Realizing quantum Ising models in tunable two-dimensional arrays of single Rydberg atoms
Henning Labuhn, Daniel Barredo, Sylvain Ravets, Sylvain de Léséleuc, Tommaso Macrì, Thierry Lahaye, Antoine Browaeys
RRealizing quantum Ising models in tunable two-dimensional arrays of single Rydberg atoms
Henning Labuhn, Daniel Barredo, Sylvain Ravets, Sylvain de L´es´eleuc, Tommaso Macr`ı, Thierry Lahaye, and Antoine Browaeys Laboratoire Charles Fabry, Institut d’Optique, CNRS, Univ Paris Sud 11,2 avenue Augustin Fresnel, 91127 Palaiseau cedex, France Departamento de F´ısica Te´orica e Experimental, Universidade Federal do Rio Grande do Norte,and International Institute of Physics, Natal-RN, Brazil.
Spin models are the prime example of simplified many-body Hamiltonians used to model complex, real-worldstrongly correlated materials . However, despite theirsimplified character, their dynamics often cannot be sim-ulated exactly on classical computers as soon as the num-ber of particles exceeds a few tens. For this reason, thequantum simulation of spin Hamiltonians using the toolsof atomic and molecular physics has become very activeover the last years, using ultracold atoms or molecules inoptical lattices, or trapped ions . All of these approacheshave their own assets, but also limitations. Here, we re-port on a novel platform for the study of spin systems, us-ing individual atoms trapped in two-dimensional arrays ofoptical microtraps with arbitrary geometries, where fill-ing fractions range from to with exact knowl-edge of the initial configuration. When excited to Rydberg D -states, the atoms undergo strong interactions whoseanisotropic character opens exciting prospects for simu-lating exotic matter . We illustrate the versatility of oursystem by studying the dynamics of an Ising-like spin- / system in a transverse field with up to thirty spins, fora variety of geometries in one and two dimensions, andfor a wide range of interaction strengths. For geometrieswhere the anisotropy is expected to have small effects wefind an excellent agreement with ab-initio simulations ofthe spin- / system, while for strongly anisotropic situa-tions the multilevel structure of the D -states has a mea-surable influence , . Our findings establish arrays of sin-gle Rydberg atoms as a versatile platform for the study ofquantum magnetism. Rydberg atoms have recently attracted a lot of interest forquantum information processing and quantum simulation .In this work, we use a system of individual Rydberg atomsto realize highly-tunable artificial quantum Ising magnets. Byshining on the atoms lasers that are resonant with the transi-tion between the ground state | g (cid:105) and a chosen Rydberg state | r (cid:105) , we implement the Ising-like Hamiltonian H = (cid:88) i (cid:126) Ω2 σ ix + (cid:88) i
Methods, along with any additional Ex-tended Data display items, are available in the online versionof the paper; references unique to these sections appear onlyin the online paper.
References and notes
1. Auerbach, A.
Interacting Electrons and Quantum Mag-netism (Springer-Verlag, New York, 1994).2. Georgescu, I.M., Ashhab, S. & Nori, F.
Quantum simu-lation . Rev. Mod. Phys. , 153 (2014).3. Bloch, I., Dalibard, J., & Nascimb`ene, S. Quantum sim-ulations with ultracold quantum gases . Nature Phys. ,267 (2012).4. Yan, B., Moses, S.A., Gadway, B., Covey, J.P., Haz-zard, K.R.A., Rey, A.M., Jin, D.S., & Ye, J. Observa-tion of dipolar spin-exchange interactions with lattice-confined polar molecules . Nature , 521 (2013).5. Blatt, R. & Roos, C.F.
Quantum simulations withtrapped ions . Nature Phys. , 277 (2012).6. Glaetzle, A.W., Dalmonte, M., Nath, R., Rousochatza-kis, I., Moessner, R., & Zoller, P. Quantum Spin-Ice andDimer Models with Rydberg Atoms . Phys. Rev. X ,041037 (2014).7. Vermersch, B., Glaetzle, A.W., & Zoller, P. Magic dis-tances in the blockade mechanism of Rydberg P and Dstates . Phys. Rev. A , 023411 (2015).8. Tresp, C., Bienias, P., Weber, S., Gorniaczyk, H., Mir-gorodskiy, I., B¨uchler, H.P., & Hofferberth, S. Dipolardephasing of Rydberg D-state polaritons . Phys. Rev.Lett. , 083602 (2015).9. Saffman, S., Walker, T.G., & Mølmer, K.
Quantum in-formation with Rydberg atoms . Rev. Mod. Phys. ,2313 (2010).10. Weimer, H., M¨uller, M., Lesanovsky, I., Zoller, P., &B¨uchler, H.P. A Rydberg quantum simulator . Nat. Phys. , 382 (2010). 11. Barredo, D., Ravets, S., Labuhn, H., B´eguin, L.,Vernier, A., Nogrette, F., Lahaye, T. & Browaeys A. Demonstration of a Strong Rydberg Blockade in Three-Atom Systems with Anisotropic Interactions . Phys. Rev.Lett. , 183002 (2014).12. B´eguin, L., Vernier, A., Chicireanu, R., Lahaye, T., &Browaeys, A.
Direct Measurement of the van der WaalsInteraction between Two Rydberg Atoms . Phys. Rev.Lett. , 263201 (2013).13. Nogrette, F., Labuhn, H., Ravets, S., Barredo, D.,B´eguin, L., Vernier, A., Lahaye, T. &. Browaeys, A.
Single-Atom Trapping in Holographic 2D Arrays of Mi-crotraps with Arbitrary Geometries . Phys. Rev. X ,021034 (2014).14. Dudin, Y.O., Li, L., Bariani, F., & Kuzmich, A. Obser-vation of coherent many-body Rabi oscillations . Nat.Phys. , 790 (2012).15. Ebert, M., Gill, A., Gibbons, M., Zhang, X., Saffman,M., & Walker, T.G. Atomic Fock state preparation us-ing Rydberg blockade . Phys. Rev. Lett. , 043602(2014).16. Zeiher, J., Schauß, P., Hild, S., Macr`ı, T., Bloch, I.,& Gross, C.
Microscopic Characterization of ScalableCoherent Rydberg Superatoms . Phys. Rev. X , 031015(2015).17. Ates, C., & Lesanovsky, I. Entropic enhancement ofspatial correlations in a laser-driven Rydberg gas .Phys. Rev. A Spatialcorrelations of Rydberg excitations in optically drivenatomic ensembles . Phys. Rev. A , 053414 (2013).19. Schauss, P., Cheneau, M., Endres, M., Fukuhara, T.,Hild, S., Omran, A., Pohl, T., Gross, C., Kuhr, S., &Bloch, I. Observation of Spatially Ordered Structuresin a Two-Dimensional Rydberg Gas . Nature , 87(2012).20. L¨ow, R., Weimer, H., Krohn, U., Heidemann, R., Bend-kowsky, V., Butscher, B., B¨uchler, H.P., & Pfau, T.
Uni-versal scaling in a strongly interacting Rydberg gas .Phys. Rev. A , 033422 (2009).21. Lester, B.J., Luick, N., Kaufman, A.M., Reynolds,C.M., & Regal, C.A. Rapid production of uniformly-filled arrays of neutral atoms . Phys. Rev. Lett. ,073003 (2015).22. Fung, Y.H., & Andersen, M.F.
Efficient collisionalblockade loading of single atom into a tight microtrap .New J. Phys. , 073011 (2015).23. Gaj, A., Krupp, A.T., Balewski, J.B., L¨ow, R., Hoffer-berth, S., & Pfau, T. From molecular spectra to a den-sity shift in dense Rydberg gases . Nature Comm. ,4546 (2014).24. Ates, C., Garrahan, J.P., & Lesanovsky, I. Thermaliza-tion of a Strongly Interacting Closed Spin System: FromCoherent Many-Body Dynamics to a Fokker-PlanckEquation . Phys. Rev. Lett. , 110603 (2012).25. Hazzard, K.R.A., van den Worm, M., Foss-Feig, M.,Manmana, S.R., Dalla Torre, E.G., Pfau, T., Kastner, M.& Rey, A.M.
Quantum correlations and entanglementin far-from-equilibrium spin systems . Phys. Rev. A ,063622 (2014).26. Barredo, D., Labuhn, H., Ravets, S., Lahaye, T.,Browaeys, A., & Adams, C.S. Coherent ExcitationTransfer in a Spin Chain of Three Rydberg Atoms . Phys.Rev. Lett. , 113002 (2015).27. Hauke, P., Cucchietti, F.M., M¨uller-Hermes, A.,Ba˜nuls, M.-C., Cirac, J.I., & Lewenstein, M.
Completedevil’s staircase and crystal-superfluid transitions in adipolar XXZ spin chain: a trapped ion quantum simu-lation . New J. Phys. , 113037 (2010).28. Senko, C., Richerme, P., Smith, J., Lee, A., Cohen,I., Retzker, A., & Monroe, C. Realization of a Quan-tum Integer-Spin Chain with Controllable Interactions .Phys. Rev. X , 021026 (2015). 29. Peter, D., Yao, N.Y., Lang, N., Huber, S.D., Lukin,M.D., & B¨uchler, H.P. Topological bands with a Chernnumber C = 2 by dipolar exchange interactions . Phys.Rev. A , 053617 (2015). Acknowledgments:
We thank H. Busche for contributionsin the early stages of the experiment, I. Lesanovsky, H.P.B¨uchler, and T. Pohl for useful discussions, and Y. Sortaisfor a careful reading of the manuscript. This work benefitedfrom financial support by the EU [FET-Open Xtrack ProjectHAIRS, H2020 FET-PROACT Project RySQ, and EU Marie-Curie Program ITN COHERENCE FP7-PEOPLE-2010-ITN-265031 (H.L.)], by the ‘PALM’ Labex (project QUANTICA)and by the Region ˆIle-de-France in the framework of DIMNano-K.
Author Contributions:
H.L. and D.B. contributed equallyto this work. H.L., D.B., S.R. and S.d.L. carried out the ex-periments, T.M. did the numerical simulations, T.L. and A.B.supervised the work. All authors contributed to the design ofthe experiments and to the data analysis. The manuscript waswritten by T.L. with input from all authors.
Author information:
The authors have no com-peting financial interests. Correspondence and re-quests for material should be addressed to T.L.([email protected]).
MethodsLoading of trap arrays
In the single-atom loading regime of optical microtraps, the probability to have a given trap filled with a single atom is p (cid:39) / . Therefore, when we monitor the number of loaded traps in view of triggering the experiment, N fluctuates in timearound a mean value N t / , with fluctuations ∼ √ N t .When the number of traps is small, we can impose, as the triggering criterion, to wait until all traps are filled. The averagetriggering time T N then increases exponentially with N , as can be seen in Extended Data Figure 1a. We used this ‘full-loadingmode’ for the data of Fig. 1 ( (cid:54) N (cid:54) ) and Fig. 3 ( N = 8 ) of main text. This exponential scaling sets a practical limit of N ∼ for fully loaded arrays. Already for N = 9 , the experimental duty cycle exceeds one minute.Due to this, for larger N t we use partially-loaded arrays. We set the triggering threshold in the tail of the binomial distributionof N , i.e. close to N t / √ N t . This allows us to keep a fast repetition rate for the experiment, on the order of − , enablingfast data collection. Extended Data Figure 1b shows the distribution of loaded traps for the ‘racetrack’ array with N t = 30 (respectively, for the N t = 7 × square array), where we set the triggering condition to N = 20 (resp. N = 30 ). Usingthis triggering procedure, we thus end up with a narrow distribution of atom numbers N = 20 ± . (resp. N = 28 ± . ),corresponding to a filling fraction of 67% (resp. 57%), significantly above the average N t / . These strongly subpoissoniandistributions of atom numbers are such that the variation in N from experiment to experiment has a negligible effect on thephysics studied in Fig. 4 of main text; however, as for each experiment the initial configuration image is saved, one can if neededpost-select experiments where an exact number of atoms was involved (this is how the data in Fig. 2 of main text for N (cid:62) were obtained).Recently, several experiments , demonstrated quasi-deterministic loading of single atoms in optical tweezers, reaching p ∼ using modified light-assisted collisions that lead to the loss of only one of the colliding atoms instead of both. Apreliminary implementation of these ideas on our setup gave p ∼ for a single trap. In future work, by using such loadingin combination with the real-time triggering based on the measured number of loaded traps, it seems realistic to reach, even in Extended Data Figure 1 | : Full and partial loading of arrays. a : Average triggering time T N when the triggering criterion is set to N = N t : achieving full loading requires an exponentially long time, limiting in practice the method to N t (cid:54) . The triggering times canvary substantially depending on the density of the magneto-optical trap used to load the array, and the data points shown here correspond totypical conditions used for the data of main text. b : Distribution of the number of loaded traps in the partially loaded regime for the 30-trap‘racetrack’ and the 49-trap square array (blue dots). The shaded distributions correspond to what would be observed with random triggering. large arrays, filling fractions in excess of 0.9, i.e. approaching those obtained in quantum gas microscope experiments usingMott insulators. Experimental parameters
Extended Data Table I summarizes the various values of the parameters of the arrays of traps and of the Rydberg states usedfor the data presented in the main text, and the resulting values of the dimensionless parameter α . It illustrates the wide tunabilityoffered by the system. Finite detection errors
Our way to detect that a given atom has been excited to a Rydberg state relies on the fact that we do not detect fluorescencefrom the corresponding trap in the final configuration image. There is however a small probability ε to lose an atom during thesequence, even if it was in the ground state, thus incorrectly inferring its excitation to a Rydberg state . These ‘false positive’detection events affect the measured populations of the N -atom system. One can show that, if P q is the observed probability tohave q Rydberg excitations, and ˜ P p the actual probability to have p Rydberg excitations, P q = q (cid:88) p =0 (cid:18) N − pq − p (cid:19) ε q − p (1 − ε ) N − q ˜ P p . (4) Trap array parameters Rydberg state parametersFigure
Spacing a N t N n
Calculated C /h Ω / (2 π ) R b α ( µ m) (GHz µ m ) (MHz) ( µ m)
2a (full) 3.0 1–9 N t − . × . .
2a (partial) 3.2 19 10–15 100 − . × . .
3a 6.3 8 8 54 − . . . .
3b 6.3 8 8 61 − . × . . .
3c 3.8 8 8 100 − . × .
95 21 5 . ± . − . × . . . ± . − . × . . . Extended Data Table I | : Experimental parameters used for the data presented in the main text.
Wide tuning of α = R b /a , over oneorder of magnitude, is achieved by a combination of changes in a and n (while Ω is kept almost constant). Extended Data Figure 2 | : Effect of detection errors. a : Experimental determination of ε . From the data of the full blockade experiments(Fig. 2 of main text), we plot the probability P to recapture all N atoms for τ = 0 . The solid line is a fit to the expected dependence (1 − ε ) N ,giving ε = 3% (the shaded area corresponds to < ε < ). b : Calculated probabilities to observe 0,1 or 2 excitations assuming a perfectblockade and ε = 3% , for atom numbers N = 3 , , . In principle, one can invert the above linear system relating the observed and actual probabilities , to correct the experimentaldata for the detection errors. Here we have chosen on the contrary to show the uncorrected populations, and to include detectionerrors on the theoretical curves instead.In order to determine the experimental value of ε , we use the initial datapoints ( τ = 0 ) of the data of Fig.2 of main text. Sinceno Rydberg pulse is sent, we have ˜ P = 1 , and from (4) the observed probability P ( τ = 0) reads (1 − ε ) N . Extended DataFigure 2a shows the variation of P (0) as a function of N , together with a fit which allows us to extract ε = (3 ± , the valuewe use for the theoretical curves in the main text (see below).Extended Data Figure 2b shows the effect of this finite value of ε on the probabilities P , P and P in the full blockaderegime, for atom numbers N = 3 , , , clearly illustrating that the ‘false positive’ detection events (i) yield non-zero (andincreasing with N ) double excitation probabilities (that oscillate in phase with P ) (ii) multiply the amplitude of P by a factor (1 − ε ) N and (iii) reduce the contrast of the P oscillations. Globally, the experimental data (see Extended Data Figure 3) showsthese features, superimposed with other imperfections such as damping, not related to the finite value of ε .Finally, let us mention the effect of the detection errors on the correlation functions. In the fully blockaded region k < α ,one ideally expects a vanishing g (2) for ε = 0 . However, to lowest order in ε , this value is increased substantially (see e.g. Fig.4c of main text) to ε/f R where f R is the Rydberg fraction. Indeed, g (2) ( k = 1) is given by an average of quantities of theform (cid:104) n i n i +1 (cid:105) / ( (cid:104) n i (cid:105)(cid:104) n i +1 (cid:105) ) . For ε = 0 , the numerator vanishes due to blockade; the only possibility to have a non-zero valuecomes from detection errors. To lowest order in ε , the probability to get a nonzero value for n i n i +1 is that either atom i is in | r (cid:105) (probability f R ) and atom i + 1 is lost (probability ε ), or vice-versa. This results in a value εf R for the numerator, while for thedenominator we can use the zeroth-order values (cid:104) n i (cid:105) = (cid:104) n i +1 (cid:105) = f R , thus giving g (1) (cid:39) ε/f R , which experimentally can beas large as 0.5. Supplementary experimental data
Full Rydberg blockade.—
Extended Data Figure 3 shows additional data in the full blockade regime (Fig. 2 of main text). InExtended Data Figure 3a, the arrays of 1 to 9 traps are fully loaded, while in Extended Data Figure 3b, the 19-trap triangulararray is partially loaded with 10 to 15 atoms. In both panels, the left column shows the time evolution of the probability P to recapture all atoms at the end of the sequence, the middle column shows P , and the right column shows P . The points inFig. 2a of main text corresponding to N = 8 and N = 9 in partially loaded arrays were taken in a similar configuration as for N = 10 to , but the array contained only N t = 17 traps. The curves (not shown here) do not show any noticeable differencewith other sets of data. • We recognize the effects of the finite detection errors ε (cid:54) = 0 on the amplitude and contrast of the collective oscillationsdiscussed in section above; • In addition, the oscillations exhibit some damping, which seems to increase with N . To quantify this, we fit the data bythe function P ( τ ) = a e − γτ (cid:0) cos (Ω N τ /
2) + b (cid:1) + c, (5)where a, b, c, γ and Ω N are adjustable parameters (solid lines). This functional form was chosen to account in a simple wayfor the asymmetry in the damping. Extended Data Figure 3c shows the damping rates γ , extracted from the probabilities P as a function of N . We observe an initial increase in the damping rates, which then saturates above N = 5 . An increasewith N of the damping rate was observed in other similar blockade experiments − . Extended Data Figure 3 | : Full dataset for the Rydberg blockade data. a : Fully loaded arrays of 1 to 9 traps ( n = 82 ). b : partially loadedarray of N t = 19 traps, containing from N = 10 to N = 15 atoms ( n = 100 ). The column on the left shows the probability P to recaptureall atoms, the center column the probability P to lose just one atom out of N , and the column on the right the probability P to lose two atomsout of N . The solid lines are fits by (5). c : Damping rate γ extracted from the P data as a function of the number of atoms in the array. Extended Data Figure 4 | : Homogeneous excitation in the 8-atom ring. a : For Ω τ = 3 . , we observe strongly contrasted oscillations inthe pair correlation function g ( k ) . b : The average density of Rydberg excitations, however, is approximately the same on every site. Thehorizontal dashed line indicates the mean over all sites. • In addition, we observe that P slowly increases over time for some specific values of N (see in particular N = 4 , , , ),corresponding to particular geometries.We do not have a full understanding of these last two observations, but they may originate from the breaking of the blockadedue to the Zeeman structure of the Rydberg states nD / (see discussion below). Extended Data Figure 4 shows that, within statistical fluctuations, the density of excitations on the 8-atom ringis homogeneous (this remains true at all times), and that the antiferromagnetic-like or crystal-like features obtained for sometimes, e.g. for Ω τ = 3 . , can only be observed in the correlation functions. This illustrates the interest of our setup in whichspin chains with PBC can be realized easily. On the contrary, in a 1D chain with open boundary conditions, ‘pinning’ of theexcitations at specific sites would occur due to edge effects. Racetrack-shaped array.—
Extended Data Figure 5a shows the full evolution of the time correlation function for the data ofFig. 4abc of the main text ( R b = 4 . a ). Extended Data Figure 5b corresponds to the same settings except for the fact that onenow has R b = 2 . a . Extended Data Figure 5 | : Full time evolution of the correlation functions for the 30-trap, racetrack-shaped chain. a : Same as forFigure 3a,b,c of main text. The right panel is the time evolution of the pair correlation function, clearly showing that, for times longer thana few Ω − , the pair correlation function does not evolve significantly anymore. The vertical dashed line indicates the value of the blockaderadius. b : The principal quantum number is now n = 57 , and the Rabi frequency Ω = 2 π × . MHz, such that R b = 2 . a . The central panelshows the time evolution of the Rydberg fraction, and the right panel the time evolution of the pair correlation function. For both a and b , f R approaches, at long times, the close-packing limit a/R b of hard rods of length R b (dashed horizontal lines). Extended Data Figure 6 | : Full time evolution of the correlation function for the × square array . One observes the blockaded regionaround ( k, l ) = (0 , , with a slight flattening reflecting the anisotropy of the interaction. After a few Ω − , the correlation function does notevolve any more. Square array of × traps.— Figure 6 shows the full time evolution of the two-dimensional Rydberg-Rydberg correlationfunction g (2) ( k, l ) for the × square lattice of Fig. 4def. Note that the two-dimensional pair correlation function is calculatedusing (3), which implies that, due to the finite size of the array, the number of terms included in the sum decreases when k, l increase. The normalization takes this variation into account. Anisotropy of the interaction
For a pair of atoms in a nD / Rydberg state with the internuclear axis not aligned with the quantization axis, the rigorousdescription of the van der Waals interaction requires to include all various Zeeman sublevels; the interaction then takes the formof a × matrix. To keep the description of a system of N atoms tractable, one can, in the blockade regime, define aneffective, anisotropic van der Waals potential reducing the previous matrix to a single scalar. For nD / states, the anisotropyreported in refs. 7,11 is well reproduced by the simple expression V eff ( r, θ ) = C (0) r (cid:18)
13 + 23 cos θ (cid:19) (6)0with θ the angle between the quantization axis and the internuclear axis, giving a reduction by a factor of three in interactionstrength when θ goes from to π/ .Due to the anisotropy in (6), the shape of the blockade volume centered on a Rydberg atom is also anisotropic. However, dueto the r -scaling of the interaction, the surface r ( θ ) defined by V eff ( r, θ ) = (cid:126) Ω is quite well approximated by a prolate spheroidwith an aspect ratio of / (cid:39) . . In the figures of the main text, the shaded regions depicting the blockade volume have thepolar equation r ( θ ) = R b (cid:0) + cos θ (cid:1) / . Numerical simulation of the dynamics
Our theoretical description of the system is based on the mapping of its dynamics into a pseudo-spin / model withanisotropic long range interactions. We therefore neglect the rich Zeeman structure of the nD / states. The numerical cal-culations rest on the solution of the Schr¨odinger equation for the Hamiltonian of Eq. (1) of the main text in a reduced Hilbertspace H . We first write the wave function | ψ (cid:105) of the system with N atoms in terms of states with fixed number of Rydberg ex-citations and ground state atoms, which correspond to the eigenstates of the Hamiltonian with vanishing Rabi frequency Ω , .Then the truncation procedure is based on two complementary steps: first we define the maximum number of Rydberg excita-tions N max r that we include in our basis, second we eliminate those states which display excitations closer than a fixed distance R . Both N max r and R are adjusted to ensure the convergence of the dynamics. For small samples (Fig. 3 of the main text)we performed simulations including all 256 basis states, whereas for the racetrack configurations we typically set R smallerthan the lattice constant but include up to N math r = 10 excitations at most, reducing the dimension of H from (cid:39) to (cid:80) N max r q =0 (cid:0) q (cid:1) (cid:39) × . For the × square array with 30 atoms, we set R = 1 . a (much smaller than the blockade radius R b = 2 . a ), thus reducing the dimension of H to (cid:39) × (the full Hilbert space is of dimension (cid:39) , and using onlythe truncation criterion on the number of excitations would reduce it to about × , still intractably large). The Schr¨odingerequation within the truncated Hilbert space is then solved with standard split-step method for the two non-commuting parts ofthe Hamiltonian of Eq. (1) of the main text. All these calculations were repeated for several realizations of the loading of thearrays ( realizations for the squared × configurations and realizations for the case with fewer traps), taking into accountthe anisotropic interparticle interaction of Eq. (6). The comparison with experimental data of the average fraction of excitations f R = (cid:80) Nq =0 qP q /N is done by including the “false positive” detection events as described by Eq. (4).The calculation of the g (2) ( k ) correlation function in Fig.3d and Fig.4c of the main text follows the definition of Eq. (7).However, contrarily to the calculation of the average fraction of the excitations it is not possible to derive an analytical formulafor g (2) ( k ) to properly take into account the detection efficiency of Rydberg excitations (unless k < α as described in sectionS.1.3.). Therefore we implement a standard Monte Carlo algorithm to perform the average of the correlation function overrandomly generated configurations which are weighted in g (2) ( k ) with the initial (quantum) probability extracted from thereal time dynamics of the Schr¨odinger equation. For example the state | r i r j (cid:105) which contains N r = 2 Rydberg excitations andamplitude c i j ( t ) can wrongly be dectected as the state | r i r j r q (cid:105) with probability p = ε (1 − ε ) N − . If the latter state is generatedfrom our sampling algorithm then its weight in the correlation function corresponds to | c i j ( t ) | . Finally we average over severalhundreds randomly generated configurations to obtain well converged results for the correlation function. Effect of partial loading of large arrays on the observed dynamics
Using the simulations described above, we explore to which extent the partial loading of our larger arrays may change theobserved dynamics as compared to the ideal case of full loading.Extended Data Figure 7 shows, for the ‘racetrack’ array of Fig. 4abc of main text, the results of simulations for the experi-mentally relevant case of partial loading (solid lines, filling fraction η (cid:39) . ) and for the ideal, full loading case (thin dashedlines): • Extended Data Figure 7a shows the time evolution of the Rydberg fraction f R . The dynamics is qualitatively similar in thetwo situations, with initial oscillations that rapidly get damped due to the dephasing of the many incommensurate eigen-energies of the Hamiltonian. Quantitatively, the initial oscillations are faster in the fully loaded case: this is expected, aseach blockade volume contains /η as many atoms, and thus, due to the scaling of the collective Rabi frequency with thenumber of atoms in a blockade volume, we expect an enhancement of the oscillation frequency by ∼ η − / (cid:39) . , closeto what we observe. In the same way, the asymptotic Rydberg fraction when τ → ∞ is reduced by a factor close to theexpected factor η .1 Extended Data Figure 7 | : Full versus partial loading for the dynamics and correlations in the case of Fig 4a,b,c of main text. a : Rydbergfraction as a function of time for the partially loaded (solid line) or fully loaded (thin dashed line) 30-trap array. b : Pair correlation function g (2) ( k ) for Ω τ (cid:39) . , for the partially loaded (solid line) or fully loaded (thin dashed line) 30-trap array. In both cases, the effect of detectionerrors ( ε = 3% ) is included. • Extended Data Figure 7b shows the pair correlation function g (2) ( k ) for Ω τ (cid:39) . . Here again, the changes are moderate,although the oscillations of the correlation function for k > α would be slightly more contrasted for the fully loaded array.Simulations for the other large array settings give similar results, allowing us to safely conclude that the partial loading ofour largest arrays does not affect significantly the observed dynamics. This conclusion would be different for other types ofexperiments, for instance the transport of a spin excitation in the case of resonant-dipole-dipole interactions. Approximative translational invariance
For the one-dimensional configurations of the main text (8-atom ring of Fig.3b and racetrack-shaped array of 30 traps ofFig.4a of main text) we plot the spatially averaged pair correlation function g (2) ( k ) = 1 N t (cid:88) i (cid:104) n i n i + k (cid:105)(cid:104) n i (cid:105)(cid:104) n i + k (cid:105) , (7)where the subscripts label sites. For a system invariant by translation, all terms in the sum are identical, and the averaging over i simply improves the signal to noise ratio. However, our systems are not translationally invariant, in particular because of theanisotropy of the interaction, and a natural question to address is whether the averaging reduces the contrast of the correlationfunctions. To answer this question, we have calculated the dynamics of the pair correlation function for the 8-atom ring, taking Extended Data Figure 8 | : Assessing the validity of the approximation of translational invariance in the 8-atom ring.
Calculated paircorrelation function g (2) ( k ) as a function of the excitation time, for the 8-atom ring. a : simulation using the experimentally relevant anisotropicinteraction, which breaks translational invariance. b : simulation with the same parameters as in a , except that the angular dependence isneglected (we replace (6) by its value for θ = 0 ), thus reestablishing translational invariance. One observes that the contrast in a is reduced, asexpected, but only in a marginal way. Effective loss mechanism arising from anisotropic interactions of D states
The agreement between our measurements and the results of the simulations is not perfect for the largest excitation times, inparticular for some settings (e.g. for some configurations in the full blockade regime, for the 8-atom ring in the partial blockaderegime, and for the × square array), where we observe a gradual increase in the number of measured Rydberg excitations.These effects could be qualitatively reproduced if the detection errors ε would increase in time. However, the main reason forthese losses is due to the fact that the microtraps are switched off during the excitation (to avoid inhomogeneous light-shifts),and as they are off for a fixed amount of time (3 µ s), independent of τ , we do not, at first sight, expect ε to increase in time. Onecould imagine however that the presence of the Rydberg excitation lasers may induce extra loss (due to off-resonant scattering forinstance), and in this case one would end up having an ε increasing with τ . We have experimentally ruled out this possibility bymeasuring the recapture probability when shining the Rydberg excitation lasers, detuned from the Rydberg line by ∼
100 MHz ,for the full 3 µ s, without measuring any detrimental effect.A second possible reason would be the motion of the atoms. Due to their finite temperature, the atoms move during free flightwith a velocity v ∼
50 nm /µ s . Now, strictly speaking, the terms corresponding to the laser coupling in Eqn. (1) of main textare not Ω σ ix , but Ωe i k · r i ( t ) σ i + + h . c . , where k is the sum of the wavevectors of the excitation lasers at 795 and 475 nm, and r i ( t ) the position of atom i . Thus, because of the motion, the phase factors of the couplings become time-dependent, which e.g.yields a dephasing of the spin wave corresponding to | W (cid:105) states. However, a numerical simulation of this effect shows that theinduced dephasing rates are negligible for our parameters.We thus believe that the cause for the observed extra losses lies in the interplay between the large number of interactingZeeman sublevels when two atoms are excited to nD / states: for θ (cid:54) = 0 all 16 pair state Zeeman sublevels are coupledtogether by the van der Waals interaction. For a large number of atoms, this may lead to an effective loss rate from the targeted | r (cid:105) states into a quasi-continuum comprising all other (weakly interacting) Zeeman states, and hence to a gradual increase ofpopulation of the Rydberg manifold. Qualitatively, this interpretation is corroborated by the fact that the observed increase inthe number of observed excitations depends quite strongly on the array geometry: for instance, the data of the racetrack-shapedarray, for which a majority of interacting atom pairs are almost aligned along the quantization axis z , are well reproduced by thesimulations even at long times, unlike in the case of the 8-atom ring or the × square array, for which many interacting pairshave their internuclear axes strongly inclined with respect to z .Achieving a quantitative understanding of these observed imperfections, using approaches similar to the ones of refs. 7,8,32, isa challenging task. However, it is an important step in view of future applications of Rydberg blockade for quantum simulation,and will thus be the subject of future work.30. Shen, C. & Duan, L.-M. Correcting detection errors in quantum state engineering through data processing . New J. Phys. , 053053 (2012).31. Schauß, P., Zeiher, J., Fukuhara, T., Hild, S., Cheneau, M., Macr`ı, T., Pohl, T., Bloch, I. & Gross, C. Crystallization inIsing quantum magnets . Science , 1455 (2015).32. Derevianko, A., Komar, P., Topcu, T., Kroeze, R.M. & Lukin, M.D.