Inside each tube, atoms effectively move along a single direction, recreating the strictly one-dimensional setting Bethe's model requires. A tunable knob let the team shift the atoms' interactions from repulsive to attractive.

That change bound the atoms. Instead of collapsing into one lump, they formed bound states of multiple sizes. Some clusters held six or more particles.

How do you prove the clusters are really bound?

"One of the simplest experiments was to let the strings expand," says Milena Horvath, one of the lead authors.

In the first test, the team let the atoms spread out while still trapped inside their one-dimensional tubes. The strings ran into one another and collided. The bound structures stayed intact.

"This is a remarkable feature of the strings: they can collide without breaking apart," Horvath says.

For the second test, the researchers released the atoms into free three-dimensional space. Bethe strings can only survive in one dimension, so removing the tubes broke the bound states apart. The binding energy converted into motion, and the atoms flew apart faster.

When the atoms were unbound — the repulsive case — both expansion modes gave the same energy reading. When Bethe strings were present, the three-dimensional expansion carried extra energy released as the strings collapsed.

Why does a controllable platform matter?

"Bethe strings were predicted almost a century ago as part of a beautiful mathematical description of quantum many-body systems," says lead author Sudipta Dhar. "Now we can create them in the laboratory, manipulate them, make them collide, and probe their remarkable collisional stability."

In solid-state magnets, physicists see only indirect fingerprints of the strings. In an ultracold atomic gas, they can instead adjust the tube geometry, the particle density, and the interaction strength with fine control.

"This opens new possibilities for studying how these collective quantum objects form and interact," says Alvise Bastianello, the project's lead theorist.

Bastianello worked with theory groups at the University of Amsterdam and the Technical University of Munich. Funding came from the Austrian Science Fund FWF through a Wittgenstein Prize grant, plus a European Research Council grant and the UK Engineering and Physical Sciences Research Council.

What's next?

The Innsbruck setup now lets physicists test predictions about how these multi-particle states form, collide, and decay in regimes theory has long described but experiment could not reach. Researchers can tune tube size, atom count, and interaction strength to map where Bethe strings hold together — and where they fall apart.