Interrelated Thermalization and Quantum Criticality in a Lattice Gauge Simulator

Gauge theory and thermalization are both topics of essential importance for modern quantum science and technology. The recently realized atomic quantum simulator for lattice gauge theories provides a unique opportunity for studying thermalization in gauge theory, in which theoretical studies have shown that quantum thermalization can signal the quantum phase transition. Nevertheless, the experimental study remains a challenge to accurately determine the critical point and controllably explore the thermalization dynamics due to the lack of techniques for locally manipulating and detecting matter and gauge fields. We report an experimental investigation of the quantum criticality in the lattice gauge theory from both equilibrium and nonequilibrium thermalization perspectives, with the help of the single-site addressing and atom-number-resolved detection capabilities. We accurately determine the quantum critical point and observe that the Néel state thermalizes only in the critical regime. This result manifests the interplay between quantum many-body scars, quantum criticality, and symmetry breaking.

Figure 1

Experimental system. (a) Schematic of the ultracold atom microscope and the prepared $|{\mathbb{Z}}_2⟩$ initial state. We combine the optical superlattices and the addressing beam generated by the digital micromirror device (DMD) to prepare the initial $|{\mathbb{Z}}_2⟩$ state [21]. The top shows an exemplary raw-data fluorescence image of the atom distribution of the initial $|{\mathbb{Z}}_2⟩$ state in a single experimental realization. (b) The physical model with bosons in a one-dimensional optical lattice with alternating deep and shallow lattice sites. Here, $U$ denotes the on-site interaction strength, $J$ denotes the hopping amplitude of bosons, $\delta$ denotes the energy offset between neighboring shallow and deep lattices, and $\mathrm{\Delta }$ denotes the linear tilt per site. The open and solid circles with $+$ or $-$ denote physical charge zero, $+1$ or $-1$ at the matter sites, and the arrows denote the electric field. (c) An Ising-type quantum phase transition by tuning $m/\tilde{t}$.

Figure 2

Adiabatic ramping and phase transition. (a),(b) Single-site-resolved atom number distribution for a range of $m/\tilde{t}$. (c) The absolute of the spatial averaged electric field $|\overline{E}|$ as a function of $m/\tilde{t}$. (d) $|\overline{E}|L^{1/8}$ as a function of $m/\tilde{t}$ for $L=5$, 7, 9. The crossing point of the three curves locates the quantum critical point. The inset verifies the critical behavior by showing $|\overline{E}|L^{\beta /\nu }$ plotted against $[m/\tilde{t}-(m/\tilde{t}{)}_c]L^{1/\nu }$, which collapse onto a single curve near the critical point [33, 34]. Error bars denote the standard error of the mean.

Figure 3

Time evolution after a quench. The real-time dynamics of $\overline{E}$ for different values of $m/\tilde{t}$ (marked at the axes of $m/\tilde{t}$). The dashed lines are the fitted longtime steady values $E_{\infty }$, and the solid lines are the theoretical thermal values $E_{\mathrm{th}}$ assuming that the initial state fully thermalizes. Here $L=7$ and each data point is averaged over 10–20 chains after the postselection. Error bars denote the standard error of the mean, and are smaller than the markers if hidden.

Figure 4

Quantum criticality. (a) The averaged KL divergence $D_{\mathrm{KL}}$ defined by Eq. (3) quantifies the overlap of three data points for each $m/\tilde{t}$ in Fig. 2. $D_{\mathrm{KL}}$ is plotted as a function of $m/\tilde{t}$. The peak of $D_{\mathrm{KL}}$ defines the point where the three curves in Fig. 2 cross each other and locates the quantum critical point. The error bars of the data points are the error of the calibrated $m/\tilde{t}$ [21]. The determined quantum critical point $(m/\tilde{t}{)}_c$ lies at the middle of the range of $m/\tilde{t}$ spanned by the shaded region. Its error bar (coming from both the Gaussian fitting error and the calibration error) is indicated by half of the horizontal width of the shaded region. (b) The data points are the steady values $E_{\infty }$ extracted from Fig. 3. The red dashed line is the theoretical steady value with the same system size as the experimental system, and the solid yellow line is the theoretical thermal value $E_{\mathrm{th}}$ assuming that the initial state obeys the eigenstate thermalization hypothesis. The horizontal error bars denote the errors of the calibrated $m/\tilde{t}$, and the vertical error bars denote the standard deviations of the fitted $E_{\infty }$. The center and the half width of the shaded region represent the quantum critical point (determined by the intersection between the solid yellow line and the fitted curve of the data points) and its error bar, respectively.