Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Programmable interactions and emergent geometry in an array of atom clouds

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

This article has been updated

Abstract

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.

This is a preview of subscription content

Access options

Buy article

Get time limited or full article access on ReadCube.

$32.00

All prices are NET prices.

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

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.

Change history

References

  1. Bloch, I., Dalibard, J. & Nascimbene, S. Quantum simulations with ultracold quantum gases. Nat. Phys. 8, 267–276 (2012).

    CAS  Google Scholar 

  2. Browaeys, A. & Lahaye, T. Many-body physics with individually controlled Rydberg atoms. Nat. Phys. 16, 132–142 (2020).

    CAS  Google Scholar 

  3. Hayden, P. & Preskill, J. Black holes as mirrors: quantum information in random subsystems. J. High Energy Phys. 2007, 120–120 (2007).

    MathSciNet  Google Scholar 

  4. Maldacena, J. & Stanford, D. Remarks on the Sachdev-Ye-Kitaev model. Phys. Rev. D 94, 106002 (2016).

    ADS  MathSciNet  Google Scholar 

  5. Bentsen, G. et al. Treelike interactions and fast scrambling with cold atoms. Phys. Rev. Lett. 123, 130601 (2019).

    ADS  CAS  PubMed  Google Scholar 

  6. Belyansky, R., Bienias, P., Kharkov, Y. A., Gorshkov, A. V. & Swingle, B. Minimal model for fast scrambling. Phys. Rev. Lett. 125, 130601 (2020).

    ADS  MathSciNet  CAS  PubMed  PubMed Central  Google Scholar 

  7. Das, A. & Chakrabarti, B. K. Colloquium: quantum annealing and analog quantum computation. Rev. Mod. Phys. 80, 1061–1081 (2008).

    ADS  MathSciNet  MATH  Google Scholar 

  8. Gopalakrishnan, S., Lev, B. L. & Goldbart, P. M. Frustration and glassiness in spin models with cavity-mediated interactions. Phys. Rev. Lett. 107, 277201 (2011).

    ADS  PubMed  Google Scholar 

  9. Strack, P. & Sachdev, S. Dicke quantum spin glass of atoms and photons. Phys. Rev. Lett. 107, 277202 (2011).

    PubMed  Google Scholar 

  10. McMahon, P. L. et al. A fully programmable 100-spin coherent Ising machine with all-to-all connections. Science 354, 614–617 (2016).

    ADS  MathSciNet  CAS  PubMed  Google Scholar 

  11. Berloff, N. G. et al. Realizing the classical XY Hamiltonian in polariton simulators. Nat. Mater. 16, 1120–1126 (2017).

    ADS  CAS  PubMed  Google Scholar 

  12. Leroux, I. D., Schleier-Smith, M. H. & Vuletić, V. Implementation of cavity squeezing of a collective atomic spin. Phys. Rev. Lett. 104, 073602 (2010).

    ADS  PubMed  Google Scholar 

  13. Barontini, G., Hohmann, L., Haas, F., Estève, J. & Reichel, J. Deterministic generation of multiparticle entanglement by quantum Zeno dynamics. Science 349, 1317–1321 (2015).

    ADS  MathSciNet  CAS  MATH  PubMed  Google Scholar 

  14. Hosten, O., Krishnakumar, R., Engelsen, N. J. & Kasevich, M. A. Quantum phase magnification. Science 352, 1552–1555 (2016).

    ADS  MathSciNet  CAS  MATH  PubMed  Google Scholar 

  15. Pedrozo-Peñafiel, E. et al. Entanglement on an optical atomic-clock transition. Nature 588, 414–418 (2020).

    ADS  PubMed  Google Scholar 

  16. Welte, S., Hacker, B., Daiss, S., Ritter, S. & Rempe, G. Photon-mediated quantum gate between two neutral atoms in an optical cavity. Phys. Rev. X 8, 011018 (2018).

    CAS  Google Scholar 

  17. Davis, E. J., Bentsen, G., Homeier, L., Li, T. & Schleier-Smith, M. H. Photon-mediated spin-exchange dynamics of spin-1 atoms. Phys. Rev. Lett. 122, 010405 (2019).

    ADS  CAS  PubMed  Google Scholar 

  18. Davis, E. J. et al. Protecting spin coherence in a tunable Heisenberg model. Phys. Rev. Lett. 125, 060402 (2020).

    ADS  CAS  PubMed  Google Scholar 

  19. Muniz, J. A. et al. Exploring dynamical phase transitions with cold atoms in an optical cavity. Nature 580, 602–607 (2020).

    ADS  CAS  PubMed  Google Scholar 

  20. Barbón, J. L. & Magán, J. M. Fast scramblers and ultrametric black hole horizons. J. High Energy Phys. 2013, 163 (2013).

    ADS  MathSciNet  MATH  Google Scholar 

  21. Gubser, S. S., Knaute, J., Parikh, S., Samberg, A. & Witaszczyk, P. p-adic AdS/CFT. Commun. Math. Phys. 352, 1019–1059 (2017).

    ADS  MathSciNet  MATH  Google Scholar 

  22. Heydeman, M., Marcolli, M., Saberi, I. & Stoica, B. Tensor networks, p-adic fields, and algebraic curves: arithmetic and the AdS3/CFT2 correspondence. Preprint at https://arxiv.org/abs/1605.07639 (2016).

  23. Qi, X.-L. Does gravity come from quantum information? Nat. Phys. 14, 984–987 (2018).

    CAS  Google Scholar 

  24. Hung, C.-L., González-Tudela, A., Cirac, J. I. & Kimble, H. J. Quantum spin dynamics with pairwise-tunable, long-range interactions. Proc. Natl Acad. Sci. USA 113, E4946–E4955 (2016).

    MathSciNet  CAS  PubMed  PubMed Central  MATH  Google Scholar 

  25. Marsh, B. P. et al. Enhancing associative memory recall and storage capacity using confocal cavity QED. Phys. Rev. X 11, 021048 (2021).

    CAS  Google Scholar 

  26. Anikeeva, G. et al. Number partitioning with Grover’s algorithm in central spin systems. PRX Quantum 2, 020319 (2021).

    ADS  Google Scholar 

  27. Masson, S. J., Barrett, M. & Parkins, S. Cavity QED engineering of spin dynamics and squeezing in a spinor gas. Phys. Rev. Lett. 119, 213601 (2017).

    ADS  PubMed  Google Scholar 

  28. Qin, Z. et al. Characterizing the multipartite continuous-variable entanglement structure from squeezing coefficients and the Fisher information. npj Quantum Inf. 5, 3 (2019).

    ADS  Google Scholar 

  29. Pezzè, L., Smerzi, A., Oberthaler, M. K., Schmied, R. & Treutlein, P. Quantum metrology with nonclassical states of atomic ensembles. Rev. Mod. Phys. 90, 035005 (2018).

    ADS  MathSciNet  Google Scholar 

  30. Omran, A. et al. Generation and manipulation of Schrödinger cat states in Rydberg atom arrays. Science 365, 570–574 (2019).

    ADS  MathSciNet  CAS  PubMed  Google Scholar 

  31. Hamley, C. D., Gerving, C., Hoang, T., Bookjans, E. & Chapman, M. S. Spin-nematic squeezed vacuum in a quantum gas. Nat. Phys. 8, 305–308 (2012).

    CAS  Google Scholar 

  32. Fadel, M., Zibold, T., Décamps, B. & Treutlein, P. Spatial entanglement patterns and Einstein-Podolsky-Rosen steering in Bose-Einstein condensates. Science 360, 409–413 (2018).

    ADS  MathSciNet  CAS  MATH  PubMed  Google Scholar 

  33. Kunkel, P. et al. Spatially distributed multipartite entanglement enables EPR steering of atomic clouds. Science 360, 413–416 (2018).

    ADS  MathSciNet  CAS  MATH  PubMed  Google Scholar 

  34. Lange, K. et al. Entanglement between two spatially separated atomic modes. Science 360, 416–418 (2018).

    ADS  MathSciNet  CAS  MATH  PubMed  Google Scholar 

  35. Islam, R. et al. Emergence and frustration of magnetism with variable-range interactions in a quantum simulator. Science 340, 583–587 (2013).

    ADS  CAS  PubMed  Google Scholar 

  36. Jurcevic, P. et al. Quasiparticle engineering and entanglement propagation in a quantum many-body system. Nature 511, 202–205 (2014).

    ADS  CAS  PubMed  Google Scholar 

  37. Vaidya, V. D. et al. Tunable-range, photon-mediated atomic interactions in multimode cavity QED. Phys. Rev. X 8, 011002 (2018).

    CAS  Google Scholar 

  38. Manovitz, T., Shapira, Y., Akerman, N., Stern, A. & Ozeri, R. Quantum simulations with complex geometries and synthetic gauge fields in a trapped ion chain. PRX Quantum 1, 020303 (2020).

    Google Scholar 

  39. Kollár, A. J., Fitzpatrick, M. & Houck, A. A. Hyperbolic lattices in circuit quantum electrodynamics. Nature 571, 45–50 (2019).

    ADS  PubMed  Google Scholar 

  40. Lücke, B. et al. Detecting multiparticle entanglement of Dicke states. Phys. Rev. Lett. 112, 155304 (2014).

    ADS  PubMed  Google Scholar 

  41. Lee, S. & Lee, K.-C. Phase transitions in the fully frustrated triangular XY model. Phys. Rev. B 57, 8472 (1998).

    ADS  CAS  Google Scholar 

  42. Gubser, S. S., Jepsen, C., Ji, Z. & Trundy, B. Continuum limits of sparse coupling patterns. Phys. Rev. D 98, 045009 (2018).

    ADS  MathSciNet  CAS  Google Scholar 

  43. Shi, Y.-Y., Duan, L.-M. & Vidal, G. Classical simulation of quantum many-body systems with a tree tensor network. Phys. Rev. A 74, 022320 (2006).

    ADS  Google Scholar 

  44. Murg, V., Verstraete, F., Legeza, Ö. & Noack, R. M. Simulating strongly correlated quantum systems with tree tensor networks. Phys. Rev. B 82, 205105 (2010).

    ADS  Google Scholar 

  45. Pastawski, F., Yoshida, B., Harlow, D. & Preskill, J. Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence. J. High Energ. Phys. 2015, 149 (2015).

    MathSciNet  MATH  Google Scholar 

  46. Amit, D. J., Gutfreund, H. & Sompolinsky, H. Spin-glass models of neural networks. Phys. Rev. A 32, 1007–1018 (1985).

    ADS  MathSciNet  CAS  Google Scholar 

  47. Aidelsburger, M. et al. Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms. Nat. Phys. 11, 162–166 (2015).

    CAS  Google Scholar 

  48. Kennedy, C. J., Burton, W. C., Chung, W. C. & Ketterle, W. Observation of Bose–Einstein condensation in a strong synthetic magnetic field. Nat. Phys. 11, 859–864 (2015).

    CAS  Google Scholar 

  49. Zeiher, J. et al. Microscopic characterization of scalable coherent Rydberg superatoms. Phys. Rev. X 5, 031015 (2015).

    Google Scholar 

  50. Evans, R. E. et al. Photon-mediated interactions between quantum emitters in a diamond nanocavity. Science 362, 662–665 (2018).

    ADS  CAS  PubMed  Google Scholar 

  51. Lee, J., Vrijsen, G., Teper, I., Hosten, O. & Kasevich, M. A. Many-atom–cavity QED system with homogeneous atom–cavity coupling. Opt. Lett. 39, 4005–4008 (2014).

    ADS  PubMed  Google Scholar 

  52. Xu, V. et al. Probing gravity by holding atoms for 20 seconds. Science 366, 745–749 (2019).

    ADS  CAS  PubMed  Google Scholar 

  53. Sørensen, A. S. & Mølmer, K. Entangling atoms in bad cavities. Phys. Rev. A 66, 022314 (2002).

    ADS  MathSciNet  Google Scholar 

  54. Stamper-Kurn, D. M. & Ueda, M. Spinor Bose gases: symmetries, magnetism, and quantum dynamics. Rev. Mod. Phys. 85, 1191 (2013).

    ADS  CAS  Google Scholar 

  55. Sinatra, A., Lobo, C. & Castin, Y. The truncated Wigner method for Bose-condensed gases: limits of validity and applications. J. Phys. B 35, 3599–3631 (2002).

    ADS  CAS  Google Scholar 

  56. Blakie, P. B., Bradley, A. S., Davis, M. J., Ballagh, R. J. & Gardiner, C. W. Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques. Adv. Phys. 57, 363–455 (2008).

    ADS  CAS  Google Scholar 

  57. Lewis-Swan, R. J., Olsen, M. K. & Kheruntsyan, K. V. Approximate particle number distribution from direct stochastic sampling of the Wigner function. Phys. Rev. A 94, 033814 (2016).

    ADS  Google Scholar 

  58. Lauritzen, S. L. Graphical Models Vol. 17 (Clarendon Press, 1996).

Download references

Acknowledgements

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

Authors

Contributions

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.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

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

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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

Supplementary information

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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). https://doi.org/10.1038/s41586-021-04156-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41586-021-04156-0

Further reading

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing