What Are Bethe Strings?

In 1931, physicist Hans Bethe proposed that particles in certain one-dimensional quantum systems could combine into collective states now known as Bethe strings. These structures are fundamentally different from familiar molecules. Instead of being held together by chemical bonds, the particles remain bound through interactions, and the resulting states can exist only in one dimension.

For most of the past century, Bethe strings were primarily a theoretical concept. Researchers from the University of Innsbruck, in collaboration with theoretical teams from the Department of Experimental Physics at the University of Amsterdam and the Technical University of Munich, have now created and detected these multiparticle bound states in ultracold gases. The study was published in Nature Communications.

Creating Bethe Strings Close to Absolute Zero

To generate the unusual quantum state, the researchers began with a cloud of cesium atoms cooled to within a few billionths of a degree of absolute zero. They then divided the cloud into thousands of extremely thin tubes.

Inside each tube, the atoms were effectively restricted to move in a single direction, creating the one-dimensional environment required for Bethe strings to form. The researchers could also precisely control the strength of the atomic interactions.

By changing those interactions from repulsive to attractive, the team was able to bind the atoms together. Rather than collapsing into a single group, the atoms formed bound states of different sizes. Some of the largest clusters contained six or more particles.

Observing Collisions Between Quantum Strings

The next challenge was to show that the particles were genuinely connected.

“One of the simplest experiments was expanding strings,” says Milena Horvath, one of the first authors.

The researchers first trapped the atoms inside a one-dimensional tube and allowed them to expand. During this process, the strings encountered and collided with one another, yet their combined structures survived.

“This is a remarkable feature of strings: they can collide without falling apart,” says Horvath.

The scientists then carried out a second experiment by removing the confinement and allowing the atoms to expand freely through three-dimensional space. Because Bethe strings can exist only in one dimension, releasing the atoms into three dimensions caused the bound states to break apart.

The energy holding the particles together was converted into motion, causing the atoms to disperse more quickly. Comparing the two expansion methods gave the researchers a clear way to identify Bethe strings.

In the unbound state, both expansion measurements produced essentially the same energy as the system with repulsive interactions. When Bethe strings were present, however, the three-dimensional expansion released additional energy as the bound states broke apart.

A New Laboratory for Quantum Many-Body Physics

“The Bethe string was predicted almost a century ago as part of a beautiful mathematical description of quantum many-body systems,” said Sudipta Dhar, another first author. “Now we can create them in the lab, manipulate them, crash them, and study their amazing crash stability.”

Bethe strings have previously been detected experimentally in solid-state magnetic systems. The new study places these unusual quantum bound states in a very different setting: an ultracold atomic gas where researchers can control the system’s shape, particle density, and interactions with high precision.

“This opens new possibilities for studying how these collective quantum objects form and interact,” says lead theorist Alvis Bastianello.

The research was funded by the Austrian Science Fund FWF through a Wittgenstein Prize grant, the European Union through an ERC grant, and the UK Engineering and Physical Sciences Research Council. Milena Horvath is a member of the FWF PhD program “Atoms, Light and Molecules (DK-ALM).”