Now, scientists at the University of Würzburg have taken a major step forward. After earlier confirmation in one-dimensional systems in 2022, the team has achieved the first experimental proof that the KPZ theory also holds in two dimensions. This marks a significant milestone in showing just how universal this model really is.

Why Growth Is So Difficult to Predict

"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."

Building an Ultracold Quantum Experiment

To test the theory, the researchers designed a highly controlled quantum setup. They cooled a semiconductor made from gallium arsenide (GaAs) to −269.15°C and continuously stimulated it with a laser. Under these conditions, unusual particles called polaritons formed inside the material.

Polaritons are hybrids of light and matter, combining photons with excitons. They exist only briefly and only under non-equilibrium conditions. Created by the laser, they disappear again within a few picoseconds, making them ideal for studying rapid growth processes.

"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.

From Theory to Experimental Proof

The concept of testing KPZ behavior in such a system was first proposed by Sebastian Diehl, a professor at the Institute for Theoretical Physics at the University of Cologne and a member of the research team. His group developed the theoretical foundation in 2015.

In 2022, researchers in Paris managed to confirm KPZ predictions experimentally, but only in a one-dimensional system. Extending this to two dimensions proved far more difficult. The new results now provide that missing piece.

"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.

Precision Materials Design Makes It Possible

A key part of the breakthrough was the ability to carefully engineer the material itself. The team created a complex structure in which mirror layers trap photons inside a central "quantum film." Within this layer, photons interact with excitons in the gallium arsenide, forming polaritons that can be observed as they evolve.

"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."