Emergence of unidirectionality and phase separation in optically dense emitter ensembles

TL;DR

This work shows that a coherently driven ensemble of two-level emitters coupled to a one-dimensional continuum exhibits a disorder-driven crossover between bidirectional Dicke dynamics and unidirectional cascaded dynamics. By formulating a bidirectional master equation and performing spiral-gauge reformulations, ensemble averaging, and mean-field plus second-order cumulant analyses, the authors map how phase separation and transmission arise as functions of optical depth, drive, and spatial disorder. In the thermodynamic limit, a sharp phase boundary emerges at a critical intensive drive tilde s (≈1 for unidirectional and ≈2 for Dicke), with phase-separated regions corresponding to saturated and ground-state emitters; this boundary remains robust against Doppler broadening. The results justify using unidirectional waveguide approaches to model a wide range of 1D light–matter systems and suggest observable signatures in elastic and inelastic transmission as well as emitter–emitter correlations, with broader implications for disordered quantum optical platforms.

Abstract

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.

Figures (8)

• Figure 1: Examples of experimental realizations in which unidirectionality and phase seperation emerge. a) Laser-cooled atoms coupled to a tapered optical nanofibers. b) Laser-cooled atoms in free space, and c) Mössbauer nuclei in a thin film cavity.
• Figure 2: Left: Waveguide model describing the experimental systems in Fig. \ref{['fig:schemFirst']} (a) - (c). Right: Sketches of the four different theoretical model used in this article to model the situation on the left. Bidirectional waveguide model (BWM), ensemble averaged waveguide model (EAM), where $\langle \ldots \rangle_\mathrm{ens}$ denotes ensemble averaging, the driven-dissipative Dicke model (DM), and the unidirectional waveguide model (UWM), details see Table \ref{['tab:models']}.
• Figure 3: First-order observables of a unidirectional waveguide system exhibiting a phase separation. (a) Elastically scattered light $s(D)$ along the chain scaled to the input saturation $s_0$ versus the inverse control parameter $D/s_0$. (b) The emitter's inversion $\expval{\hat{\sigma}_z(D)}$ along the chain versus the inverse control parameter $D/s_0$. (c) The mean polarization $j_z$ versus the control parameter $\tilde{s}$ for a fixed optical depth. The points indicate the input parameters for which in (a) and (b) the evolution along the chain is shown.
• Figure 4: (a) Inversion $\expval{\hat{\sigma}_i^z}$ of an ensemble of bidirectional waveguide systems (EAM) versus $\eta$ and $D_i$ for fixed input saturation $s_0=20$, total emitter number $N=2000$ and coupling strength $\beta=0.005$. For $\eta\geq 0.1$ the dynamics is effectively a unidirectional one. (b)-(d) Cuts of the density plot in (a) at $\eta=0.1, 0.01, 0.001$, compared to results of a single realizations of the system (BWM) and the limits of dissipative Dicke (DM) and unidirectional (UWM) dynamics.
• Figure 5: (a) Difference $\expval{\Delta\hat{\sigma}_i^z}$ between the EAM and 20 realization of the BWM with respect to $\hat{\sigma}_i^z$ for the same set of parameters as in Fig. \ref{['fig:BiWG_sigz']}, i.e. total emitter number $N=2000$, fixed input saturation $s_0=20$ and coupling strength $\beta=0.005$. Averaging at the level of solutions to the BWM predicts a phase separations which are not as sharp as in the case of the EAM but with the same critical values. From $\eta=0.1$ on there is no difference between averaging at the level of equation of motions, that is the EAM, and averaging at the level of solutions to these equations. (b) variance $\expval{\Delta\hat{\sigma}^z_i}^2$ between the EAM and $20$ realizations of the BWM for the same set of parameters as (a). The variance gets maximal along the border of the phase separation and for disorder values of $\eta\sim0.02$.