Physicists have observed “Bethe strings” in an ultracold gas nearly a century after Hans Bethe predicted them, providing a controlled setting for further study.
In 1931, physicist Hans Bethe predicted that particles in certain quantum systems could bind together into groups called Bethe strings. These groups could exist only in one dimension, where particles move along a line. Unlike ordinary molecules, which are held together by chemical bonds, Bethe strings would form purely through interactions between the particles.
Bethe strings remained primarily theoretical for decades before experiments detected them in solid-state magnetic systems. Researchers have now created and observed them in an ultracold gas, a setting that lets scientists adjust the conditions in which these groups form and interact.
The research, published in Nature Communications, brings together experimental researchers from the University of Innsbruck and theory teams from the Department of Experimental Physics at the University of Amsterdam and the Technical University of Munich.
“Bethe strings were predicted almost a century ago as part of a beautiful mathematical description of quantum many-body systems,” says Sudipta Dhar, one of the lead authors. “Now we can create them in the laboratory, manipulate them and make them collide and probe their remarkable collisional stability.”
Ultracold atoms bind into Bethe strings
To create the one-dimensional conditions the strings need, the team divided a cloud of cesium atoms, cooled to just a few billionths of a degree above absolute zero, among several thousand narrow tubes. Each tube restricted the atoms’ movement essentially to a single direction along its length.
The researchers then switched the interactions between the atoms from repulsive, which pushed them apart, to attractive, which drew them together. Rather than simply collapsing, the atoms formed bound groups of different sizes, including clusters containing six or more particles.
Along with these interactions, the gas experiment allows researchers to precisely control the system’s geometry and the density of its particles. “This opens new possibilities for studying how these collective quantum objects form and interact,” says lead theorist Alvise Bastianello.
Bethe strings break in three dimensions
To establish that the atoms had actually bound together, the team compared how they moved when confined to their tubes with how they moved after that confinement was removed.
“One of the simplest experiments was to let the strings expand,” says Milena Horvath, another lead author.
The researchers first allowed the atoms to spread out along the tubes, where the strings encountered one another as they moved. Those collisions tested whether the groups could stay intact when they met. “This is a remarkable feature of the strings: they can collide without breaking apart,” says Horvath.
When the researchers removed the confinement, the atoms could move freely in three dimensions, where Bethe strings cannot exist. The bound groups broke apart, converting the energy that had held their particles together into motion and causing the atoms to fly apart more rapidly.
For unbound atoms with repulsive interactions, expansion along the tubes and expansion in three dimensions yielded essentially the same energy. When Bethe strings were present, however, the three-dimensional expansion carried additional energy released by the clusters’ breakup, providing a clear signature that the particles had been bound together.
Reference: “Probing Bethe strings in an attractive one-dimensional Bose gas” by Milena Horvath, Alvise Bastianello, Sudipta Dhar, Rebekka Koch, Yanliang Guo, Jean-Sébastien Caux, Manuele Landini and Hanns-Christoph Nägerl, 29 August 2026, Nature Communications.
DOI: 10.1038/s41467-026-76018-0
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.
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