Hans Bethe Predicted in 1931 That Particles Could Bind Into 'Strings' Held Together by Nothing but Their Own Interactions. Innsbruck Just Made Them Out of Cesium Atoms and Watched Them Collide Without Breaking.

In 1931, a 25-year-old Hans Bethe wrote down an exact solution for a chain of interacting quantum spins, a piece of mathematics now known as the Bethe ansatz that would become one of the foundations of many-body physics. Buried in it was a prediction: in certain one-dimensional systems, particles could bind together into multi-particle states held together not by chemical bonds but purely by their mutual interactions, and these "Bethe strings" could exist only in one dimension. Ninety-five years later, physicists at the University of Innsbruck have built them out of cesium atoms, let them collide, and watched them survive.

The work, published in Nature Communications by a team led by Milena Horvath and Sudipta Dhar, with theory collaborators at the University of Amsterdam and the Technical University of Munich, begins with a cloud of cesium atoms cooled to a few billionths of a degree above absolute zero. Laser light then slices the cloud into several thousand narrow tubes, each so tightly confining that the atoms can move only along a single axis. Inside those tubes, the strength and sign of the interaction between atoms can be dialed with a magnetic field.

Turn the interaction repulsive and the atoms keep their distance. Turn it attractive and something unusual happens. Rather than collapsing into a dense clump, as attracting atoms would in three dimensions, the one-dimensional gas organizes itself into bound clusters of different sizes, including strings of six or more atoms. These are Bethe's strings: bound states that exist because the geometry forbids the atoms from escaping sideways.

Proving the atoms were really bound, rather than simply close together, required two experiments. "One of the simplest experiments was to let the strings expand," Horvath said. First the team released the atoms along the tubes while keeping the one-dimensional confinement in place. The strings moved, ran into each other and came out intact. "This is a remarkable feature of the strings: they can collide without breaking apart," she said. Then the researchers switched off the confinement entirely and let the gas expand in three dimensions. Since Bethe strings cannot exist outside one dimension, the bound states disintegrated, and the binding energy that had held them together turned into kinetic energy. The atoms flew apart faster than they otherwise would.

The difference between those two expansions is the fingerprint. For unbound atoms in the repulsive regime, the one-dimensional and three-dimensional expansions release essentially the same energy. When strings are present, the three-dimensional expansion carries extra energy, exactly the amount liberated when the strings break. The experiment measured that surplus directly.

Bethe strings had previously been glimpsed in solid-state magnets, where their signatures appear in neutron-scattering and terahertz spectra of certain spin-chain crystals. The Innsbruck platform is different in kind. "Bethe strings were predicted almost a century ago as part of a beautiful mathematical description of quantum many-body systems," Dhar said. "Now we can create them in the laboratory, manipulate them, make them collide and probe their remarkable collisional stability." In an atomic gas, the density, the geometry and the interaction strength are all knobs the experimenter controls.

Lead theorist Alvise Bastianello said the result "opens new possibilities for studying how these collective quantum objects form and interact." The deeper payoff is that one-dimensional systems with exact solutions are among the very few places where the full quantum many-body problem can be checked against theory without approximation. Bethe's 1931 mathematics said the strings had to be there. The cesium says he was right.