Semiclassical Phase-Space Dynamics of Emitter Ensembles with Local Dissipation

We establish that permutation-invariant emitter ensembles with local dissipation admit a semiclassical description on the four-dimensional phase space of a variable-length collective spin, within a perturbative truncated Wigner approximation (TWA). The TWA captures nonlinear dynamics and nonclassical signatures without prescribing closures of correlations. We cast a class of spatially extended emitter systems as interacting ensembles tractable in the TWA to show how quantum fluctuations reshape directional cooperative emission beyond mean-field predictions in a pumped chain containing tens of thousands of emitters, using hundreds of phase-space variables. Our results offer a unified description of collective dynamics beyond the classical limit in emitter ensembles.