Nearly one hundred years after Nobel laureate Hans Bethe first theorized their existence, scientists have successfully created and observed unusual quantum structures known as "Bethe strings" by utilizing ultracold atoms. Led by prominent quantum physicist Hanns-Christoph Nägerl, this groundbreaking experiment offers researchers an unprecedented, highly controllable platform to explore these complex quantum many-body states in a laboratory setting.
Back in 1931, physicist Hans Bethe proposed a mathematical description suggesting that particles within specific quantum systems restricted entirely to one dimension could join together into distinct collective states. These unique structures are now referred to as Bethe strings. They differ fundamentally from the familiar molecules that make up the everyday physical world. Rather than being connected through traditional chemical bonds, the particles in a Bethe string remain bound purely because of their quantum interactions with one another. Furthermore, the resulting states are strictly confined to a one-dimensional existence, unable to maintain their structural integrity if expanded into higher dimensions.
For much of the past century, Bethe strings remained largely a theoretical concept, fascinating generations of physicists without yielding to direct laboratory observation in this specific context. Now, an international team of researchers has changed that landscape. Led by experimental groups at the University of Innsbruck, the team successfully created and detected these multiparticle bound states within an ultracold gas. They worked in close collaboration with theory teams from the Department of Experimental Physics at the University of Amsterdam and the Technical University of Munich. Their findings, which mark a significant milestone in modern quantum physics, were recently published in the prestigious journal Nature Communications.
Creating Bethe Strings Near Absolute Zero
To produce these elusive quantum states, the research team began with a dense cloud of cesium atoms. They subjected this cloud to advanced cooling techniques, bringing the temperature down to within just a few billionths of a degree above absolute zero—the theoretical temperature at which molecular motion halts. Once the atoms were sufficiently chilled, the team used specialized laser setups to separate the cloud into several thousand extremely narrow, parallel tubes.
Within the confines of each individual tube, the cesium atoms are effectively restricted to moving along a single spatial direction. This strict confinement creates the exact one-dimensional environment required for Bethe strings to form and persist. Moreover, the experimental setup grants the researchers the ability to precisely adjust how strongly the atoms interact with one another by tuning external magnetic fields.
By shifting those atomic interactions dynamically from repulsive—where the atoms push away from each other—to attractive, the research team successfully coaxed the atoms into binding together. Rather than simply collapsing into a single, undifferentiated clump, the atoms formed organized bound states comprising several different sizes. Some of the larger clusters observed during the experiment contained six or more particles operating in unison.
Watching Quantum Strings Collide
Once the bound states were successfully formed, the next major challenge for the experimental team was demonstrating unequivocally that these particles were truly bound together rather than just temporarily clustered.
"One of the simplest experiments was to let the strings expand," explains Milena Horvath, one of the lead authors of the study.
To accomplish this, the researchers first allowed the atoms to spread out while maintaining their strict confinement inside the one-dimensional tubes. During this controlled expansion, the individual strings inevitably encountered one another and collided. To the delight of the research team, the delicate bound structures survived these energetic encounters intact.
"This is a remarkable feature of the strings: they can collide without breaking apart," notes Milena Horvath, emphasizing the robustness of the quantum states despite their unusual nature.
To confirm the true nature of the binding, the scientists then performed a second, contrasting version of the experiment. In this iteration, they removed the spatial confinement entirely, allowing the atoms to expand freely outward through three-dimensional space. Because Bethe strings inherently require a one-dimensional environment to exist, releasing the atoms from their tubular constraints caused the bound states to instantly fall apart.
As the structures dissolved, the internal energy that had previously kept the particles bound together was suddenly transformed into kinetic motion, causing the individual atoms to spread apart at a noticeably faster rate. Comparing the data from these two distinct types of expansion provided the researchers with a clear, unambiguous method to identify and verify the presence of Bethe strings.
When the particles were unbound—such as during phases when the interactions were set to be repulsive—both expansion measurements yielded essentially the same energy profile. However, when Bethe strings were actively present in the system, the subsequent three-dimensional expansion contained a measurable amount of extra energy, which was released precisely as the bound states disintegrated.
A New Laboratory for Quantum Many-Body Physics
"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 work places the unusual quantum bound states into an entirely different and vastly more versatile environment: an ultracold atomic gas. In this gaseous medium, researchers retain the exceptional ability to control the system’s underlying geometry, particle density, and atomic interactions with remarkable precision.
"This opens new possibilities for studying how these collective quantum objects form and interact," adds lead theorist Alvise Bastianello, highlighting the broader implications for the field of quantum mechanics.
Funding for this extensive research project was provided by several prominent scientific organizations. These include the Austrian Science Fund FWF through a prestigious Wittgenstein Prize grant, the European Union via a dedicated European Research Council (ERC) grant, and the UK Engineering and Physical Sciences Research Council. Additionally, lead author Milena Horvath is an active member of the FWF-sponsored doctoral program known as Atoms, Light and Molecules, designated as DK-ALM.
