Probing critical phenomena in open quantum systems using atom arrays

Abstract

At continuous phase transitions, quantum many-body systems exhibit scale-invariance and complex, emergent universal behavior. Most strikingly, at a quantum critical point, correlations decay as a power law, with exponents determined by a set of universal scaling dimensions. Experimentally probing such power-law correlations is extremely challenging, owing to the complex interplay between decoherence, the vanishing energy gap, and boundary effects. Here, we employ a Rydberg quantum simulator to adiabatically prepare critical ground states of both a one-dimensional ring and a two-dimensional square lattice. By accounting for and tuning the openness of our quantum system, which is well-captured by the introduction of a single phenomenological length scale, we are able to directly observe power-law correlations and extract the corresponding scaling dimensions. Moreover, in two dimensions, we observe a decoupling between phase transitions in the bulk and on the boundary, allowing us to identify two distinct boundary universality classes. Our work demonstrates that direct adiabatic preparation of critical states in quantum simulators can complement recent approaches to studying quantum criticality using the Kibble-Zurek mechanism or digital quantum circuits.

Figures (14)

• Figure 1: Experimental protocols for initialization and state preparation. (A) Fluorescence images of stochastically-loaded $^{133}$Cs atoms in optical tweezer arrays for 1D (top) and 2D (bottom). Open circles show the locations of unloaded optical tweezers. (B) Defect-free atom arrays after rearrangement. (C) Upper: Schematic of the 1D phase diagram. The dashed arrow denotes the adiabatic state preparation trajectory with the star indicating the critical point. Lower: Optimized ramp profile for adiabatic preparation of the critical state. Here $\Omega$ and $\Delta$ correspond to the two-photon Rabi frequency and the detuning of the driving field, respectively. $R_b$ is the blockade radius. (D) Fluorescence images of the array after adiabatic state preparation. The open circles denote the absence of an atom, which is inferred to be in the Rydberg state.
• Figure 2: Critical correlations in a 1D ring geometry. Squares (40-atom), circles (24-atom), stars (24-atom at the critical point) denote experimental measurements, with shaded regions representing 1-$\sigma$ bootstrap errors. Gray solid, dotted, and dashed lines respectively indicate ground state, unitary under optimized ramp, and stochastic wave-function simulation results using experimental parameters. Horizontal and vertical gray lines mark the location of the gap minimum for the 24-atom system. (A) Rydberg population density across the phase transition at different sweep rates. Color gradients indicate the sweep rates variation. Inset: Linear ramp profiles at different sweep rates. (B) Peak location of susceptibility $\chi$ at different sweep rates. (C) $\sigma$ field correlation measurements around the critical point, with red stars representing the critical correlations. Inset: 1D phase diagram and gap profile with red markers denoting the locations of the measurements. (D) Measured order parameter $\langle\hat{O}\rangle$ as a function of the tuning parameter $\Delta/\Omega$. (E) Fitted decoherence-induced length scale $\xi_d/a$ at the critical point and in the AFM phase. The uncertainties in the measurements correspond to a combination of 1-$\sigma$ bootstrap error and fitting error. Inset: Measured $\langle\sigma_0\sigma_j\rangle$ in the AFM phase for two values of $\Delta/\Omega$, with exponential fits to extract $\xi_d/a$. (F) Measured $\langle\sigma_0\sigma_j\rangle$ at critical points divided by $\mathrm{exp}(-\delta_j/\xi_d)$ as a function of $\delta_j$. The results yield a power-law decay with an exponent of 2$\Delta_\sigma^{{}^{{}_{\text{1D}}}}$(solid red line). The uncertainty in $\Delta_\sigma^{{}^{{}_{\text{1D}}}}$ includes both 1-$\sigma$ bootstrap error and fitting error. Inset: Raw measurements of $\langle\sigma_0\sigma_j\rangle$. The solid pink line represents the expected power-law decay for (1+1)d Ising universality class.
• Figure 3: Tuning quantum criticality in an open system. Markers (24-atom) with different shapes are experimental measurements at the critical point under different experimental conditions, with shaded regions denoting the 1-$\sigma$ bootstrap errors. Error bars in fitted $\xi_d/a$ include 1-$\sigma$ bootstrap error and fitting error. (A) $\sigma$ field correlation measurements at two different ramp times. Upper inset: Fitted $\xi_d/a$. Lower inset: Corresponding ramp profiles. (B) $\sigma$ field correlation measurements at three different intensity configurations of Rydberg beams at fixed $\Omega$. Upper inset: Fitted $\xi_d/a$. Lower inset: A schematic showing the single photon rabi frequencies $\Omega_{455}$ and $\Omega_{1062}$ for each configuration. (C) Correlation measurements in (A) and (B) divided by the exponential $\mathrm{exp}(-\delta_j/\xi_d)$, displaying a collapse onto a power-law decay with an exponent of 2$\Delta_\sigma^{{}^{{}_{\text{1D}}}}$ (solid black line), acquired via a simultaneous fit to the power-law exponential model for all scenarios. The uncertainty in $\Delta_\sigma^{{}^{{}_{\text{1D}}}}$ includes both 1-$\sigma$ bootstrap error and fitting error.
• Figure 4: Probing critical Ising correlations and surface transitions in a 2D square lattice. Circles (7 x 7), stars (7 x 7 at the critical point) and squares (9 x 9) denote experimental measurements, with shaded regions representing 1-$\sigma$ bootstrap errors. (A) A schematic for the 2D phase diagram, adapted to the finite system size. The shaded area labeled boundary marks the region of a boundary-ordered phase with a disordered bulk. Red stars indicate the location of the critical point along the two cuts. Gray dashed line indicates $\Delta/\Omega = 1.5$. Inset: The $\sigma$ fields live on lattice bonds, both along the boundary (purple) and within the bulk (orange). $\delta_{mn}$ denotes the Euclidean distance between two nearest $\sigma$ fields. (B) $\sigma$ field correlation measurements around the critical point along the ordinary cut, with measured ground state population (i) and $\sigma$ field (ii) across the phase transition. The critical spatial correlation is highlighted by the star marker. Dashed lines are guides to the eye. Inset: 2D phase diagram schematic with markers indicating the measurements' locations. (C) Peak location of susceptibility $\chi$ along the ordinary cut at different sweep rates with boundary and bulk analyzed separately. Gray dashed line marks the location of gap minimum. (D) The measured critical $\sigma$ field correlation within the bulk exhibits a power-law decay with an exponent of 2$\Delta_{\sigma}$, shown by the orange solid line. The gray solid line represents the ground state simulation using experimental parameters. (E, F) Order parameter analyzed independently for the bulk ($\langle \hat{O} \rangle$) and the boundary ($\langle\hat{O}\rangle_\partial$) across the phase transition along the ordinary cut (E) and the surface cut (F). (G, H) Measured $\sigma$ field spatial correlations along the boundary (G) at the critical point for the two cuts (marked by red stars in (A)), and within the bulk (H) at $\Delta/\Omega$ = 1.5 (marked by the gray dashed line in (A)). Dashed lines are guides to the eye.
• Figure S1: Rearrangement procedure in 1D for a 40-atom ring. Blue squares denote locations of SLM traps, and orange circles denote the locations of atoms. Each target site on the ring is associated with a reservoir row or column. (A) Row by row, we shuffle atoms horizontally to columns that need additional atoms to reach their targets. Green arrows denote atoms that are moved in order to fill the deficient red column. Orange arrows denote atoms that are moved to more central columns to contribute more atoms to the next step. We note that if all columns have a sufficient number of atoms, this step can be skipped. Furthermore, not all rows will be moved. (B) Column by column (within the green rectangle), we shuffle atoms vertically to their target sites, ejecting any unneeded atoms and also filling rows that need additional atoms. Importantly all atoms are now in the central rows which will be targeted in the next step. (C) Row by row (within the red rectangle), we place the remaining atoms into the target sites, while ejecting unneeded atoms. (D) After the previous rearrangement steps, a defect free ring is achieved.