🌀 nonlinear sciences

Experimental study of coupled quantum billiards with integrable and chaotic classical dynamics and test of a special Rosenzweig-Porter model

This paper experimentally demonstrates that a special Rosenzweig-Porter model accurately describes the spectral properties of two coupled quantum billiards—one integrable and one chaotic—by showing how increasing coupling strength induces eigenstate mixing and a transition toward fully chaotic dynamics.

Original authors: Xiaodong Zhang, Jiongning Che, Barbara Dietz

Published 2026-11-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Xiaodong Zhang, Jiongning Che, Barbara Dietz

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

In the vast landscape of physics, there is a fundamental divide between order and chaos. For centuries, scientists have understood that some systems, like a perfectly swinging pendulum or the predictable orbit of a planet, follow strict, unchanging rules. These are called integrable systems, and their behavior is easy to forecast. On the other side of the divide lie chaotic systems, where tiny changes in starting conditions lead to wildly different outcomes, making long-term prediction impossible. For decades, researchers have been fascinated by how the quantum world—the realm of atoms and subatomic particles—mimics these classical behaviors. They discovered that the energy levels of a quantum system that is classically chaotic are distributed in a very specific, statistical way, while those of an orderly system follow a different pattern. The big question has always been: what happens when you take a piece of an orderly system and connect it to a piece of a chaotic one? Does the chaos infect the order, or does the order calm the chaos?

A team of researchers set out to answer this by building a physical model of two such worlds colliding. They constructed a flat, circular microwave chamber, essentially a giant, hollow drum made of metal, which acts as a laboratory for studying quantum behavior without needing to manipulate individual atoms. Inside this chamber, they placed a wall to split the space into two distinct halves. One half was a perfect semicircle, a shape known to produce orderly, predictable wave patterns. The other half was slightly distorted and filled with four circular metal disks, a design that forces waves to bounce around in a chaotic, unpredictable manner. The key to their experiment was the wall itself. They did not make it a solid, impenetrable barrier. Instead, they left a single opening in the wall, allowing the waves to leak from one side to the other. By adjusting the size of this opening, they could control how strongly the two halves talked to each other, ranging from a whisper to a shout.

The researchers cooled their entire apparatus to a temperature just above absolute zero, roughly minus 268 degrees Celsius, to ensure the metal walls became superconducting. This extreme cold eliminated electrical resistance and noise, allowing them to hear the faintest echoes of the waves inside the chamber. They fired microwaves into the box and listened for the specific frequencies at which the waves resonated, or "sang" loudly. These resonant frequencies are the fingerprints of the system's energy levels. Even when the opening was effectively closed, there was still a weak coupling between the two halves due to microwave leakage, meaning the two sides were never perfectly isolated. As they widened the opening, the sound changed. The waves from the chaotic side began to seep into the orderly side, and vice versa. This mixing caused the orderly patterns to break down, and the entire system began to behave more like a fully chaotic one, even though half of it was still shaped to be perfectly orderly.

To understand exactly what was happening, the team compared their physical measurements to a mathematical model known as the Rosenzweig-Porter model. This model is a way of simulating how two different types of systems interact using random numbers to represent the chaotic parts and simple lists for the orderly parts. The researchers found that this model was an excellent match for their real-world data. It successfully predicted how the energy levels shifted and how the patterns mixed as the connection between the two halves grew stronger. This confirmed that the model is a powerful tool for measuring the strength of the coupling between two quantum systems. The study showed that even a very weak connection is enough to start changing the long-range relationships between the energy levels, a phenomenon that was previously only seen in systems with specific, rare defects.

The experiment demonstrated that the transition from order to chaos is not a sudden switch but a gradual blending. As the opening in the wall grew, the distinct identities of the two halves faded. The orderly side lost its perfect symmetry, and the chaotic side's randomness became the dominant feature of the whole. The researchers found that for sufficiently large openings, the entire system behaved as if it were fully chaotic, regardless of the fact that half of it was still a perfect semicircle. This finding is significant because it provides a clear, experimental path to understanding how complex, chaotic behavior emerges from the interaction of simpler, orderly components. It also validates the use of specific random matrix models to describe these transitions, offering a new way to quantify how strongly different parts of a quantum system are linked. By watching the waves mix in a simple metal box, the team has illuminated a fundamental process that likely occurs in many complex quantum systems, from the nuclei of atoms to the behavior of electrons in new materials.

Technical Summary: Experimental Study of Coupled Quantum Billiards and the Rosenzweig-Porter Model

Problem Statement
The paper addresses the spectral properties of quantum systems composed of two coupled subsystems: one exhibiting integrable classical dynamics and the other exhibiting chaotic classical dynamics. While the spectral statistics of purely integrable systems (Poisson statistics) and purely chaotic systems (Wigner-Dyson statistics, specifically the Gaussian Orthogonal Ensemble or GOE) are well-established, the behavior of systems where these two distinct dynamical regimes interact remains a subject of investigation. Specifically, the authors aim to understand how the coupling between an integrable billiard and a chaotic billiard alters spectral properties, ranging from very weak to moderate coupling strengths. Previous studies have often induced transitions from integrable to chaotic dynamics by introducing random potentials; this work instead investigates the transition induced by the physical coupling of two distinct billiard geometries.

Methodology
The study employs a combination of microwave billiard experiments and Random Matrix Theory (RMT) modeling.

Experimental Setup: The experiments utilize flat, cylindrical microwave resonators (microwave billiards) operating at superconducting temperatures ( $T \lesssim 5$ K) to minimize absorption and achieve high quality factors ( $Q \gtrsim 10^4$ ). The resonator is divided by a wall into two sections:
1. A semicircular section, which possesses integrable classical dynamics.
2. A nearly-semicircular section containing four circular scatterers, which possesses non-integrable (chaotic) classical dynamics.
Three configurations (MB1, MB2, MB3) were tested by varying the width of the opening in the dividing wall (0 mm, 12 mm, and 15 mm, respectively). This variation controls the coupling strength between the two sections. Resonance frequencies were measured using a Vector Network Analyzer (VNA) connected to ten antenna ports arranged in a spiral pattern to ensure comprehensive mode excitation.
Theoretical Model: The authors propose a specific variant of the Rosenzweig-Porter (RP) model to describe the system. The Hamiltonian is constructed as a block matrix:
$H = \begin{pmatrix} H_1 & 0 \\ 0 & H_2 \end{pmatrix} + \alpha \begin{pmatrix} 0 & V \\ V^T & 0 \end{pmatrix}$
Here, $H_1$ is a diagonal matrix with Gaussian-distributed random entries, modeling the integrable (semicircular) part with Poisson statistics. $H_2$ is a GOE random matrix, modeling the chaotic (scattered) part. The coupling matrix $V$ connects the two blocks, with its strength tuned by the parameter $\alpha$ , which is rescaled to a dimensionless parameter $\lambda$ .
Data Analysis: The experimental resonance frequencies were "unfolded" to remove system-specific density variations, allowing for the analysis of universal fluctuation properties. The authors analyzed:
- Short-range correlations: Nearest-neighbor spacing distributions $P(s)$ and the integrated distribution $I(s)$ .
- Ratio statistics: The distribution of ratios of consecutive level spacings ( $r$ ).
- Long-range correlations: Number variance $\Sigma^2(L)$ , spectral rigidity $\Delta_3(L)$ , spectral form factor $K(t)$ , and power spectrum $s(\tau)$ .

Key Results

Coupling-Induced Mixing: The experiments demonstrate that even extremely weak coupling significantly affects long-range spectral correlations. As the opening size increases, eigenmodes from the integrable section penetrate into the chaotic section (and vice versa), leading to a mixing of eigenstates and a breaking of the symmetry inherent in the integrable dynamics.
Model Validation: The special Rosenzweig-Porter model (Eq. 1) successfully reproduces the experimental spectral statistics across all coupling regimes.
- For weak coupling (MB1), the statistics deviate from pure Poisson behavior but do not yet reach GOE.
- For moderate and strong coupling (MB2 and MB3), the spectral properties approach those of a fully chaotic system (GOE).
Quantification of Coupling: The study establishes that the dimensionless parameter $\lambda$ in the RP model can be used to quantitatively determine the strength of the physical coupling between the two billiard sections. The experimental data aligns with RMT predictions for specific values of $\lambda$ (e.g., $\lambda \approx 0.03$ for weak, $0.23$ for moderate, and $0.35$ for strong coupling).
Comparison to Previous Work: Unlike previous studies where a random potential was introduced into a single billiard to induce complexity, this work isolates the effect of coupling two distinct dynamical regimes. The results show that the transition from integrable to chaotic statistics occurs via the overlap of eigenmodes penetrating the coupling aperture.

Significance and Claims
The paper claims that the specific Rosenzweig-Porter model variant presented is a suitable and effective tool for describing the spectral properties of coupled quantum systems with mixed integrable and chaotic dynamics. Its primary contribution is demonstrating that this model can be employed to determine the strength of the coupling in such physical systems based on spectral data.

The authors note that the transition to chaotic statistics is driven by the increasing overlap of eigenmodes penetrating through the opening, which breaks the symmetry of the integrable subsystem. The findings suggest that for sufficiently large coupling strengths, the system's spectral properties converge to those of a fully chaotic system. The study provides an experimental benchmark for RMT models describing transitions between universality classes, specifically validating the applicability of the Rosenzweig-Porter model in a physical setting involving coupled billiards.

Technical Summary: Experimental Study of Coupled Quantum Billiards and the Rosenzweig-Porter Model

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