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Programmable interactions and emergent geometry in an array of atom clouds

An Author Correction to this article was published on 11 March 2022

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Interactions govern the flow of information and the formation of correlations between constituents of many-body quantum systems, dictating phases of matter found in nature and forms of entanglement generated in the laboratory. Typical interactions decay with distance and thus produce a network of connectivity governed by geometry—such as the crystalline structure of a material or the trapping sites of atoms in a quantum simulator1,2. However, many envisioned applications in quantum simulation and computation require more complex coupling graphs including non-local interactions, which feature in models of information scrambling in black holes3,4,5,6 and mappings of hard optimization problems onto frustrated classical magnets7,8,9,10,11. Here we describe the realization of programmable non-local interactions in an array of atomic ensembles within an optical cavity, in which photons carry information between atomic spins12,13,14,15,16,17,18,19. By programming the distance dependence of the interactions, we access effective geometries for which the dimensionality, topology and metric are entirely distinct from the physical geometry of the array. As examples, we engineer an antiferromagnetic triangular ladder, a Möbius strip with sign-changing interactions and a treelike geometry inspired by concepts of quantum gravity5,20,21,22. The tree graph constitutes a toy model of holographic duality21,22, in which the quantum system lies on the boundary of a higher-dimensional geometry that emerges from measured correlations23. Our work provides broader prospects for simulating frustrated magnets and topological phases24, investigating quantum optimization paradigms10,11,25,26 and engineering entangled resource states for sensing and computation27,28.

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Fig. 1: Engineering distance-dependent interactions.
Fig. 2: Pair creation at programmable distance.
Fig. 3: Geometry extracted from correlations.
Fig. 4: Treelike geometry.

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Data availability

All data displayed in Figs. 14 and Extended Data Figs. 14 are available from the corresponding author upon reasonable request.

Code availability

All code used for simulation and analysis is available from the corresponding author upon reasonable request.

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We thank S. Gubser for discussions that inspired our exploration of non-Archimedean geometry. We also acknowledge discussions with G. Bentsen, A. Daley, I. Bloch, B. Lev, N. Berloff, A. Deshpande, B. Swingle and P. Hayden. This work was supported by the DOE Office of Science, Office of High Energy Physics and Office of Basic Energy Sciences under grant no. DE-SC0019174. A.P. and E.S.C. acknowledge support from the NSF under grant no. PHY-1753021. We additionally acknowledge support from the National Defense Science and Engineering Graduate Fellowship (A.P.), the NSF Graduate Research Fellowship Program (E.J.D. and E.S.C.), the Hertz Foundation (E.J.D.) and the German Academic Scholarship Foundation (J.F.W.).

Author information

Authors and Affiliations



A.P., E.S.C., P.K., J.F.W. and E.J.D. performed the experiments. A.P., E.S.C., P.K. and M.S.-S. analysed the experimental data and developed supporting theoretical models. A.P., E.S.C., P.K. and M.S.-S. wrote the manuscript. All authors contributed to the discussion and interpretation of results.

Corresponding author

Correspondence to Monika Schleier-Smith.

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The authors declare no competing interests.

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Peer review information Nature thanks Robert Lewis-Swan and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Fig. 1 Coupling graphs.

Sketch of couplings \(J(i-j)\) for the model in equation (4) with local interactions (s = − 1, left) or treelike interactions (s = 1, right). The strengths of the interactions are indicated by the thickness and transparency of the red lines. For s = 1, reordering the sites according to the Monna map makes the couplings more local, corroborating the treelike geometry.

Extended Data Fig. 2 Experimental sequence and imaging.

a Schematic of experimental sequence for measurements of \({F}_{i}^{x}\). After driving the cavity to induce interactions, we apply spin rotations sequentially to the M sites of the array and subsequently perform state-sensitive readout via fluorescence imaging. b Fluorescence images after spin rotation, showing the signal for the F = 2 manifold and the three magnetic substates for the case of interactions at distance r = 3 with periodic boundary conditions. c Transverse magnetization \({F}_{i}^{x}\) and structure factor \({\tilde{F}}_{k}^{x}\) extracted from the image in b.

Extended Data Fig. 3 Effect of finite statistics.

Left, correlation plot reproduced from Fig. 1, showing \({C}^{{\rm{pm}}}\) obtained from 50 realizations of the experiment with interactions at distance r = 10. Right, simulation results obtained from a truncated Wigner approximation, where we either choose the same number of realisations as in the experiment or increase the number of realisations by a factor of 10 to reduce statistical uncertainty. The simulations indicate that residual correlations in the experimental data are mainly due to the finite sample size.

Extended Data Fig. 4 Comparison between measured structure factor and simulation results.

The left graph shows the measured structure factor after T = 3 Bloch periods of evolution, which is also shown in Fig. 2c. The two plots at right show results of a truncated Wigner simulation with and without periodic boundary conditions. For the simulated data we used 100 realizations of the TWA simulation, which is four times higher than the number of experimental realizations to reduce statistical fluctuations. For open boundary conditions, we find that the simulation has an offset with respect to the theoretical prediction (blue line). We attribute this offset to the finite system size, as the model is exact only for an infinite system or a system with periodic boundary conditions. Repeating the same simulation with a pulsed drive shown on the right shows that in this case the TWA simulation is consistent with the analytical model. The error bars indicate the standard error of the mean.

Extended Data Table 1 Experimental parameters

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Periwal, A., Cooper, E.S., Kunkel, P. et al. Programmable interactions and emergent geometry in an array of atom clouds. Nature 600, 630–635 (2021).

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