Observing the Space- and Time-Dependent Growth of Correlations in Dynamically Tuned Synthetic Ising Models with Antiferromagnetic Interactions

We explore the dynamics of artificial one- and two-dimensional Ising-like quantum antiferromagnets with different lattice geometries by using a Rydberg quantum simulator of up to 36 spins in which we dynamically tune the parameters of the Hamiltonian. We observe, in a region in parameter space, the onset of antiferromagnetic (AF) ordering, albeit with only finite-range correlations. We study systematically the influence of the ramp speeds on the correlations and their growth in time. We observe a delay in their buildup associated to the finite speed of propagation of correlations in a system with short-range interactions. We obtain a good agreement between experimental data and numerical simulations, taking into account experimental imperfections measured at the single-particle level. Finally, we develop an analytical model, based on a short-time expansion of the evolution operator, which captures the observed spatial structure of the correlations, and their buildup in time.

Popular Summary

The ability to create and control ensembles of interacting quantum objects, such as atoms and ions, paves the way to quantum simulators, which could solve difficult computational problems that cannot be tackled by simulation on traditional digital computers. For some such tasks, one can vary in time the parameters describing the system and then let the system evolve to a state where the different particles are correlated; that is, the state of one particle depends on that of the others. However, these quantum correlations need time to build up, which places a limit on how fast one can tune the system. Here, we experimentally observe this fundamental limit and develop a theory to explain it, both of which are prerequisites for many applications of quantum simulators.

We arrange up to 36 rubidium atoms in two-dimensional arrays of optical tweezers with tunable geometries. When illuminated by lasers, the atoms are promoted to highly excited states known as Rydberg states, where one electron is excited to high energy. The system can then be described as an ensemble of interacting spins such that, for some range of parameters, any two neighboring spins want to align in opposite directions, giving rise to what is called antiferromagnetic order. When varying the laser parameters, we observe the progressive buildup of these correlations in space and time over a few microseconds, and we are able to model it theoretically, thus understanding the current limits of the platform.

Our results are an important step towards developing more complex quantum simulations of correlated systems.

Figure 1

Studying the AF Ising model on 1d and 2d systems. (a) Examples of single-shot fluorescence images of single-atom arrays used in our experiments: a 24-atom 1d chain with periodic boundary conditions, a $6\times 6$ square array, and a 36-atom triangular array. Each atom is used to encode a spin-$1/2$, whose internal states $|\uparrow ⟩$ and $|\downarrow ⟩$ are coupled with Rabi frequency $\mathrm{\Omega }$ and detuning $\delta$. (b) Time dependence of the Rabi frequency $\mathrm{\Omega }(t)$ and detuning $\delta (t)$ used to probe the buildup of correlations. (c) Sketched ground-state phase diagrams of the Ising model in Eq. (1), in the nearest-neighbor interaction limit, for a 1d chain, a 2d square lattice, and a 2d triangular lattice. In the figure, AFM stands for antiferromagnetic, PM for paramagnetic, and OBD for order by disorder. (d) Typical experimental correlation functions obtained for these geometries (see text). For the 1d chain, the correlation length $\xi =1.5$ sites (bottom left panel).

Figure 2

Exploring the $\mathrm{\Omega }=0$ line of the phase diagram for an $L\times L$ square lattice with $L=4$ and $L=6$. (a) Experimental spin-spin correlation function $g^{(2)}(k,l)$ for different values of the final detuning. (b) Corresponding experimental histograms of the populations of the two Néel sublattices $A$ and $B$, to be contrasted with those for uncorrelated random states with the same Rydberg density (the dotted circles indicate the positions where the random state histogram falls off to a value of $5\times {10}^{-4}$). (c) Simulated histograms (without dephasing) for $L=4$ as a comparison. (d) Average Rydberg density $n$, and (e) Néel structure factor $S_{\mathrm{N}é\mathrm{el}}$ as a function of the final detuning. The shaded areas highlight the region of the AF phase in the phase diagram for $\mathrm{\Omega }=0$. In (d) and (e), the error bars (sometimes smaller than the symbol size) denote the standard error on the mean (s.e.m).

Figure 3

Buildup and decay of correlations in a 2d square lattice antiferromagnet for different ramp durations $t_{\mathrm{tot}}$. (a) Experimental results for the 2d correlation function $g^{(2)}(k,l)$ for four different values of $t_{\mathrm{tot}}$, showing short-range AF ordering, with a pronounced sweep duration dependence. (b) The correlation function $g^{(2)}$ (averaged within a Manhattan shell $m=|k|+|l|$) decays exponentially with the Manhattan distance $m$, with a correlation length $\xi$ of about 1.4 lattice sites. (c) Néel structure factor $S_{\mathrm{N}é\mathrm{el}}$ as a function of $t_{\mathrm{tot}}$. The green solid line shows the result of a numerical simulation using exact diagonalization for a $4\times 4$ system without dephasing, while the yellow dashed line denotes a numerical simulation including dephasing (see Appendix pp3). (d) Dependence of the correlations on $t_{\mathrm{tot}}$ for nearest, second-nearest, third-nearest, and fourth-nearest neighbors. One observes a dependence not only on the Manhattan distance $|k|+|l|$, but also on $k-l$ (see Sec. 5c). The dashed lines represent a numerical simulation including dephasing (the shaded area corresponds to the s.e.m. obtained therein).

Figure 4

Time evolution of a $6\times 6$ square array along a ramp with $t_{\mathrm{tot}}=1.0\mu \mathrm{s}$. (a) Buildup of correlations along the ramp. (b) Observation of a time delay between the buildup of significant correlations in increasing Manhattan shells $m$. This is the manifestation of a finite propagation speed of correlations in their buildup between distant sites (see text). (c) Néel structure factor and (d) correlation length $\xi$. All figures: The dotted (dashed) lines correspond to the result of a numerical simulation for a $4\times 4$ lattice without (with) dephasing.

Figure 5

Spatial structure of the spin-spin correlations on the 36-site square (a),(b) and on the 36-site triangular lattice (c),(d). Experimental data after a sweep is shown in (a) and (c), together with the values of $(k,l)$ for the first three shells $|k|+|l|$ [see Fig. 1 for the full range]. The numbers in brackets [$c$] give the number of linking paths contributing to the short-time expansion. (b) and (d) Dependence of the correlation on $(k,l)$ for several Manhattan distances with a qualitative comparison to binomial coefficients (green dotted lines) and leading-order short-time expansion (red dashed lines); see text for details. The shaded region corresponds to the s.e.m. of the experimental data.

Figure 6

Influence of the sign of $C_6$ on the many-body spectrum. Eigenenergies of a small $2\times 2$ square system as a function of $\delta$ for $\hslash \mathrm{\Omega }/U\simeq 0.5$. (a) For a repulsive interaction $C_6>0$, the ground state of $H$ is antiferromagnetic for $0<\hslash \delta <2U$. It can be reached from ${|\downarrow ⟩}^{\otimes 4}$ by adiabatically following the ground state starting from negative values of $\delta$ (trajectory in dashed arrows). (b) For an attractive interaction $C_6<0$, the antiferromagnetic state is the most excited state, and the sign of the detuning needed to reach it from ${|\downarrow ⟩}^{\otimes 4}$ is reversed.

Figure 7

Classical ($\mathrm{\Omega }=0$) phase diagrams for arrays with open boundary conditions, with Rydberg density $n$ as a function of $x=\hslash \delta /U$ for the $N=36$ square and triangular arrays used in the experiment. Blue lines correspond to open boundary conditions, while red dashed lines correspond to periodic boundary conditions (bulk phase diagram). The numbers indicate the degeneracy of the classical configurations for the distinct plateaus.

Figure 8

(a) Single-atom Rabi oscillations, averaged over a typical square array (dark red symbols). We perform a fit (dashed line) to the analytical model including detection deficiency to obtain the dephasing rate $\gamma$ and the Rabi frequency $\mathrm{\Omega }$. The value of $\gamma$ thus obtained is then used for the simulations of the experiments with ${\mathrm{\Omega }}_{\max }=\mathrm{\Omega }$ [cf. Figs. 3 and 3]. (b),(c) Simulation of the many-body real-time evolution as in Figs. 4 and 4, with different values of the dephasing rate $\gamma$ for (b) the Néel structure factor and (c) the correlation length. The rate $\gamma =3\mu {\mathrm{s}}^{-1}$ used in the main text as obtained in (a) is indicated by a solid line.

Figure 9

The Lieb-Robinson bounds. “Significant” correlations spread along a light cone $d=2vt$. Along and inside the light cone (grey shaded area), correlations of order $\mathcal{O}(1)$ can develop. Outside the light cone, $d>2vt$ correlations are immediately built up for each $t>0$, but are exponentially suppressed in $d-2vt$.

Figure 10

Quasiparticle velocity along the ramp used in Fig. 4 obtained by linear spin-wave theory ($a$ denotes the lattice spacing). The maximal velocity is reached when the ramp crosses the phase boundary at $t\approx 0.5\mu \mathrm{s}$.

Figure 11

Illustration of the different paths connecting two sites with a distance $(k,l)$ contribution to the leading-order short-time expansion on (a) the square lattice and (b) the triangular lattice. The number in brackets shows the binomial coefficient $C_k^{|k|+|l|}$ (assuming $k,l\ge 0$) corresponding to the number of distinct paths. The arrows show the potentially different couplings along paths contributing to the short-time expansion. The grey lines show the three distinct paths contributing to the correlation indicated by a grey square, as an example.

Figure 12

Correlations on the $N=36$-site triangular array (only the first four Manhattan shells are shown). (a) Experimentally measured correlations after a sweep on the triangular array featuring a pronounced Manhattan shell structure $g^{(2)}(k,l)\propto (-1{)}^m$. The numbers give the value of $m=|k|+|l|$. (b) Correlations of the equal superposition of the classical ground states for ${\delta }_{\text{final}}$ as reached at the end of the sweep used in (a). These correlations are in qualitative agreement with the correlations expected in the $1/3$ AF state in the right panel of Fig. 1.