When materials undergo major transformations—such as water boiling into steam or a permanent magnet losing its magnetic properties at high temperatures—the microscopic details of the physical system can vary drastically. Yet, physicists have long observed a remarkable phenomenon: very different physical systems can sometimes start behaving according to identical mathematical rules.
"Physicists call this trait universality—the messy, microscopic details wash out and only a few essential features survive," explains Jason Alicea, the William K. Davis Professor of Theoretical Physics at the California Institute of Technology (Caltech).
To describe this universal behavior, researchers traditionally rely on an advanced mathematical framework known as conformal field theory. For decades, this framework has allowed scientists to model critical points in physical systems on paper. However, validating these theoretical predictions through direct experimentation has remained an extraordinary challenge.
Now, in a landmark study published in the journal Nature, researchers have carried out the first experiments of their kind on two different conformal field theories using advanced quantum simulators. These specialized systems are simpler and more task-specific than general-purpose quantum computers, designed explicitly to reproduce particular quantum behaviors rather than run arbitrary algorithms.
The successful collaboration brought together the experimental physics group of Caltech’s Manuel Endres, Alicea’s theoretical physics group, and theoretical researchers at Université Paris-Saclay and the Technical University of Munich. Using newly developed quantum simulator technology, the international team made the first direct measurements of energy levels in synthetic quantum matter predicted by the Ising and tricritical Ising conformal field theories. The term "Ising" references Ernst Ising, a pioneering physicist who, back in the 1920s, solved an early mathematical model of magnetism.
Both of these theoretical models describe universal behavior that can emerge when a quantum system—possessing unusual properties such as quantum entanglement and superposition—reaches a critical tipping point between two distinct states. In these scenarios, one state is typically more ordered than the other, and the transition between them yields universal traits that transcend the specific material makeup of the system.
Unlike an ordinary phase transition like water becoming steam, this quantum transition is not driven by thermal energy or temperature changes. Instead, it arises entirely from subtle quantum effects operating at temperatures close to absolute zero. At this delicate critical point, specialized lasers can excite the quantum system into a sequence of specific, highly ordered energy states. The researchers naturally compare those quantized levels to the individual rungs of a ladder.
"The energy levels predicted by these theories are important because they encode profound information about the theories themselves," Alicea notes.
For roughly 40 years, theoretical physicists have used conformal field theories to calculate precisely how far apart those energy rungs ought to be. According to the math, these levels are expected to appear in exact, rigid ratios. But until this breakthrough experiment, those theoretical predictions had never been directly observed or measured in a physical laboratory setting.
"Our new tools borrow from quantum computing platforms," says Xiangkai Sun, a co-lead author of the new study and a graduate student working in the Endres lab. "Over the past 10 years, people have been learning to control these systems, and now we are at the point where we can use them to do fundamental physics research."
The experiment directly builds upon sophisticated hardware technology that the Endres lab also utilizes for developing scalable quantum computers. The underlying system relies on complex arrays of neutral atoms held stably in place by tightly focused lasers known as optical tweezers. In fact, a related neutral atom platform developed in the same lab recently reached a major technological milestone by successfully trapping 6,100 individual atoms in a single synchronized array.
Although optical tweezer arrays were originally developed largely to advance the field of quantum computing, the researchers turned the technology inward to investigate a fundamental, long-standing question in theoretical physics.
To conduct the measurements, the team arranged individual strontium atoms in a clean, orderly line using optical tweezers. Additional lasers were then introduced to push the atoms into highly excited energy states known as Rydberg states. In these unique states, neighboring atoms interact with one another very strongly, creating robust correlations across the entire chain.
Those powerful interactions caused the linear chain of atoms to behave collectively as a unified quantum system rather than as a collection of separate, independent particles. The researchers then meticulously adjusted the laser controls until the entire atomic system reached the precise critical tipping point they aimed to study.
To detect the predicted energy levels that emerged at this threshold, the researchers developed an innovative experimental technique called many-body modulation spectroscopy.
Using this method, they gently disturbed the entire atomic chain by modulating the lasers at a specific frequency and then carefully measuring how strongly the atoms responded to the perturbation. By scanning continuously through a wide range of frequencies and looking for prominent peaks in the collective atomic response, the researchers could definitively identify and catalog the different energy levels.
The underlying physical intuition is remarkably similar to running a wet finger around the smooth rim of a wine glass. When the physical motion matches the natural resonant frequency of the glass, the object resonates loudly and produces a clear tone. At other frequencies, little to no response occurs.
"We repeated the experiment on chains of up to 35 atoms, and the rungs came out as predicted by the Ising conformal field theory: the spectra collapsed onto a single universal curve once rescaled for size," Sun explains. "We then tuned to the tricritical point and measured the lowest levels of its distinct spectrum, which came out in the different ratios theory predicts."
Because the scientists maintained individual control over each atom within the chain, they were able to perform high-resolution measurements that would be exceptionally difficult, if not impossible, to execute using conventional materials found in nature.
By classifying the quantum excitations according to their inherent symmetry, the team successfully exposed a second set of energy rungs that had remained entirely hidden during the initial, broader measurements. Furthermore, they altered the behavior of the atoms located at either end of the linear chain. Doing so successfully rearranged the entire energy ladder, producing distinct patterns that closely matched the theoretical predictions derived from the tricritical Ising theory.
"Even though we believed these theories to be true, it’s important to have an experimental realization, something you can poke and prod," Alicea says. "To see those predictions borne out is a beautiful thing."
Building upon this success, the researchers now plan to expand their experimental framework to larger, more complex quantum systems. Rather than restricting their atomic arrays to a simple one-dimensional line, they hope to study intricate grids and two-dimensional planes of atoms.
"In two dimensions, the conformal field theories are not as well understood, so this is an exciting opportunity," Sun says.
Expanding into higher dimensions could allow scientists to investigate complex quantum systems whose behavior cannot yet be calculated precisely by human intellect alone, potentially opening doors to exploring physical problems that remain far too difficult for classical computers to simulate.
"What excites me is that the technique doesn’t require knowing the answer in advance. Here we could check our measurements against exact predictions," Endres concludes. "The next step is to point this at systems where nobody knows the response of the system quantitatively—including regimes that classical computers can’t reach."
The research study, titled "Observation of conformal field theory spectra in a quantum simulator," was supported by funding from the US Department of Energy, including its Quantum Systems Accelerator and its Quantum Science Center; the National Science Foundation, including the Institute for Quantum Information and Matter at Caltech (IQIM); the Army Research Office; the Defense Advanced Research Projects Agency; the Air Force Office of Scientific Research; the Gordon and Betty Moore Foundation; and the Deutsche Forschungsgemeinschaft.
Additional Caltech co-authors include Yuan Le, Stephen Naus, Richard Bing-Shiun Tsai, and Lewis Picard, who is now with the Caltech-linked startup Oratomic. Other contributors include Sara Murciano of Université Paris-Saclay, who previously served as a postdoctoral scholar at Caltech, and Michael Knap of the Technical University of Munich and the Munich Center for Quantum Science and Technology.
