Experimental observation of conformal field theory spectra

This study experimentally observes the universal energy excitation spectra of emergent Ising and tricritical Ising conformal field theories in a Rydberg atom quantum simulator, utilizing advanced modulation and local control techniques to recover characteristic energy ratios and probe dynamical correlations.

The Gist

Imagine you have a giant, complex orchestra made of tiny atoms. Each atom is an instrument, and they are all playing together to create a single, massive symphony. In the world of physics, when these atoms are tuned to a very specific, critical moment—like the exact second ice turns to water—they start behaving in a way that follows a hidden, universal rulebook called a Conformal Field Theory (CFT).

Think of a CFT as the "sheet music" for the universe at its most chaotic and critical moments. It predicts exactly how the notes (energy levels) should sound and how they relate to each other, regardless of the size of the orchestra. For decades, physicists have known this sheet music exists on paper, but they've never been able to hear the actual music played by a real quantum system.

The Experiment: Tuning the Quantum Orchestra
In this paper, a team of researchers at Caltech and other institutions used a programmable quantum simulator. Imagine this as a digital conductor that can arrange 19 to 35 individual atoms (specifically, Rydberg atoms) in a line and control them with lasers.

They didn't just listen to the music; they built a special tool to "tap" the orchestra and see how it responded. This tool is called modulation spectroscopy.

• The Analogy: Imagine tapping a bell. If you tap it at just the right rhythm, it rings loudly. If you tap at the wrong rhythm, it stays quiet. The researchers "tapped" their line of atoms with laser pulses at different frequencies.
• The Result: When the tap frequency matched the natural energy of an excited state in the atoms, the atoms "sang back" (absorbed energy). By measuring which frequencies made the atoms sing, they mapped out the entire energy spectrum.

What They Found: The Universal Ratios
The most exciting part is that the "notes" the atoms played matched the theoretical sheet music perfectly.

• The Ising CFT: In one setup, the atoms behaved like a specific type of theory called the "Ising" model. The theory predicted that the energy levels should follow a simple ratio: 2:4:6:8. When the researchers measured the energy of the excited atoms, they found exactly those ratios. It was like hearing a chord where the notes were perfectly spaced, confirming the theory.
• The Tricritical Ising (TCI) CFT: They then tweaked the setup to reach a more complex state called the "Tricritical" point. Here, the "sheet music" changes. The theory predicted a different set of ratios (like 4/3 or 10/3). By adjusting the "boundary conditions" (essentially changing how the ends of the atom chain were held or pinned), they could switch between different versions of this music. They successfully measured these new, fractional ratios, proving that the atoms were indeed following the complex rules of the TCI theory.

Why This Matters
Before this, these specific energy patterns were only mathematical predictions. You couldn't see them in solid materials because the signals were too messy or the systems too hard to control.

• The Breakthrough: This experiment is the first time scientists have directly "heard" these universal energy patterns in a controlled quantum system.
• The Tool: They developed a new way to "diagnose" these systems. Even if you don't know what kind of physics is happening in a new material, you can use this modulation technique to listen to its energy spectrum and figure out which "universal rulebook" (universality class) it is following.

In Summary
The researchers built a line of atoms, tuned them to a critical point, and used laser "taps" to listen to their energy levels. They discovered that the atoms sang in perfect harmony with the complex mathematical predictions of Conformal Field Theory, revealing the hidden, universal "sheet music" that governs how matter behaves at the edge of phase transitions. They didn't just confirm the theory; they gave us a new microphone to listen to the quantum world.

Technical Summary

Problem and Motivation
Conformal field theories (CFTs) are fundamental frameworks in high-energy physics, statistical mechanics, and condensed matter physics, governing the universal properties of systems at quantum phase transitions (QPTs). These properties include entanglement structures, correlations, and low-energy excitation spectra. While CFTs predict rich structures, such as specific operator contents and universal energy ratios determined by scaling dimensions, much of this structure remains unobserved experimentally. Existing probes in quantum materials (e.g., neutron scattering) or quantum simulators have largely focused on ground-state preparation or dynamical scaling (e.g., Kibble-Zurek scaling), but a direct measurement of the finite-size excitation spectra of emergent CFTs has been outstanding.

Methodology
The authors utilize a programmable neutral atom quantum simulator consisting of a one-dimensional chain of interacting Rydberg atoms. The system is described by an effective Hamiltonian that reduces to the Fendley-Sengupta-Sachdev (FSS) model in the Rydberg blockade regime, capable of realizing both Ising and tricritical Ising (TCI) universality classes depending on the interaction strength and detuning.

To probe the excitation spectra, the team developed a modulation spectroscopy technique involving a "modulation-ramp-probe" sequence:

1. Initialization: The system is quasi-adiabatically prepared in the ground state near a critical point ($\Delta_c$).
2. Modulation: A global or local detuning modulation is applied to couple the ground state to excited states. The modulation operator $\hat{K}$ can be symmetric (global) or antisymmetric (local) under reflection to target specific parity sectors.
3. Readout: The system is ramped into a symmetry-broken (Z2 ordered) or disordered phase where the Rydberg occupation number is measured. The change in population ($\delta n$) relative to the ground state indicates transitions to excited states.

The authors employ two variations of this technique:

• Peak-resolved regime: Using long modulation times to resolve individual energy levels and measure their ratios.
• Continuum regime: Using short modulation times to measure the dynamical structure factor, which encodes the universal scaling function of field correlations.

Additionally, the authors implement local detuning control at the chain edges to tune boundary conditions, allowing them to switch between "fixed" (strong edge order) and "free" (suppressed edge order) boundary conditions in the TCI phase.

Key Results

1. Ising CFT Spectrum: For chains tuned to the Ising critical point, the experiment resolves the even-parity excitation spectrum. The measured energy ratios of the first four levels follow the universal prediction of $2:4:6:8$. By varying system sizes ($L=7$ to $35$) and scaling the frequencies by $L$, the data collapses onto the universal CFT prediction, confirming the inverse scaling of energy levels with system size.
2. Parity Resolution: By applying antisymmetric local modulation, the authors access the odd-parity sector, observing energy ratios of $3:5:7$. The combined even and odd spectra yield the full $2:3:4:5:6:7:8$ pattern predicted by the Ising CFT.
3. Tricritical Ising (TCI) Transition: The authors experimentally traverse the transition from the Ising to the TCI CFT by tuning the next-nearest-neighbor interaction to attractive values. They observe a collapse of the edge charge density wave (CDW) order, signaling a transition to "free" boundary conditions.
4. TCI Boundary Conditions:
   • Free Boundary: At the TCI point with free boundaries, the energy ratio $E_2/E_1$ is measured to be $\approx 1.45$, consistent with the TCI prediction of $4/3$.
   • Fixed Boundary: By applying local detunings to pin edge order, the system realizes fixed boundary conditions, yielding an $E_2/E_1$ ratio of $\approx 2.0$, consistent with the TCI prediction of $2$.
   • Intermediate Boundary: An unstable intermediate fixed point is identified numerically and experimentally, where the ratio $E_2/E_1 \approx 2.5$, matching the prediction of $10/3$.
5. Dynamical Structure Factor: In the continuum limit for large systems ($L=35$), the authors measure the dynamical structure factor of the energy density operator ($\epsilon$). The response is found to be constant at low frequencies, consistent with the CFT prediction for the scaling dimension $\Delta_\epsilon = 1$.

Significance and Claims
The paper claims to provide the first direct experimental observation of the finite-size excitation spectra of emergent CFTs at quantum phase transitions. By recovering universal energy ratios and scaling behaviors, the work validates the operator content and scaling dimensions predicted by Ising and TCI CFTs in a quantum simulator.

The authors emphasize that their modulation spectroscopy technique serves as a powerful diagnostic tool for identifying a priori unknown universality classes in future experiments. Furthermore, the ability to control boundary conditions and measure dynamical structure factors demonstrates a pathway to studying general QPTs, including regimes where classical simulations are challenging. The work establishes a method to probe the emergence of CFT features, such as linear dispersion of emergent fermions and specific boundary fixed points, in programmable quantum systems.