Nearly a century after they were predicted by Nobel laureate Hans Bethe, a team led by quantum physicist Hanns-Christoph Nägerl has observed so-called "Bethe strings" in ultracold atoms. The ultracold gas provides an ideal platform for further investigation of these quantum many-body states.
In 1931, physicist Hans Bethe predicted that in certain one-dimensional quantum systems par- ticles can bind together to form multi-particle states known as Bethe strings. Unlike ordinary molecules, which are held together by chemical bonds, Bethe strings arise purely from the interactions between particles and only exist in one dimension. For decades, Bethe strings remained primarily a theoretical concept. Now, almost a century after Bethe's prediction, researchers from the University of Innsbruck, in collaboration with theory teams from the Department of Experimental Physics at the University of Amsterdam and the Technical University of Munich, have created and observed these multi-particle-bound-states in an ultracold gas. Their findings are published in Nature Communications.
The experiment begins with a cloud of caesium atoms cooled to temperatures only a few bil- lionths of a degree above absolute zero. The researchers then divide the cloud into several thousand narrow, one-dimensional tubes. Inside these tubes, the atoms can move essentially only along a single direction. The interactions between the atoms can be precisely controlled. By tuning the in- teractions between the atoms from repulsive to attractive, they can make the atoms bind together. Instead of simply collapsing, the atoms form bound states of different sizes, including larger clusters containing six or more particles.
The researchers then asked a simple question: how can we tell that the particles are really bound together? "One of the simplest experiments was to let the strings expand," says Milena Horvath, one of the lead authors. First, the researchers allow the atoms to expand while keeping them confined to their one-dimensional tubes. As they move the strings collide, but remain intact. "This is a remarkable feature of the strings: they can collide without breaking apart" says Milena Horvath. In a second step, the researchers remove the confinement and let the atoms expand freely in three dimensions. Because Bethe strings can only exist in one dimension, this sudden change causes the bound states to break apart. The energy that held the particles together is converted into motion, causing the atoms to fly apart more rapidly. The difference between the two expansions provides a clear signature of the Bethe strings. For unbound particles, as in the repulsive interaction regime, the two measurements give essentially the same energy. When strings are present, however, the three-dimensional expansion carries additional energy released when the bound states break apart.
"Bethe strings were predicted almost a century ago as part of a beautiful mathematical description of quantum many-body systems," says Sudipta Dhar another lead author. "Now we can create them in the laboratory, manipulate them and make them collide and probe their remarkable collisional stability." Bethe strings have also been observed experimentally in solid-state magnetic systems. The present work brings these unusual quantum-bound-states into a different setting: an ultracold atomic gas, where the system geometry, density of particles and interactions can be controlled with exceptional precision. "This opens new possibilities for studying how these collective quantum objects form and interact" says lead theorist Alvise Bastianello.
The research has been funded by the Austrian Science Fund FWF through a Wittgenstein Prize grant, by the European Union through an ERC grant, and by the UK Engineering and Physical Sciences Research Council. Milena Horvath is a member of the FWF doctoral program Atoms, Light and Molecules (DK-ALM) .
Publication: Probing Bethe strings in an attractive one-dimensional Bose gas. Milena Horvath, Alvise Bastianello, Sudipta Dhar, Rebekka Koch, Yanliang Guo, Jean-Sébastien Caux, Manuele Landini, Hanns-Christoph Nägerl. Nat Commun (2026). DOI: 10.1038/s41467-026-76018-0
