Emergence of unidirectionality and phase separation in optically dense emitter ensembles

The transmission of light through an ensemble of two-level emitters in a one-dimensional geometry is commonly described by one of two emblematic models of quantum electrodynamics (QED): the driven-dissipative Dicke model or the Maxwell-Bloch equations. Both exhibit distinct features of phase transitions and phase separations, depending on system parameters such as optical depth and external drive strength. Here, we explore the crossover between these models via a parent spin model from bidirectional waveguide QED, by varying positional disorder among emitters. Solving mean-field equations and employing a second-order cumulant expansion for the unidirectional model -- equivalent to the Maxwell-Bloch equations -- we study phase diagrams, the emitter's inversion, and transmission depending on optical depth, drive strength, and spatial disorder. We find in the thermodynamic limit the emergence of phase separation with a critical value that depends on the degree of spatial order but is independent of Doppler broadening effects. Even far from the thermodynamic limit, this critical value marks a special point in the emitter's correlation landscape of the unidirectional model and is also observed as a maximum in the magnitude of inelastically transmitted photons. We conclude that a large class of effective one-dimensional systems without tight control of the emitter's spatial ordering can be effectively modeled using a unidirectional waveguide approach.