Understanding how surfaces grow has long been one of the most important challenges in physics. In 1986, researchers introduced the Kardar-Parisi-Zhang (KPZ) equation, a theory designed to explain growth across a wide range of systems. Over time, this framework has been applied to everything from crystal formation and population dynamics to flame surfaces and even machine learning. This idea is simple but powerful. Quite different systems may grow to follow the same underlying rules.
Now, scientists at the University of Würzburg have taken a major step forward. After previously being confirmed in a one-dimensional system in 2022, the team achieved the first experimental proof that the KPZ theory also holds in two dimensions. This is an important milestone in demonstrating how universal this model really is.
Why predicting growth is so difficult
“When a surface grows, be it a crystal, bacteria, or a flame surface, the process is always nonlinear and random. In physics, such systems are described as being out of equilibrium,” explains Siddhartha Damm, postdoctoral researcher at the Würzburg Dresden Cluster of Excellence ctd.qmat at the Department of Technical Physics at the University of Würzburg. “Designing systems that can simultaneously measure how non-equilibrium processes evolve in space and time is extremely difficult, especially since these processes unfold over ultra-short timescales. That is why validating the KPZ model in two dimensions took so long. We have now succeeded in controlling non-equilibrium quantum systems in the laboratory, something that has only recently become technically feasible.”
Construction of ultralow temperature 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 light-matter hybrids that combine photons and excitons. They exist only temporarily and under non-equilibrium conditions. Created by lasers, 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 light into the system, they are created and grow. Using advanced experimental techniques, we were able to quantify the spatial and temporal evolution of this growing quantum system, and we found that it follows the KPZ model,” Damm 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 were able to experimentally confirm KPZ’s predictions, but only in a one-dimensional system. Extending this to two dimensions proved much more difficult. The new results provide the missing piece.
“The experimental demonstration of the universality of KPZ in two-dimensional material systems highlights how fundamental this equation is for real non-equilibrium systems,” Diehl comments on the Wurzburg team’s work.
Precise material design enables
A key part of this advancement was the ability to carefully design the materials themselves. The researchers created a complex structure that traps photons within a “quantum film” with a mirror layer in the middle. Within this layer, photons interact with excitons within the gallium arsenide, forming polaritons that can be observed as they evolve.
“By precisely controlling the thickness of the individual material layers using molecular beam epitaxy, we were able to tune their optical properties and fabricate the highly reflective mirrors needed under ultra-high vacuum conditions,” explains Simon Widman, a postdoctoral fellow in the Department of Engineering Physics who conducted the experiments with Siddhartha Dam. “We can control how the material grows atom by atom and fine-tune all the experimental parameters, such as the laser that needs to excite the sample with micrometer precision. This level of control was essential to successfully demonstrate the universality of KPZ.”
