Emergent spin order and steady-state superradiance in one-dimensional baths

Spontaneous collective decay in incoherently driven atomic ensembles generates macroscopic coherence in the steady state, as exemplified by superradiant lasing in single-mode cavities. Whether spontaneous order persists in multimode reservoirs with competing collective decay channels and light propagation remains an open question. We address this problem by analyzing atoms coupled to one-dimensional electromagnetic baths through two models: a ring cavity with two bright decay channels, and a bidirectional waveguide where, in addition to competition between channels, propagation induces Hamiltonian dipole-dipole interactions. For suitable pumping strengths, both models enter a synchronization window leading to steady-state phase ordering and superradiant emission, scaling as $N^2$. The resulting order is not described by a single macroscopic dipole: in the ring cavity spontaneous chirality emerges at the level of individual trajectories, while the waveguide develops a local chirality with different orders dominating opposite ends of the array. The spectral analysis of the emitted light seems to indicate line narrowing with increased system size in the ring cavity. Line narrowing in the waveguide remains inconclusive within accessible numerics. Our results demonstrate how competition and propagation shape emergent order beyond the Dicke limit.