Experimental observation of conformal field theory spectra

Abstract

Conformal field theories (CFTs) feature prominently in high-energy physics, statistical mechanics, and condensed matter. For example, CFTs govern emergent universal properties of systems tuned to quantum phase transitions, including their entanglement, correlations, and low-energy excitation spectra. Much of the rich structure predicted by CFTs nevertheless remains unobserved in experiment. Here we directly observe the energy excitation spectra of emergent CFTs at quantum phase transitions -- recovering universal energy ratios characteristic of the underlying field theories. Specifically, we develop and implement a modulation technique to resolve a Rydberg chain's finite-size spectra, variably tuned to quantum phase transitions described by either Ising or tricritical Ising CFTs. We also employ local control to distinguish parities of excitations under reflection and, in the tricritical Ising chain, to induce transitions between distinct CFT spectra associated with changing boundary conditions. By utilizing a variant of the modulation technique, we furthermore study the dynamical structure factor of the critical system, which is closely related to the correlation of an underlying Ising conformal field. Our work not only probes the emergence of CFT features in a quantum simulator, but also provides a technique for diagnosing a priori unknown universality classes in future experiments.

Figures (12)

• Figure 1: Many-body spectroscopy and the Ising CFT spectrum.(a) Phase diagram of the Fendley-Sengupta-Sachdev Hamiltonian, reproduced from Ref. slagle_microscopic_2021. Arrows represent the adiabatic sweep trajectory, ending at critical points probed in this work. (b) Modulation spectroscopy protocol (top) and corresponding schematic spectral evolution (bottom). We adiabatically evolve an initial disordered state to Ising criticality and then apply modulation to target a low-lying excited state; the readout of the change of the number of atoms in $| 0 \rangle$, $\delta n$, is performed following a subsequent detuning sweep to the $\mathbb{Z}_2$-ordered / disordered phase. (c) Modulation spectroscopy implementation setup. We use acousto-optic deflectors (AODs) and an acousto-optic modulator (AOM) to control the global and the local detuning modulation with the Rydberg laser frequency and the tweezer-induced light shift. (d) Spectroscopy of the many-body Hamiltonian on a 19-atom array with modulation applied at various detunings $\Delta$. Red solid lines are numerical calculations of the excited state energies of Eq. \ref{['eq:masterH']}, with the opacity indicating the transition strength between the ground state and each excited state. The dashed line represents the numerically determined critical detuning $\Delta_c$. (e) Spectroscopy at the critical detuning $\Delta_c$ of a 19-atom chain. We observe an increase in the number of excitations after the sweep ($\delta n$) (blue, top axis) at frequencies coinciding with excited state energies. The solid line is a fit with a sum of four Gaussians and indicate fit range. Dashed lines indicate the CFT predicted energy ratio. Red circles are numerical calculations of transition strength $|\langle g |\hat{K}| e \rangle|^2$ (bottom axis). The color code represents clusters of states with close eigenenergies which will become degenerate in the thermodynamic limit as predicted by the Ising CFT. (f) The predicted even-parity Ising CFT spectrum, from exact analytical calculations. The Ising CFT energy levels have the universal ratio 2:4:6:8.
• Figure 1: Many-body Rabi oscillation and coherence on a 7-atom array.(a) Rabi oscillation between the ground and the first excited state. We modulate the system with the frequency being the energy difference between the two states and measure the final change of ground state population, $\delta n$, after ramping into the $\mathbb{Z}_2$ phase. When $\delta n = 0 / 1$, it indicates that the system is in its ground / excited state. The solid line is a fit to a decaying sinusoidal oscillation. (b) Ramsey interrogation of the energy gap. Using the many-body Rabi frequency determined from (a), we apply two $\pi/2$ rotations, separated by a time $t$. After the Ramsey sequence, the system is ramped into $\mathbb{Z}_2$ phase for readout. We observe a damped oscillation, whose oscillation frequency is consistent with the energy gap. The solid line is a fit to a decaying sinusoidal oscillation.
• Figure 2: Scaling of the finite-size spectrum.(a) Sketch of scaling modulation frequency with system size. When multiplying the frequency with the system size, the eigenvalues for different system sizes collapse and yield a universal ratio predicted by the CFT. (b) Spectroscopy at the critical detuning $\Delta_c$ at various system sizes. In the horizontal axis the modulation frequency $f$ is scaled by system size $L$. Dashed lines are numerically extrapolated eigenvalues, corresponding to the even-parity Ising CFT spectrum with the universal ratio 2:4:6:8. Solid lines are multi-Gaussian fits to the data and indicate fit ranges. (c) First four spectral peak positions $f$, rescaled by $L$, for each system size. Solid lines are the numerically calculated rescaled even-parity state energies with the same colors as in Fig. \ref{['fig:isingspectroscopy']}e. Dashed lines are the extrapolated rescaled energies.
• Figure 2: Locating the Ising and the tricritical Ising critical point.(a) Locating the Ising critical point via the finite-size scaling of the curve crossing of $\sigma_{RS}$ for (top) $V_2/\Omega = +0.51$ and (bottom) $V_2/\Omega = -0.51$. Each data point is the location of $\Delta/\Omega$ where $\sigma_{RS}$ has the same value for chains of length $L-2$ and $L+2$. Using a quadratic fit, we estimate the critical point in the thermodynamic limit at $1/L \to 0$, $\Delta_c(L\to\infty)/\Omega$. For $V_2/\Omega=+0.51,-0.51$, $\Delta_c(L\to\infty)/\Omega \approx 1.6975(4), -0.142(2)$, respectively. (b) Curve crossing of the energy ratio between the second and the first excited state close to the tricritical point. We plot the energy ratio versus detuning $\Delta$ for $\Omega = 2\pi \times 5.5~\rm{MHz}$ and $V_2 = 2\pi \times (-9.0)~\rm{MHz}$ ($V_2/\Omega = -1.63$) for various system sizes $L$. We find the intersection $(\Delta_X, E_2/E_1)$ for two adjacent odd-system sizes $L-1$ and $L+1$ and plot them in (c) and (d). (c) System-size dependence of the energy ratio. We plot the intersecting $E_i/E_1$ for $i \in [2,6]$ and linearly extrapolate to $L\rightarrow\infty$. At this $V_2$, we find best agreements of the first two energy ratio with the TCI spectrum with free boundary condition, $4/3$ and $5/3$, respectively. (d) System-size dependence of the intersecting detunings $\Delta_X$. We plot the intersecting $\Delta_X$ for the low-lying energies and quadratically extrapolate to $L\rightarrow\infty$. We find them collapse to $\Delta_c = 2\pi\times (-8.3)~\rm{MHz}$, which we determine to be the critical detuning.
• Figure 3: Parity-resolved Ising CFT spectrum with local control.(a) The applied odd-parity local detuning modulation profile with a wavevector of $k=\pi/18$. (b) Even- and odd-parity spectra of a 19-atom array tuned to Ising criticality. The measured spectra are consistent with the universal energy ratio, 2:3:4:5:6:7:8, predicted by the Ising CFT. Solid lines are Gaussian fits to the data and indicate fit ranges. Dashed lines are rescaled Ising CFT predictions. (c) Momentum-resolved spectrum. The previous measurements can be interpreted as the even-sector at $k=0$ and the odd-sector at $k=\pi/18$. The slope of the dashed line represents the numerically determined light-cone velocity.