Ultracold Atoms Form a Quantum “Necklace,” Offering New Evidence for Bethe Strings

A necklace-like chain of ultracold atoms has provided new evidence supporting a nearly 100-year-old prediction in quantum physics.

In 1931, physicist Hans Bethe studied a line of interacting electrons whose spins could point up or down. His calculations showed that groups of reversed spins could bind together and travel along the line as a single unit.

Decades later, physicists found that Bethe’s mathematical method could also describe larger particles moving along a line. When these particles attract one another, the theory predicts that they form clusters known as Bethe strings.

Now, researchers have found evidence for these clusters in a gas of ultracold atoms. Their findings, published in Nature Communications, could help scientists better understand how large quantum systems behave when many particles interact.

What are Bethe strings?

Bethe strings are clusters of particles that form when the particles attract one another in a one-dimensional system. Although the particles remain separate, they behave mathematically as a bound group.

Researchers have previously observed tiny Bethe strings in magnetic crystals and superconducting quantum circuits. However, those experiments captured individual strings one at a time.

Theory predicts that densely packed strings of different sizes could change the behavior of an entire quantum system. Studying that possibility requires creating many strings at once, something that has been difficult to achieve with atoms.

Atoms that attract one another usually collapse into clumps. In the new experiment, however, the researchers appear to have created multiple strings of different sizes, including strings containing at least six atoms.

How the atomic necklace was created

Creating an atomic necklace is difficult because attractive atoms tend to collapse in three dimensions. As the atoms clump together and form molecules, the process releases energy. That energy can force atoms out of the experiment, causing the gas to disappear quickly.

In a one-dimensional system, the atoms behave differently. “They’re actually like pearls on a string, right? Like a necklace, you can’t pass them through,” study co-author Hans-Christoph Nägerl, an experimental physicist at the University of Innsbruck, told Live Science.

“Everything collapses in 3D. But in 1D it doesn’t collapse,” Nägerl said.

The team began by cooling tens of thousands of cesium atoms. At temperatures near absolute zero, the atoms formed a Bose-Einstein condensate — a state of matter in which atoms share a single quantum state.

A laser beam then divided the condensate into thousands of thin tubes. Each tube contained dozens of atoms arranged in a line.

At first, the atoms repelled one another. The researchers used a magnetic field to gradually reduce the repulsive force to zero before turning it into an attractive force.

Measuring strings the researchers could not see

The team could not observe the Bethe strings directly. Instead, the researchers released the atoms and measured how quickly the resulting cloud expanded.

When the atoms were allowed to expand only along their tube, the strings remained intact. When the atoms were released in all three dimensions, the strings broke apart and released their binding energy as motion.

Three-dimensional clouds spread with roughly three times more energy than one-dimensional clouds. By comparing the expansion patterns, the researchers estimated how tightly the atoms were bound together.

The results indicated the presence of a string containing at least six atoms. The measurements also agreed well with predictions from a theory called generalized fluid mechanics.

This approach describes gases as fluids made up of different types of particles, including strings of different sizes. Rather than tracking every atom individually, it models how each type of particle flows over time.

Nägerl emphasized that the experiment does not prove Bethe strings exist in the system. “The fact that a theory fits the data is not evidence that something exists,” he said. The findings provide additional evidence supporting the theory.

A short-lived quantum state

The atomic strings do not last indefinitely. Although the atoms are constrained to a line during the experiment, they still experience their three-dimensional environment. Eventually, the strings collapse into ordinary cesium molecules.

In the coldest experiment, the team lost about 40% of its atoms.

“We create these Bethe string states temporarily,” Nägerl said. “But I don’t think they will live long.”

Future experiments could make the strings more stable. Physicists have recently learned how to use microwaves to shield ultracold molecules from close-range collisions. In principle, strings made from shielded molecules could last long enough to be studied directly, although that idea has not yet been tested.

Another possibility is a quantum gas microscope, which can image atoms one by one. That could provide more direct evidence of individual Bethe strings.

Why Bethe strings matter

For Nägerl, studying Bethe strings is part of a broader effort to understand how many-particle quantum systems behave and stabilize over time.

Everything from magnets to computer chips is made of enormous numbers of interacting particles. Because tracking every particle is so difficult, ultracold atoms provide a clean and tunable way to investigate complex quantum systems.

Nägerl’s long-term goal is an experiment that would allow physicists to control how individual particles move and how their spins flip. Such an approach could allow model materials to be assembled piece by piece, “like Lego bricks.”

The current study represents another step toward that goal. Although Bethe strings were not the team’s only focus, they cannot be ignored: whenever atoms attract one another in a one-dimensional system, these unusual quantum clusters can form.

Reference: Horvath, M., Bastianello, A., Dhar, S., Koch, R., Guo, Y., Caux, J.-S., Landini, M., and Nägerl, H.-C. (2026). “Explore Bethe strings with fascinating one-dimensional Bose gas.” Nature Communications, 17, 10330.