In a landmark achievement for the field of condensed matter physics, a multi-institutional research team led by the California Institute of Technology (Caltech) has successfully utilized quantum simulators to provide the first direct experimental verification of energy levels predicted by conformal field theories (CFT). The study, published in the journal Nature, marks a significant bridge between abstract mathematical frameworks and physical reality, confirming predictions that have remained untested for nearly four decades. By manipulating individual strontium atoms with high-precision lasers, the researchers observed the "universal" behavior of quantum systems as they reached a critical tipping point, revealing a hidden "ladder" of energy states that governs the transition between different phases of matter.
The Principle of Universality and Conformal Field Theory
To understand the magnitude of this discovery, one must first grasp the concept of universality. In the physical world, materials are composed of a chaotic array of microscopic components—atoms, electrons, and molecules—each interacting in complex ways. However, when a system undergoes a phase transition, such as water boiling into steam or a piece of iron losing its magnetic properties, these "messy" details often disappear. What remains are a few essential mathematical features that are identical across vastly different systems.
"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 Caltech. This phenomenon allows scientists to use the same set of equations to describe the behavior of a magnet and the behavior of a fluid near its critical point.
The mathematical framework used to describe this universal behavior is known as conformal field theory. Developed largely in the 1970s and 1980s, CFT provides a rigorous way to calculate the properties of systems at their critical points. One of its most famous predictions involves the existence of specific energy levels, or "rungs" on an energy ladder, which should appear in precise, predictable ratios. While these ratios have been a cornerstone of theoretical physics for forty years, they had never been measured directly in a laboratory setting until now.
Quantum Simulators: A Specialized Tool for Fundamental Physics
The breakthrough was made possible by the rapid advancement of quantum simulators. Unlike general-purpose quantum computers, which are designed to perform a wide variety of computational tasks using quantum bits (qubits), quantum simulators are specialized systems. They are "bespoke" machines engineered to mimic specific quantum phenomena that are too complex for classical computers to model accurately.
The experimental setup was led by Manuel Endres, a professor of physics at Caltech, whose lab specializes in neutral atom platforms. The team also included theorists from Alicea’s group at Caltech, as well as collaborators from Université Paris-Saclay and the Technical University of Munich.
The researchers employed a technology known as optical tweezers—tightly focused beams of laser light—to trap and arrange strontium atoms in a precise one-dimensional line. By utilizing these tweezers, the scientists could control the position of every single atom in the chain, a level of precision that is impossible to achieve with conventional solid-state materials. This "bottom-up" approach to building matter allowed them to create a synthetic environment where quantum effects could be isolated and observed with unprecedented clarity.
The Chronology of the Experiment: From Absolute Zero to the Critical Point
The experiment began by cooling the strontium atoms to temperatures nearly reaching absolute zero. At these extreme temperatures, the thermal fluctuations that drive everyday phase transitions (like melting ice) are suppressed, allowing quantum mechanical effects—such as entanglement and superposition—to take center stage.
Once the atoms were positioned in a line, the researchers used additional lasers to push them into "Rydberg states." In a Rydberg state, an atom’s outermost electron is pushed very far from the nucleus, making the atom much larger and highly sensitive to its neighbors. These highly excited atoms interact strongly with one another, causing the entire chain to behave as a single, collective quantum system rather than a collection of individual particles.
The next phase involved carefully tuning the laser parameters to bring the system to its "critical point." This is a delicate tipping point between two different phases of matter—one where the atoms are more ordered and another where they are more disordered. It is at this precise juncture that the laws of conformal field theory take over and the universal energy ladder is expected to emerge.
Measuring the Hidden Energy Ladder: Many-Body Modulation Spectroscopy
To detect the predicted energy levels, the team developed a novel technique called "many-body modulation spectroscopy." Xiangkai Sun, a co-lead author of the study and a graduate student in the Endres lab, compares the process to a classic acoustic experiment.
"The basic idea is similar to running a wet finger around the rim of a wine glass," Sun explains. "When the motion matches the glass’s natural frequency, the glass resonates and produces a sound. At other frequencies, little happens."
In the quantum simulator, the researchers "disturbed" the atomic chain by modulating the intensity of the lasers at specific frequencies. They then measured how the system responded. By scanning through a range of frequencies and identifying the peaks in the response, they were able to map out the energy levels of the system.
The results were a near-perfect match for the Ising and tricritical Ising conformal field theories. The Ising model, named after physicist Ernst Ising who solved a foundational model of magnetism in the 1920s, predicts a specific set of energy ratios. When the researchers rescaled their data for the size of the atomic chain (testing up to 35 atoms), the energy spectra collapsed onto a single universal curve, exactly as the theory predicted.
Historical Significance and Theoretical Validation
The validation of these theories is a major milestone for the physics community. For decades, CFT has been used as a "gold standard" in theoretical physics, influencing everything from string theory to the study of high-temperature superconductors. However, because it describes the behavior of systems at the limit of infinite size and perfect criticality, finding a real-world material that fits the model perfectly has been a challenge.
"Even though we believed these theories to be true, it’s important to have an experimental realization, something you can poke and prod," says Alicea. "To see those predictions borne out is a beautiful thing."
The team’s ability to tune the system to the "tricritical point"—a more complex state where three different phases of matter meet—was particularly significant. By changing the boundary conditions at the ends of the atomic chain, they were able to observe how the energy ladder rearranged itself, providing a secondary layer of confirmation for the tricritical Ising theory.
Implications for the Future of Quantum Computing and Materials Science
While the primary goal of this research was to answer fundamental questions about the nature of matter, the implications extend far into the realm of technology. The tools developed for this experiment—such as the high-precision control of Rydberg atoms and many-body spectroscopy—are the same tools required to build more powerful and error-resistant quantum computers.
A related platform in the Endres lab recently achieved a milestone by trapping 6,100 atoms in a single array, demonstrating the scalability of this neutral-atom approach. As these systems grow larger, they will allow scientists to explore regimes that are completely inaccessible to even the most powerful classical supercomputers.
The researchers are already looking toward the next frontier: two-dimensional systems. "In two dimensions, the conformal field theories are not as well understood, so this is an exciting opportunity," says Sun. Moving from a one-dimensional chain to a two-dimensional grid increases the complexity of the interactions exponentially, potentially revealing new types of quantum behavior that have not yet been theorized.
Conclusion: Entering the Era of Discovery
The successful measurement of CFT spectra represents a transition in the field of quantum simulation. We are moving from an era where simulators are used to "check" known answers to an era where they will be used to discover new ones.
"What excites me is that the technique doesn’t require knowing the answer in advance," says Manuel Endres. "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."
By bridging the gap between high-level mathematics and experimental physics, this research provides a roadmap for future investigations into the "quantum advantage." As scientists continue to "poke and prod" these synthetic systems, they are likely to uncover the fundamental rules that govern not just magnets and boiling water, but the very fabric of quantum reality.
Funding and Collaboration Acknowledgments:
The study was supported by a diverse array of institutions, reflecting the global and interdisciplinary nature of the research. Funding was provided by the U.S. Department of Energy (including the Quantum Systems Accelerator and the Quantum Science Center), the National Science Foundation, the Army Research Office, DARPA, the Air Force Office of Scientific Research, and the Gordon and Betty Moore Foundation. International support came from the Deutsche Forschungsgemeinschaft and the Munich Center for Quantum Science and Technology. Additional authors on the paper include researchers from Caltech, the Caltech-linked startup Oratomic, Université Paris-Saclay, and the Technical University of Munich.
