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Noise-robust distinguishability of collective coherence via Jensen–Shannon divergence in a cavity

This paper demonstrates that a local scaling exponent derived from steady-state photon emission serves as a robust, noise-resilient experimental probe for identifying collective coherence and dissipative criticality in bad-cavity superradiant systems, overcoming limitations of traditional photon-number measurements.

Original authors: Muhammad Tayyab Raza, Daeho Yang

Published 2026-10-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Muhammad Tayyab Raza, Daeho Yang

Original paper licensed under CC BY 4.0 (https://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

In the vast, humming world of quantum physics, scientists often look for signs of order emerging from chaos. Imagine a crowd of people in a dark room. If each person acts alone, the noise they make is just a random collection of individual voices. But if they begin to listen to one another and start speaking in unison, a powerful, collective sound emerges. This shift from individual action to synchronized unity is a fundamental concept in physics, known as collective behavior. In the microscopic realm, atoms trapped inside a box of light can do something similar. They can act as independent emitters, or they can lock into a rhythm, radiating energy together in a burst of coordinated brilliance. This phenomenon, called superradiance, is a hallmark of how quantum systems organize themselves. However, spotting this transition in a real experiment is notoriously difficult. The signals are often messy, and the difference between a crowd of individuals and a synchronized chorus can be hidden within the same measurements, making it hard to tell when the true moment of unity has arrived.

A team of researchers at Gachon University in South Korea has found a new way to see this hidden transition clearly, even when the data is noisy. They focused on a specific setup where atoms are trapped inside an optical cavity, a device that bounces light back and forth. In this "bad cavity" environment, the light leaks out very quickly, but the atoms can still store a collective memory of their synchronized state. The researchers realized that simply counting how many photons, or particles of light, are emitted is not enough. A large number of photons could come from a few atoms acting together, or from many atoms acting alone. To solve this, they stopped looking at the total brightness and started looking at how that brightness changes as you add more atoms to the system. They developed a method to measure the "scaling" of the light. If the atoms are acting independently, adding more of them increases the light output in a steady, predictable way. But if the atoms have synchronized, adding more of them causes the light output to surge much more dramatically, growing at a different, faster rate.

By tracking this change in growth rate, the researchers discovered a sharp, clear signal that marks the exact moment the atoms switch from acting alone to acting together. They called this a collective response scaling. In their simulations, which involved groups of up to forty atoms, they watched this scaling factor shift abruptly. When the atoms were uncorrelated, the factor was close to one, indicating a simple, additive relationship. As they tuned the system toward the critical point where synchronization begins, this factor jumped sharply toward two, revealing that the light was now being driven by a collective force. This jump was so distinct that it acted like a perfect switch, separating the two states of matter. The researchers compared this new method against older ways of measuring the same phenomenon, such as looking at the total light intensity or calculating the fraction of atoms that were synchronized. They found that those traditional methods often blurred the line between the two states, showing a smooth, gradual change that made it hard to pinpoint exactly when the transition happened.

To prove that their new method was robust, the team tested it against a realistic challenge: measurement noise. In any real experiment, instruments have a background hum or a "noise floor" that can obscure subtle details. The researchers simulated this by adding a layer of random statistical noise to their data, similar to static on a radio. They found that the traditional measurements became indistinguishable under this noise; the signals for the synchronized and unsynchronized states overlapped so much that they looked identical. However, the new scaling method remained sharp and clear. Even with the noise present, the jump in the scaling factor stood out, allowing the researchers to identify the transition point with high confidence. They used a mathematical tool called Jensen-Shannon divergence to quantify this difference, which essentially measures how easy it is to tell two sets of data apart. For the traditional methods, this measure dropped to zero, meaning the data was indistinguishable. For their new scaling method, the measure stayed at its maximum, proving that the signal of collective coherence was still perfectly visible.

The study also connected this observation to the fundamental speed at which the system settles down. In physics, when a system is near a critical transition, it often slows down, taking a long time to reach a stable state. The researchers found that the moment their scaling factor jumped was exactly the same moment when the system's relaxation time stretched out the most. This confirmed that they were observing a genuine critical event, a point where the system is fundamentally reorganizing itself. Their work suggests that by simply watching how the brightness of light changes as you add more atoms, scientists can detect the birth of collective quantum behavior without needing complex, microscopic measurements that are often impossible to perform in a real lab. This approach offers a practical, noise-resistant way to study how quantum systems synchronize, opening a clearer window into the strange and beautiful world where individual particles learn to move as one.

Technical Summary: Noise-Robust Distinguishability of Collective Coherence via Jensen–Shannon Divergence in a Cavity

Problem Statement
In driven-dissipative many-body systems, a central challenge is identifying collective organization from experimentally accessible observables. While phenomena such as dissipative phase transitions and superradiant emission arise from strong collective correlations, standard measurements often fail to distinguish between independent spontaneous emission and collective coherence. In bad-cavity superradiant systems, the steady-state photon number is a "branch-composite" signal containing both a linear contribution from independent emitters ( $N$ ) and a quadratic contribution from collective coherence ( $N^2$ ). Consequently, the magnitude of the photon number, or even conventional normalizations (e.g., photon number per $N^2$ ), can vary smoothly across a critical crossover, obscuring the microscopic origin of the response and failing to directly reveal the development of collective organization.

Methodology
The authors propose a diagnostic approach based on the local scaling exponent of the steady-state photon number with respect to the system size (atom number $N$ ). This metric, denoted as $q_N$ , is defined as the logarithmic derivative of the mean intracavity photon number with respect to $N$ :
$q_N(N, g, w) \equiv \frac{\partial \log(\langle n \rangle_{ss})}{\partial \log(N)}$
where $g$ is the atom-cavity coupling and $w$ is the incoherent pump rate.

The study employs the following analytical and numerical tools:

Master Equation Analysis: The cavity field is adiabatically eliminated to derive a reduced master equation for the atomic ensemble, allowing the extraction of steady-state density matrices ( $\rho_{ss}$ ) and the calculation of $q_N$ .
Finite-Size Scaling: The authors perform numerical simulations using Permutation-Invariant Quantum Simulations (PIQS) for finite atom numbers ( $N$ up to 40) to analyze the behavior of $q_N$ near the critical coupling $g_c$ . They test for data collapse using the scaling form $q_N(g, N) = F[(g-g_c)N^{1/\nu}]$ .
Liouvillian Gap Analysis: To validate the critical nature of the transition, the spectral gap of the Liouvillian superoperator ( $\delta_L$ ) is computed. The gap closing ( $\delta_L \to 0$ ) serves as a benchmark for the quantum dissipative critical point.
Information-Theoretic Distinguishability: To assess robustness against experimental noise, the authors apply the Jensen–Shannon Divergence (JSD). They model measurement uncertainty using Gaussian distributions with a finite noise floor and calculate the JSD between probability distributions of observables on either side of the critical boundary ( $g_c \pm \epsilon$ ).

Key Results

Scaling Exponent as a Phase Marker: The metric $q_N$ acts as a branch-selective physical marker. In the normal phase (independent emission), $q_N \approx 1$ . In the superradiant phase (collective coherence), $q_N \approx 2$ . Unlike the photon number magnitude or the collective fraction ( $F_{coll}$ ), which change smoothly across the crossover, $q_N$ exhibits a sharp step-like transition, effectively separating the linear and quadratic scaling regimes.
Finite-Size Scaling and Susceptibility: The collective response susceptibility ( $\chi_{q_N} = \partial q_N / \partial g$ ) exhibits finite-size divergence, with peak height increasing and width narrowing as $N$ increases. The data for different system sizes collapses onto a single universal curve using an effective critical exponent $1/\nu \approx 0.89$ , confirming the identification of a genuine critical point.
Noise Robustness via JSD: When subjected to a finite Gaussian noise floor, conventional observables like the normalized photon number ( $\Phi_{SR}$ ) and the collective fraction ( $F_{coll}$ ) become statistically indistinguishable across the critical boundary (JSD $\to$ 0). In contrast, the scaling response $q_N$ maintains a sharp distinction, with JSD approaching the theoretical maximum of 1.0 bit. This indicates that the information-theoretic signature of the phase transition remains robust against measurement noise.
Liouvillian Gap Coincidence: The peak in the collective response $q_N$ and the pronounced minimum (closing) of the Liouvillian gap $\delta_L$ occur at the exact same critical coupling $g_c$ . This coincidence validates that the scaling behavior of the photon number directly encodes the underlying quantum dissipative criticality and the slowing down of relaxation dynamics.

Significance and Claims
The paper establishes that local photon-number scaling ( $q_N$ ) is an experimentally accessible probe for collective coherence and dissipative criticality in cavity QED and open quantum many-body systems. The authors claim that this approach offers distinct advantages over traditional order parameters:

Directness: It utilizes the photon number, an observable naturally recorded in experiments, without requiring the reconstruction of microscopic correlations or full quantum-state tomography.
Sensitivity: It transforms the diagnostic from measuring magnitude at a fixed size to analyzing the atom-number dependence, thereby revealing the "branch content" (independent vs. collective) of the emission.
Robustness: The enhanced distinguishability of the scaling response, quantified by Jensen–Shannon divergence, persists even in the presence of significant measurement noise, where other metrics fail.

The work suggests that this method provides a viable route to identify non-equilibrium steady-state phase transitions and study non-equilibrium universality in systems such as strontium or ytterbium atoms in optical resonators, where adiabatic elimination of the cavity mode is applicable.

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