With this semiconductor sample, approximately 20 micrometers in size, Würzburg researchers from the Cluster of Excellence ctd.qmat have, for the first time ever, demonstrated KPZ universality in a two-dimensional system in space and time.
Image Credit: Jochen Thamm / think-design
Scientific Frontline: Extended "At a Glance" Summary: Kardar-Parisi-Zhang (KPZ) Equation
The Core Concept: The Kardar-Parisi-Zhang (KPZ) equation is a universal mathematical framework used to describe the nonlinear and random growth of surfaces and interfaces in systems that operate out of thermodynamic equilibrium.
Key Distinction/Mechanism: The KPZ model mathematically captures the complex spatial and temporal evolution of growing boundaries. Recently, researchers experimentally verified its application in a two-dimensional quantum system by continuously exciting an engineered gallium arsenide semiconductor with a laser. This created polaritons—highly dynamic hybrid particles of light and matter—allowing scientists to precisely track the growth and decay of a non-equilibrium system in real time.
Origin/History: The theoretical foundation for the KPZ equation was established by three physicists in 1986. While the model was first experimentally confirmed for one-dimensional systems in 2022 by a research group in Paris, the world's first experimental proof for two-dimensional surfaces and interfaces was published in April 2026 by researchers from the Würzburg–Dresden Cluster of Excellence (ctd.qmat).
Major Frameworks/Components:
- Non-Equilibrium Dynamics: The behavior of open systems driven away from thermal equilibrium, characterized by continuous energy input and nonlinear development.
- Polaritons: Hybrid quasiparticles formed by coupling photons (light) and excitons (matter), which only exist under non-equilibrium conditions and were utilized to measure the quantum system's growth.
- Molecular Beam Epitaxy: An advanced materials engineering technique used to fabricate ultra-thin, highly reflective semiconductor mirror layers atom-by-atom to successfully confine photons.
Branch of Science: Quantum Physics, Condensed Matter Physics, and Materials Science.
Future Application: Validating the 2D KPZ equation provides a robust tool for advanced materials engineering, understanding cell and population growth in biology, refining the dynamic mathematical models underlying machine learning algorithms, and predicting complex physical phenomena like crystal formation and flame front expansion.
Why It Matters: Successfully demonstrating KPZ universality in a two-dimensional quantum system provides crucial experimental proof for a long-standing theoretical model. It confirms that the KPZ equation is extremely fundamental for understanding real-world, non-equilibrium growth processes across vastly different scientific disciplines.
The question of how surfaces grow is one of the most fundamental problems in physics. In 1986, three physicists laid the foundation for a universal theory of growth with the Kardar–Parisi–Zhang (KPZ) equation—a framework with wide-ranging applications across physics, mathematics, biology, and computer science. From the dynamics of crystal formation and mathematical system analysis to the growth of cells, populations, and flame fronts—and even the development of machine-learning algorithms—the KPZ universality class applies wherever growth processes are modeled.
After the model was first experimentally confirmed for one-dimensional systems based on polaritons in 2022, a Würzburg research team has now tested this powerful framework again in the lab, delivering the world’s first experimental proof for two-dimensional systems and interfaces.
Würzburg researchers achieve breakthrough in 2D quantum system
“When surfaces grow—whether crystals, bacteria, or flame fronts—the process is always nonlinear and random. In physics, we describe such systems as being out of equilibrium,” explains Siddhartha Dam, a postdoctoral researcher in the Würzburg–Dresden Cluster of Excellence ctd.qmat at the University of Würzburg’s Chair of Technical Physics. “Engineering a system capable of simultaneously measuring how a non-equilibrium process evolves in space and time is extremely challenging – especially because these processes unfold on ultrashort timescales. That’s why verifying the KPZ model in two dimensions has taken so long. We have now succeeded in controlling a non-equilibrium quantum system in the laboratory – something that has only recently become technically feasible.”
To achieve this, the researchers cooled a semiconductor sample based on gallium arsenide (GaAs) to –269.15°C and continuously excited it with a laser. Through precise materials engineering, polaritons—hybrid particles made of photons (light) and excitons (matter)—formed within a specific layer of the structure. Polaritons exist only under non-equilibrium conditions: they are generated by laser excitation and decay within just a few picoseconds before leaving the system.
“We can precisely track where the polaritons are in the material. When we pump the system with light, polaritons are created—they grow. Using advanced experimental techniques, we were able to quantify both the spatial and temporal evolution of this growing quantum system and found that it follows the KPZ model,” Dam explains.
The key idea—testing a universal theory of growth in a quantum system using polaritons, which themselves exist only within a highly dynamic growth process—was developed by Sebastian Diehl, a professor at the Institute for Theoretical Physics at the University of Cologne and a member of the research team. The theoretical groundwork dates back to 2015. In 2022, a research group in Paris provided the first experimental evidence of KPZ behavior—but only in a one-dimensional system. “The experimental demonstration of KPZ universality in two-dimensional material systems highlights just how fundamental this equation is for real non-equilibrium systems,” says Diehl, commenting on the Würzburg team’s achievement.
Targeted materials design enables injection of polaritons
To inject polaritons into the material, the researchers engineered a highly complex sample structure. Mirror layers confine photons within a central “quantum film” layer where they can couple with excitons in the gallium arsenide to form polaritons, grow, and be measured.
“By precisely controlling the thickness of individual material layers using molecular beam epitaxy, we were able to tune their optical properties and hence fabricate the necessary highly reflective mirrors under ultra-high vacuum conditions,” explains Simon Widmann, a doctoral researcher at the Chair of Engineering Physics, who conducted the experiments together with Siddhartha Dam. “We control how the material grows atom by atom and can fine-tune all experimental parameters—for example, the laser, which must excite the sample with micrometer precision. This level of control was essential for successfully demonstrating KPZ universality.”
Published in journal: Science
Title: Observation of Kardar-Parisi-Zhang universal scaling in two dimensions
Authors: Simon Widmann, Siddhartha Dam, Johannes Düreth, Christian G. Mayer, Romain Daviet, Carl Philipp Zelle, David Laibacher, Monika Emmerling, Martin Kamp, Sebastian Diehl, Simon Betzold, Sebastian Klembt, and Sven Höfling
Source/Credit: Universität Würzburg
Reference Number: qs041026_01
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