When physical systems undergo radical transformations—such as water flashing into steam or a magnet losing its ferromagnetic properties due to thermal agitation—the chaotic, microscopic intricacies often wash away, leaving behind a profound mathematical harmony. In these moments, vastly different physical substances begin to behave according to identical mathematical governance. Physicists refer to this overarching phenomenon as universality, a foundational concept in statistical mechanics and condensed matter physics that suggests nature relies on a surprisingly sparse set of rules when systems reach critical thresholds.
For decades, theoretical physicists have modeled this universal behavior using a sophisticated mathematical framework known as conformal field theory (CFT). While these equations have successfully described everything from particle physics to critical phase transitions on paper, putting them to a direct experimental test has remained an elusive goal. Now, in a landmark study published in the prestigious journal Nature, an international collaboration of researchers has successfully carried out the first direct measurements of energy levels in synthetic quantum matter predicted by specific conformal field theories. Utilizing advanced quantum simulators—specialized hardware designed specifically to mimic quantum mechanics rather than perform general-purpose computations—the team has bridged a decades-old gap between theoretical predictions and empirical reality.
The breakthrough collaboration brought together the experimental physics group led by Manuel Endres at the California Institute of Technology (Caltech), the theoretical physics group of Caltech’s Jason Alicea, and collaborating theorists at Université Paris-Saclay and the Technical University of Munich. By harnessing neutral-atom quantum simulator technology, the research team successfully observed and measured the exact energy spectra predicted by the Ising and tricritical Ising conformal field theories, named in honor of Ernst Ising, the German physicist who pioneered foundational mathematical models of magnetism in the 1920s.
The Historical Evolution of Quantum Simulations and CFTs
The journey toward this milestone spans nearly a century of theoretical physics developments paired with rapid hardware advancements over the last decade. In the 1920s, Ernst Ising introduced his theoretical model to explain phase transitions in magnets. While his initial one-dimensional model was solved analytically, scaling such concepts to complex quantum systems involving entanglement and superposition quickly outgrew the capabilities of classical mathematics.
By the 1980s, theoretical physicists had begun heavily relying on conformal field theories to calculate the structural properties of systems at critical points. Central to these calculations are energy levels that theoretically dictate how a quantum system behaves when disturbed. Physicists realized that these energy rungs should appear in exceptionally precise mathematical ratios. However, for roughly forty years, these predictions remained an untestable abstraction in many regimes because conventional materials did not offer the precise control required to isolate them.
The turning point arrived with the rapid development of programmable quantum platforms over the past ten years. Laboratories worldwide began mastering the art of manipulating isolated quantum particles using electromagnetic fields and laser arrays. While most of these engineering efforts were channeled into building general-purpose quantum computers for cryptographic and optimization tasks, visionary research groups realized that these exact platforms could serve as pristine analog simulators. Instead of processing bits or qubits for algorithmic computations, these systems could recreate complex quantum Hamiltonians from the ground up, allowing scientists to study fundamental physics in a controlled laboratory environment.
Trapping Atoms with Lasers: The Experimental Methodology
The experimental framework deployed by the Endres lab relies on advanced neutral-atom technology, utilizing arrays of optical tweezers—tightly focused, high-intensity laser beams capable of grasping and holding individual atoms with microscopic precision. This specific platform has demonstrated remarkable scalability; recently, a related neutral-atom setup in the Endres laboratory achieved a historic milestone by trapping an unprecedented 6,100 individual atoms within a single cohesive array.
For this specific study on conformal field theories, the researchers arranged a clean, linear chain of strontium atoms using the optical tweezer grid. Once the atoms were locked into position, secondary lasers were introduced to excite the strontium particles into highly excited energy states known as Rydberg states. In these unique states, the electronic orbitals of the atoms expand massively, causing neighboring atoms to exert exceptionally strong van der Waals-like interactions upon one another.
These powerful, localized interactions forced the individual atoms to abandon their independent behaviors, compelling the entire chain to act as a unified, collective quantum entity. By carefully tuning the parameters of the controlling lasers, the experimentalists could drive the entire atomic assembly precisely toward a quantum tipping point—a critical juncture separating two distinct quantum phases, one more ordered than the other. Unlike classical phase transitions driven by thermal energy, this quantum phase transition occurs near absolute zero, governed entirely by quantum fluctuations and Heisenberg uncertainty principles.
Measuring the Hidden Energy Ladder via Many-Body Modulation Spectroscopy
To probe the system and detect the theoretical energy levels predicted by the Ising conformal field theories, the Caltech-led team engineered a novel diagnostic protocol designated as many-body modulation spectroscopy.
In practice, the researchers gently and periodically perturbed the entire atomic chain by modulating the governing lasers at a specific frequency, subsequently recording how strongly and resiliently the atoms responded. By systematically scanning across a broad spectrum of frequencies and searching for pronounced resonance peaks in the atomic response, the team mapped out the complete energy spectrum of the system.
This detection mechanism is conceptually analogous to running a moistened finger along the rim of a crystal wine glass. When the motion matches the structural and physical dimensions of the glass, it produces a clear, resonant tone; at any other frequency, the glass remains silent and unresponsive.
By applying this spectroscopic technique to chains containing up to 35 individual atoms, the researchers observed remarkable results. "The rungs came out as predicted by the Ising conformal field theory: the spectra collapsed onto a single universal curve once rescaled for size," noted Xiangkai Sun, a co-lead author of the study and a graduate student in the Endres lab. Following this initial success, the team adjusted their laser controls to target the tricritical Ising point, successfully measuring the lowest energy levels of its distinct spectrum, which manifested in the exact mathematical ratios anticipated by theory.
Furthermore, because the experimental setup allowed for individual, site-resolved control over every single atom in the array, the team performed complex measurements that would be virtually impossible in conventional solid-state materials. By classifying excitations based on their inherent spatial symmetries, the researchers exposed a secondary set of energy rungs that had remained completely hidden during initial, less granular observations. They also manipulated the boundary conditions of the atomic chain by altering the behavior of the atoms situated at either extreme end, demonstrating that such modifications successfully rearranged the energy ladder in precise accordance with tricritical Ising predictions.
Official Responses and Perspectives from the Research Team
The successful validation of these four-decade-old theoretical frameworks has generated considerable enthusiasm within the global physics community, affirming both the reliability of conformal field theories and the viability of quantum simulators as instruments for discovery.
"Even though we believed these theories to be true, it’s important to have an experimental realization, something you can poke and prod," remarked Jason Alicea, the William K. Davis Professor of Theoretical Physics at Caltech and a co-author of the study. "To see those predictions borne out is a beautiful thing."
The ability to directly interact with mathematical predictions that were previously untestable provides experimentalists with an unprecedented feedback loop. Rather than relying solely on abstract calculations or indirect inferences, researchers can now physically interrogate quantum states that exhibit universal properties.
"What excites me is that the technique doesn’t require knowing the answer in advance," added Manuel Endres, professor of physics at Caltech. "Here we could check our measurements against exact predictions. 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."
Broader Impact, Future Implications, and Funding
The successful observation of conformal field theory spectra marks a definitive turning point for quantum simulation, transitioning the field from a phase of hardware benchmarking to an era of genuine scientific discovery. Having successfully demonstrated the technique on one-dimensional chains of up to 35 atoms, the research collaboration is already setting its sights on higher-dimensional systems.
Looking forward, the team aims to expand their optical tweezer arrays from linear chains into two-dimensional grids and lattices. Exploring conformal field theories in two spatial dimensions introduces significantly richer mathematical complexity, as these higher-dimensional CFTs are considerably less understood than their one-dimensional counterparts. Such geometries could grant scientists direct empirical access to complex quantum states and phase transitions whose behaviors utterly defy analytical calculation on classical computing architectures.
By deploying quantum simulators into uncharted regimes where classical algorithms fail—such as highly entangled, frustrated magnetic systems or non-equilibrium quantum matter—researchers hope to decode the deepest mysteries of quantum mechanics. These advancements may eventually inform the design of novel quantum materials, shed light on high-temperature superconductivity, and expand our fundamental understanding of quantum field theories that govern everything from tabletop condensed matter systems to cosmological phenomena.
The published study, titled "Observation of conformal field theory spectra in a quantum simulator," received extensive financial backing from major scientific funding bodies. Support was provided by the U.S. Department of Energy through its Quantum Systems Accelerator and Quantum Science Center initiatives; the National Science Foundation, including the Institute for Quantum Information and Matter (IQIM) at Caltech; the Army Research Office; the Defense Advanced Research Projects Agency (DARPA); the Air Force Office of Scientific Research; the Gordon and Betty Moore Foundation; and the Deutsche Forschungsgemeinschaft (German Research Foundation).
In addition to Alicea, Endres, and lead author Sun, the Caltech research team included Yuan Le, Stephen Naus, Richard Bing-Shiun Tsai, and Lewis Picard. Additional contributions to the theoretical and experimental design were provided by 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.
