Stability of macroscopic spin ensembles against inhomogeneous dephasing

Spin ensembles play a pivotal role in various quantum applications such as metrology and simulations of many-body physics. Recent research has proposed utilizing spin cat states to encode logical quantum information, with logical lifetimes potentially on the order of seconds, achieved via enhanced collective interactions that scale with system size. We investigate the dynamics of spin cat states under inhomogeneous broadening, revealing a phenomenon termed ``parity-sensitive inhomogeneous dephasing'': For small amplitudes, odd cat states are significantly more susceptible to inhomogeneous dephasing than even cat states due to the difference in parity symmetry. This discrepancy between even and odd cat states vanishes at large amplitudes and behaves similarly to a spin coherent state with the same amplitude. To analyze the stability of the spin coherent state, we perform a mean-field analysis of the driven-dissipative dynamics, from which we identify a synchronization phase transition wherein the ensemble becomes completely dephased beyond a critical inhomogeneous linewidth. The mean-field analysis suggests that the dissipative stabilization can suppress the decoherence effects from inhomogeneous broadening. We argue that the stability of the mean-field model provides a reasonable estimate for that of spin cat states with a large amplitude in the full quantum model. Our findings shed light on the stability of collective spin states, which is important for advancing quantum technologies.