Control how the material grows atom by atom
For forty years, a mathematical theory sat at the edge of proof — elegant in its predictions, elusive in its confirmation. In May 2026, physicists at the University of Würzburg crossed that threshold, demonstrating experimentally that the KPZ theory, which describes how chaotic surfaces grow, holds true in two-dimensional quantum systems. Using light-matter hybrid particles cooled to the edge of absolute zero, they revealed what may be a universal law threading through crystals, flames, bacteria, and quantum physics alike — a single hidden grammar beneath the apparent disorder of growth itself.
- A 40-year-old theoretical prediction about how random, nonlinear surfaces grow had never been experimentally confirmed in two dimensions — a gap that left one of physics' most tantalizing universality claims incomplete.
- The core difficulty was brutal: growth processes unfold in picoseconds, across both space and time simultaneously, demanding experimental precision that has long exceeded what any lab could engineer.
- The Würzburg team built their way through the obstacle — cooling gallium arsenide to near absolute zero, firing continuous laser light into it, and coaxing photons and matter to fuse into fleeting hybrid particles called polaritons that exist just long enough to be measured.
- When the data came in, it matched — the KPZ equation held in two dimensions, extending a confirmation that had only reached one dimension as recently as 2022.
- The finding now points outward: if crystals, bacteria, flames, and quantum systems all obey the same growth law, science may be holding a master key to predicting and engineering complex, unpredictable systems across biology, materials science, and physics.
For four decades, physicists have suspected that a crystal forming, bacteria spreading, and a flame advancing might all obey the same hidden mathematical rule. In May 2026, researchers at the University of Würzburg turned that suspicion into experimental fact — becoming the first to prove that the KPZ theory, which describes how surfaces grow in chaotic, random ways, holds true in two dimensions.
The challenge was formidable. Growth processes are fundamentally nonlinear and out of equilibrium, unfolding on ultrashort timescales that make simultaneous measurement across space and time extraordinarily difficult. As postdoctoral researcher Siddhartha Dam explained, that difficulty is precisely why two-dimensional verification took so long.
To solve it, the team built a quantum system of exceptional precision. They cooled gallium arsenide to just above absolute zero and bombarded it with laser light until photons and matter fused into hybrid particles called polaritons — fleeting, existing for only picoseconds, but ideal for studying rapid growth. Using molecular beam epitaxy, they constructed the material atom by atom, trapping photons inside a quantum film where they could interact with the semiconductor and form the particles needed for observation. Doctoral researcher Simon Widmann credited this atomic-level control as essential to the result.
The theoretical scaffolding had been built in 2015 by Sebastian Diehl and colleagues at the University of Cologne, and a one-dimensional confirmation had arrived from Paris in 2022 — a proof of concept, but incomplete. The leap to two dimensions required solving new technical problems, which Würzburg's team did.
When they measured their polariton system's evolution, the data aligned with the KPZ model. The implication reaches far beyond the lab: if the same mathematical rule governs growth across quantum systems, biology, and materials science, it may represent a fundamental law of nature — and a new tool for predicting and engineering the complex, unpredictable systems that shape the world.
For four decades, physicists have suspected that wildly different things—a crystal forming, bacteria spreading, a flame advancing—might all obey the same hidden mathematical rule. Now, researchers at the University of Würzburg have moved that suspicion into experimental fact. They have become the first to prove that the KPZ theory, a model describing how surfaces grow in chaotic, random ways, actually works in two dimensions. The confirmation arrived in May 2026, capping a long hunt that began with theoretical prediction and moved through one-dimensional proof just four years earlier.
The puzzle has been stubborn because growth itself is stubborn. When surfaces expand—whether you're watching crystals form, bacteria colonize a surface, or flames consume material—the process is fundamentally nonlinear and random. Physicists call this being "out of equilibrium," a state where the normal rules of balance don't apply. The challenge isn't just understanding what happens; it's building an experiment that can actually measure how a system evolves across both space and time simultaneously, all while the action unfolds in picoseconds. "Engineering a system capable of simultaneously measuring how a non-equilibrium process evolves in space and time is extremely challenging," explains Siddhartha Dam, a postdoctoral researcher at Würzburg's Chair of Technical Physics. "These processes unfold on ultrashort timescales. That's why verifying the KPZ model in two dimensions has taken so long."
To crack the problem, the Würzburg team built a quantum system of extraordinary precision. They took gallium arsenide, a semiconductor, and cooled it to −269.15°C—just a few degrees above absolute zero. Then they bombarded it continuously with laser light. Under these extreme conditions, something unusual happened: light and matter fused into hybrid particles called polaritons. These are fleeting creatures, existing for only a few picoseconds before vanishing. But in that brief window, they are perfect for studying rapid growth. As the laser pumped energy into the material, polaritons were born and multiplied, creating a growing system the researchers could track with precision.
The theoretical groundwork had been laid years earlier. Sebastian Diehl, a professor at the University of Cologne's Institute for Theoretical Physics, and his colleagues developed the theoretical framework in 2015 for testing KPZ behavior in exactly this kind of quantum setup. In 2022, researchers in Paris managed to confirm KPZ predictions experimentally, but only in a one-dimensional system—a crucial proof of concept, but incomplete. The jump to two dimensions required solving new technical problems. The Würzburg team succeeded by engineering the material itself with atomic precision. Using molecular beam epitaxy, they built a complex structure with mirror layers that trap photons inside a central quantum film. Within that film, photons interact with excitons in the gallium arsenide, forming the polaritons they needed to observe. "We control how the material grows atom by atom and can fine-tune all experimental parameters," says Simon Widmann, a doctoral researcher who conducted the experiments alongside Dam. "This level of control was essential for successfully demonstrating KPZ universality."
When they measured the spatial and temporal evolution of their growing polariton system, the data aligned with the KPZ model. The equation held. The implication is profound: if the same mathematical rule governs growth in crystals, bacteria, flames, and quantum light-matter systems, it may be a fundamental law of nature itself. The discovery suggests that seemingly unrelated processes across biology, materials science, and physics might all be expressions of the same underlying principle. That realization opens new possibilities for predicting and engineering complex systems that grow and evolve in unpredictable ways.
Notable Quotes
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.— Siddhartha Dam, postdoctoral researcher at University of Würzburg
The experimental demonstration of KPZ universality in two-dimensional material systems highlights just how fundamental this equation is for real non-equilibrium systems.— Sebastian Diehl, Institute for Theoretical Physics, University of Cologne
The Hearth Conversation Another angle on the story
Why did it take forty years to prove this in two dimensions when one dimension worked in 2022?
The jump from one to two dimensions isn't just adding another direction—it's exponentially more complex. You have to measure growth happening in multiple directions at once, on timescales measured in picoseconds. The technical barriers were immense until very recently.
What exactly are polaritons, and why are they the right tool for this?
They're hybrids of light and matter—photons bound to excitons in the material. They exist only briefly, only under non-equilibrium conditions. That's exactly what you need to study rapid growth processes. They're born from the laser, they grow, they vanish. Perfect window into the phenomenon.
So you're saying bacteria spreading and a crystal forming follow the same equation?
That's what the data suggests. The KPZ model describes any surface growing in a nonlinear, random way. Whether it's biological, physical, or quantum—if it's out of equilibrium and growing, the same hidden rule seems to apply.
How does cooling something to near absolute zero help you see growth?
At that temperature, quantum effects dominate. The polaritons behave in ways that let you track them precisely. You eliminate thermal noise that would blur the measurement. You get a clean signal of the growth process itself.
What comes next? Does this change how we design materials or predict biological systems?
That's the frontier now. If this universality holds across domains, it could reshape how we approach problems in materials engineering, disease modeling, even flame dynamics. We're just beginning to understand what this means in practice.