🔬 atomic physics

Emergence of unidirectionality and phase separation in optically dense emitter ensembles

This paper investigates the crossover between bidirectional and unidirectional light-matter interaction models in optically dense emitter ensembles, demonstrating that spatial disorder induces a critical phase separation point independent of Doppler broadening, thereby validating the unidirectional waveguide approach for modeling systems with imperfect spatial ordering.

Original authors: Kasper J. Kusmierek, Max Schemmer, Sahand Mahmoodian, Klemens Hammerer

Published 2026-10-14

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CC BY 4.0

Original authors: Kasper J. Kusmierek, Max Schemmer, Sahand Mahmoodian, Klemens Hammerer

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Light passing through a cloud of atoms is one of the most fundamental experiments in quantum physics, a simple setup that reveals how matter and energy interact on the smallest scales. Imagine a stream of photons, the particles of light, moving through a line of tiny, two-level atoms. Depending on how these atoms are arranged and how strongly they are pushed by an external laser, the light can behave in dramatically different ways. Sometimes the atoms act like a single, unified team, responding in perfect unison. Other times, they act as individuals, absorbing and re-emitting light in a chaotic, independent fashion. For decades, physicists have used two main mathematical frameworks to describe these scenarios: one that assumes the atoms are perfectly ordered and talking to each other in both directions, and another that assumes the light only moves forward, with the atoms reacting in a strict sequence. The question has always been whether these two models are truly distinct or if they are just different ends of the same spectrum, and how real-world imperfections, like atoms being slightly out of place, change the picture.

A team of researchers has now mapped out the entire journey between these two extremes. By simulating a long line of atoms coupled to a one-dimensional channel for light, they explored what happens when the perfect order of the atoms is gradually disturbed. In their idealized, perfectly ordered setup, the atoms are spaced at exact intervals, allowing light to bounce back and forth between them, creating a collective behavior where the whole group acts as one giant atom. This is a scenario where light can get trapped or amplified in complex ways. However, as the researchers introduced disorder—shifting the atoms slightly from their perfect spots—the physics began to shift. They found that even a small amount of disorder is enough to destroy the delicate back-and-forth communication. The light stops bouncing backward and instead flows strictly forward, like water down a one-way pipe. In this disordered state, the atoms no longer act as a single unit but rather as a chain of individuals, where each atom only feels the light that has passed through its neighbors.

The most striking discovery is that this shift from a collective, two-way interaction to a one-way, sequential flow happens surprisingly quickly. The researchers found that you do not need a chaotic mess of atoms to see this one-way behavior; a modest amount of disorder is sufficient to suppress the backward-moving light. Once this threshold is crossed, the system enters a regime where the light intensity drops not in the smooth, exponential way predicted by standard laws of absorption, but in a much sharper, power-law fashion. This change marks a distinct boundary in the system's behavior. On one side of this boundary, the atoms at the front of the line are fully excited by the laser, absorbing all the energy. On the other side, the atoms at the back remain in their calm, ground state, untouched by the light. This creates a sharp divide, a phase separation, where the cloud of atoms is split into two distinct regions: a saturated, active front and a quiet, passive rear.

This division is not just a theoretical curiosity; it appears to be a robust feature of many real-world systems. The researchers showed that this sharp boundary exists even when the atoms are not perfectly identical, such as when they are moving at different speeds, which causes a blurring effect known as Doppler broadening. The transition point remains clear, suggesting that the one-way model is a powerful tool for describing a wide variety of experimental setups, from atoms trapped near a thin fiber to nuclei in a solid film. Furthermore, by looking deeper than the average behavior of the atoms, the team discovered that this critical point is also where the atoms' internal correlations change most dramatically. The fluctuations in how the atoms interact with each other peak right at the boundary between the two phases, and the amount of light scattered in a messy, inelastic way also reaches a maximum there.

The work suggests that for a large class of systems where the exact positioning of atoms cannot be perfectly controlled, the complex, two-way models are often unnecessary. Instead, a simpler, one-way description captures the essential physics with high accuracy. This finding bridges the gap between two major theories in quantum optics, showing that the messy reality of disordered atoms naturally leads to the clean, sequential behavior of a one-way waveguide. It provides a clearer understanding of how light propagates through dense matter, offering a more intuitive and effective way to model experiments that have previously been difficult to describe. The study confirms that the sharp separation of phases is a real, observable phenomenon that persists even in the presence of disorder and thermal motion, offering a new lens through which to view the interaction of light and matter in one dimension.

Technical Summary: Emergence of Unidirectionality and Phase Separation in Optically Dense Emitter Ensembles

Problem Statement
The transmission of light through optically dense ensembles of two-level emitters in one-dimensional geometries is traditionally described by two distinct theoretical frameworks: the driven-dissipative Dicke model (DM) and the unidirectional (chiral/cascaded) waveguide model, which is equivalent to the Maxwell-Bloch equations. While both models predict phase transitions and phase separations, their regimes of validity and the relationship between them remain unclear. Specifically, experimental systems such as free-space atomic clouds or nanofiber-coupled atoms often possess varying degrees of positional disorder. It is not transparently obvious when a system should be modeled by the permutationally symmetric DM (requiring precise Bragg ordering) versus the unidirectional waveguide model (UWM), particularly given that disorder typically breaks the symmetry required for the DM. This paper addresses the crossover between these models by investigating how positional disorder influences the emergence of unidirectionality and phase separation.

Methodology
The authors employ a parent spin model derived from a bidirectional waveguide master equation (BWM), which describes $N$ emitters coupled to forward and backward propagating modes. The position of the emitters is treated as a variable, with disorder parametrized by $\eta$ , representing the standard deviation of inter-emitter distances relative to the Bragg spacing ( $\lambda/2$ ).

The study utilizes three primary theoretical approaches:

Ensemble Averaging (EAM): The master equation is averaged over the distribution of emitter positions. This yields a position-independent model that interpolates between the DM (limit $\eta \to 0$ ) and the UWM (limit $\eta \to \infty$ ).
Mean-Field Theory (MFT): The authors solve the mean-field equations for the EAM and for single realizations of the BWM. This allows for the calculation of emitter inversion, transmission, and reflection across varying optical depths ( $D$ ) and drive strengths ( $s_0$ ).
Second-Order Cumulant Expansion (CE): To go beyond mean-field theory, specifically for the UWM, the authors employ a second-order cumulant expansion. This method leverages the effective linearity of the unidirectional model to compute two-body correlation functions and inelastic scattering components, which are computationally prohibitive in the bidirectional case due to stiffness and non-linearity.

The analysis covers the thermodynamic limit and finite system sizes, examining the dependence on optical depth, drive strength, and the disorder parameter $\eta$ . The effects of Doppler broadening are also included numerically within the mean-field framework.

Key Contributions and Results

Crossover from Bidirectional to Unidirectional Dynamics: The study demonstrates that the transition from DM dynamics to UWM dynamics is driven by positional disorder. For $\eta \gtrsim 0.08$ , the random phases acquired by light traveling between emitters lead to destructive interference of the left-propagating mode. Consequently, the system effectively becomes unidirectional, with the left-propagating field acting as an additional decay channel. The authors find that even modest disorder ( $\eta \sim 0.01$ ) is sufficient to induce significant fluctuations and shift the system toward unidirectional behavior.
Phase Separation and Critical Values:
- Unidirectional Limit: In the UWM limit, a sharp phase separation emerges in the thermodynamic limit. The system separates into a saturated region (front of the ensemble) and a ground-state region (back of the ensemble). The critical point for this transition occurs at a scaled saturation parameter $\tilde{s} = s_0/D = 1$ .
- Dicke Limit: In the ordered DM limit ( $\eta \to 0$ ), the phase transition (bistability) occurs at a higher critical value, approximately $\tilde{s} \approx 2$ (specifically $s_0 \approx 1.77 D_N$ for finite systems).
- Intermediate Regime: As disorder is varied, the transition point shifts continuously. The authors observe that the critical value $\tilde{s}=1$ remains a special point in the correlation landscape even for finite systems and is independent of Doppler broadening effects.
Correlations and Fluctuations: Using the second-order cumulant expansion, the authors reveal that the critical point $\tilde{s}=1$ in the UWM marks a transition in the emitter's correlation landscape. Far from the thermodynamic limit, this point separates regions of strong correlations from regions of weak correlations. Furthermore, the magnitude of inelastically transmitted photons (and integrated quadrature fluctuations) peaks at this critical value.
Validity of the Unidirectional Model: The results indicate that a large class of effective one-dimensional systems with imperfect spatial ordering can be accurately modeled using the unidirectional waveguide approach. The UWM provides a more intuitive and accurate description for systems like elongated cigar-shaped atomic clouds in free space compared to the DM.
Doppler Broadening: Numerical inclusion of Doppler broadening shows that while it smooths the "kink" in the transmission curve, it does not eliminate the signatures of phase separation or alter the critical value $\tilde{s}=1$ at the mean-field level.

Significance and Claims
The paper claims to provide a comprehensive analysis of the transmission properties of driven ensembles by bridging the gap between the driven-dissipative Dicke model and the Maxwell-Bloch equations. The primary significance lies in establishing that:

Robustness of Unidirectionality: Unidirectional dynamics emerge naturally in the presence of relatively small positional disorder, justifying the use of the UWM (and Maxwell-Bloch equations) for a broader range of experimental setups than previously assumed, including those without tight control over emitter spatial ordering.
Universality of Phase Separation: Phase separation is a feature common to both the DM and the UWM, though the critical control parameters differ. The UWM exhibits a phase separation at $\tilde{s}=1$ , which is distinct from the DM's bistability region.
Correlation Signatures: The critical point in the UWM is not merely a mean-field artifact but corresponds to a distinct feature in the two-body correlation landscape and a maximum in inelastic scattering, observable even in finite-sized systems.

The authors conclude that their work offers a firmer theoretical motivation for using semiclassical unidirectional descriptions in optically dense media and highlights the critical role of disorder in determining the effective dynamics of light-matter interaction in one-dimensional geometries.

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