Nearly a century after Nobel laureate Hans Bethe first mapped out their theoretical existence on paper, physicists have successfully created and observed unusual quantum structures known as "Bethe strings" in a laboratory setting. The breakthrough experiment, spearheaded by quantum physicist Hanns-Christoph Nägerl and his team, provides researchers with a highly controllable, pristine platform to explore these rare quantum many-body states with unprecedented precision.
The roots of this discovery stretch back to 1931, when the pioneering physicist Hans Bethe proposed a mathematical description suggesting that particles within certain quantum systems restricted strictly to one dimension could join together into collective states. Today, these formations are called Bethe strings. These structures differ fundamentally from familiar molecules encountered in standard chemistry. Rather than being connected through traditional chemical bonds involving shared or transferred electrons, the particles in a Bethe string remain bound purely because of their quantum mechanical interactions with one another. Furthermore, these unique states can exist exclusively in a strictly one-dimensional environment.
For the vast majority of the past hundred years, Bethe strings remained largely a theoretical idea—elegant mathematics describing a fascinating corner of quantum mechanics, but notoriously difficult to realize in physical experiments. That landscape has now shifted thanks to a collaborative effort centered at the University of Innsbruck, where researchers created and detected these multiparticle bound states inside an ultracold gas. The experimental team worked in close partnership with theory groups from the Department of Experimental Physics at the University of Amsterdam and the Technical University of Munich. Their findings have been formally detailed and published in the scientific journal Nature Communications.
Creating Bethe Strings Near Absolute Zero
To produce these elusive and unusual quantum states, the research team at the University of Innsbruck had to engineer an environment where extreme quantum effects could dominate without thermal interference. They began with a dense cloud of cesium atoms, which they cooled down using sophisticated laser and evaporative cooling techniques until the cloud reached a temperature within only a few billionths of a degree above absolute zero. At this near-absolute-zero threshold, thermal motion is almost completely eradicated, allowing delicate quantum phenomena to emerge.
Once the cesium cloud was sufficiently cold, the researchers separated it into several thousand extremely narrow, parallel tubes using optical lattices—spatial patterns of light created by intersecting laser beams. Within each of these microscopic tubes, the atoms are effectively restricted in their movement, confined to slide back and forth along a single spatial dimension. This spatial constraint successfully engineered the precise one-dimensional environment required for Bethe strings to theoretically form and persist.
A crucial capability of this experimental setup is the researchers’ ability to tune the system dynamically. They can precisely and rapidly adjust how strongly the cesium atoms interact with one another by utilizing magnetic field controls known as Feshbach resonances. By changing those atomic interactions from repulsive—where atoms push away from each other—to attractive—where they pull toward one another—the team triggered a dramatic shift in the system. Rather than simply collapsing into a single, unstructured clump of matter, the atoms organized themselves into stable bound states of several distinct sizes. Some of these newly formed clusters grew quite large, containing six or more particles bound together in the string configuration.
Watching Quantum Strings Collide
With the bound states successfully created inside their microscopic tubes, the next major challenge for the experimental team was proving definitively that these particles were truly bound together rather than merely passing close to one another by coincidence.
"One of the simplest experiments was to let the strings expand," explains Milena Horvath, one of the lead authors of the study, describing the methodology used to interrogate the fragile quantum structures.
To execute this, the researchers first allowed the confined atoms to spread out longitudinally while keeping them strictly trapped inside their one-dimensional tubes. During this controlled expansion, the individual Bethe strings encountered one another and underwent direct collisions. Remarkably, despite the violent nature of these subatomic impacts at such low temperatures, the bound structures survived intact.
"This is a remarkable feature of the strings: they can collide without breaking apart," notes Milena Horvath, highlighting the unexpected robustness of the collective states.
Following the confinement tests, the scientists performed a second version of the experiment designed to act as a definitive diagnostic test. In this phase, they abruptly removed the optical confinement altogether, allowing the cesium atoms to expand freely outward through ordinary three-dimensional space. Because the foundational laws governing Bethe strings dictate that they can only exist and remain stable in one dimension, releasing the atoms instantly into three dimensions caused the bound states to fall apart.
The internal energy that had previously kept the constituent particles bound together was suddenly transformed into kinetic energy, forcing the atoms to spread apart faster in open space. By carefully comparing the expansion data from the two types of experiments—one where the particles remained strictly one-dimensional and one where the dimensional constraints were removed—the researchers gained a clear, unambiguous method to identify the presence of Bethe strings.
When the particles were unbound, which was the case when the interatomic interactions were set to be repulsive, both expansion measurements yielded essentially the same energy profile. When Bethe strings were present under attractive interactions, however, the three-dimensional expansion data revealed distinct extra energy. This surplus energy was directly released as the fragile bound states were torn apart by the transition into three dimensions.
A New Laboratory for Quantum Many-Body Physics
The successful observation of these states marks a major milestone for modern physics, bridging a nearly century-old gap between abstract theory and empirical reality.
"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 of the research. "Now we can create them in the laboratory, manipulate them and make them collide and probe their remarkable collisional stability."
While Bethe strings have previously been detected experimentally in certain solid-state magnetic systems, this new body of work places these unusual quantum bound states into a completely different physical environment: an ultracold atomic gas. This gaseous platform offers advantages that solid-state materials cannot match, granting researchers the ability to control the system’s geometry, particle density, and interatomic interactions with exceptional precision and clarity.
"This opens new possibilities for studying how these collective quantum objects form and interact," adds lead theorist Alvise Bastianello, pointing toward future avenues of exploration in the field of quantum many-body physics.
The collaborative research undertaking received substantial financial backing from several prominent scientific funding bodies. Support was provided by the Austrian Science Fund FWF through a prestigious Wittgenstein Prize grant, by the European Union through an advanced ERC grant, and by the UK Engineering and Physical Sciences Research Council. Additionally, lead author Milena Horvath is a participating member of the FWF doctoral program known as Atoms, Light and Molecules, or DK-ALM, which supports emerging researchers in the field of atomic and quantum physics.
