Nearly a century after Nobel laureate Hans Bethe first theorized their existence on paper, scientists have successfully created and observed unusual quantum structures known as "Bethe strings" in a laboratory setting. Utilizing ultracold atoms, an international team of researchers led by quantum physicist Hanns-Christoph Nägerl has opened up an entirely new and highly controllable pathway to explore these complex, multi-particle quantum states that have remained largely in the realm of theoretical physics for decades.
In 1931, the renowned physicist Hans Bethe proposed that particles within specific quantum systems—specifically those strictly limited to a single dimension—could join together into distinct collective states. Today, these are referred to as Bethe strings. These fascinating structures differ fundamentally from the familiar chemical molecules that make up our everyday world. Rather than being linked together through traditional chemical bonds, the particles in a Bethe string remain bound strictly because of their mutual quantum interactions with one another. Furthermore, the mathematical and physical conditions required for these states to exist dictate that they can only manifest in one dimension.
For the vast majority of the past hundred years, Bethe strings were treated primarily as a theoretical concept, serving as a brilliant mathematical solution for understanding complex quantum behavior without an accessible experimental counterpart. That barrier has now been broken by researchers at the University of Innsbruck, who successfully created and detected these elusive multi-particle bound states within an ultracold atomic gas. To accomplish this feat, the Innsbruck team worked in close collaboration with theoretical physics groups from the Department of Experimental Physics at the University of Amsterdam and the Technical University of Munich. Their groundbreaking results have now been published in the scientific journal Nature Communications.
Creating Bethe Strings Near Absolute Zero
To produce these unusual quantum states in a physical laboratory, the researchers had to push matter to its absolute limits. They started with a dense cloud of cesium atoms, which they cooled 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 extreme temperature, thermal motion is almost entirely eliminated, allowing quantum mechanical effects to dominate the behavior of the particles.
Once the cesium gas was sufficiently chilled, the experimental team separated the single cloud into several thousand extremely narrow, microscopic tubes using an optical lattice—a grid formed by intersecting laser beams. Within each of these isolated tubes, the cesium atoms are effectively restricted to moving along a single direction, successfully creating the strict one-dimensional environment that is fundamentally required for Bethe strings to form and persist. Moreover, the experimental setup grants the researchers the ability to precisely tune and adjust how strongly the atoms interact with one another in real time.
By dynamically changing these atomic interactions from repulsive to attractive, the research team successfully compelled the atoms to bind together. Rather than simply collapsing into a single indiscriminate clump or forming standard molecular pairs, the atoms organized themselves into bound states comprising several different sizes. In some of the larger clusters observed during the experiments, the bound structures contained six or more individual particles moving in a coordinated quantum harmony.
Watching Quantum Strings Collide
Once the Bethe strings were successfully created inside their microscopic tubes, the next major hurdle for the experimental team was definitively demonstrating that these particles were truly bound together rather than just passing near 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 and a doctoral researcher tied to the Austrian Science Fund FWF doctoral program Atoms, Light and Molecules.
To execute this, the researchers initially allowed the confined atoms to spread out longitudinally while keeping them strictly trapped inside their one-dimensional tubes. During this controlled expansion, the individual strings inevitably encountered one another and collided. To the surprise and delight of the research team, the bound structures proved remarkably resilient, surviving the impacts intact.
"This is a remarkable feature of the strings: they can collide without breaking apart," notes Horvath.
Following this initial test, the scientists performed a second version of the experiment designed to prove the unique dimensionality of the states. They abruptly removed the optical confinement entirely, allowing the cesium atoms to expand freely through three-dimensional space. Because fundamental physical laws dictate that Bethe strings can only exist in a one-dimensional environment, releasing the atoms into unrestricted three dimensions caused the delicate bound states to immediately fall apart.
During this dissolution, the internal binding energy that had previously kept the constituent particles locked together was suddenly transformed into kinetic motion, forcing the atoms to spread apart at a distinctly faster rate. By comparing the measurements and outcomes of these two distinct types of expansion, the researchers obtained a clear, unambiguous method to identify and verify the presence of Bethe strings.
When the particles were unbound—such as when the atomic interactions were tuned to be repulsive—both expansion measurements yielded essentially the same energy profile. However, when Bethe strings were present in the initial one-dimensional tubes, the subsequent three-dimensional expansion contained measurable extra energy that was released specifically as the bound states disintegrated into free particles.
A New Laboratory for Quantum Many-Body Physics
The successful observation of these structures marks a major milestone in modern physics, bridging a nearly century-old gap between abstract theory and empirical observation.
"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 study. "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 solid-state magnetic systems, this new research places the unusual quantum bound states into an entirely different and highly advantageous environment: an ultracold atomic gas. In this gaseous platform, researchers enjoy unprecedented experimental control over the system’s geometry, the density of the particles, and the strength of their quantum interactions with exceptional precision.
"This opens new possibilities for studying how these collective quantum objects form and interact," adds lead theorist Alvise Bastianello, highlighting the broader implications of the work for theoretical and experimental physics alike.
The foundational research was supported by significant international and national funding sources, including the Austrian Science Fund FWF through a prestigious Wittgenstein Prize grant, the European Union through an active ERC grant, and the UK Engineering and Physical Sciences Research Council. Milena Horvath continues her research as an active member of the FWF doctoral program Atoms, Light and Molecules, contributing to a growing body of work that continues to unpack the deepest mysteries of quantum many-body physics.
