axis twisting. At the end of the protocol the entanglement can also be efficiently transferred directly onto the long-lived clock states by turning the drive off. For experimentally realistic parameters, we theoretically show that more than 25 dB of squeezing can be generated for 10⁵ atoms, closely saturating the ideal one-axis twisting scaling ξ_min² propto N^-2/3. At fixed cooperativity, the optimized squeezing remains broadly comparable to the unresolved-regime implementation, while the resolved-regime implementation reaches comparable squeezing on a substantially shorter physical timescale. These results establish symmetry-protected dissipative dynamics as a practical route to beyond the standard-quantum-limit performance in optical-clock platforms.