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Signatures of self-organized criticality in an ultracold atomic gas

A Publisher Correction to this article was published on 11 March 2020

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Abstract

Self-organized criticality is an elegant explanation of how complex structures emerge and persist throughout nature1, and why such structures often exhibit similar scale-invariant properties2,3,4,5,6,7,8,9. Although self-organized criticality is sometimes captured by simple models that feature a critical point as an attractor for the dynamics10,11,12,13,14,15, the connection to real-world systems is exceptionally hard to test quantitatively16,17,18,19,20,21. Here we observe three key signatures of self-organized criticality in the dynamics of a driven–dissipative gas of ultracold potassium atoms: self-organization to a stationary state that is largely independent of the initial conditions; scale-invariance of the final density characterized by a unique scaling function; and large fluctuations of the number of excited atoms (avalanches) obeying a characteristic power-law distribution. This work establishes a well-controlled platform for investigating self-organization phenomena and non-equilibrium criticality, with experimental access to the underlying microscopic details of the system.

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Fig. 1: SOC in an ultracold atomic gas excited to Rydberg states by a laser field.
Fig. 2: Self-organization: above a threshold value, the remaining total atom density nt is attracted to the same stationary-state density independent of the initial conditions.
Fig. 3: Scale invariance of the self-organized stationary state as a function of the driving intensity Ω2.
Fig. 4: Observation of power-law distributed excitation avalanches.

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Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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References

  1. Bak, P., Tang, C. & Wiesenfeld, K. Self-organized criticality: an explanation of the 1/f noise. Phys. Rev. Lett. 59, 381–384 (1987).

    ADS  CAS  PubMed  Google Scholar 

  2. Sornette, A. & Sornette, D. Self-organized criticality and earthquakes. EPL 9, 197–202 (1989).

    ADS  MATH  Google Scholar 

  3. Rhodes, C. J. & Anderson, R. M. Power laws governing epidemics in isolated populations. Nature 381, 600–602 (1996).

    ADS  CAS  PubMed  Google Scholar 

  4. Malamud, B. D., Morein, G. & Turcotte, D. L. Forest fires: an example of self-organized critical behavior. Science 281, 1840–1842 (1998).

    ADS  CAS  PubMed  Google Scholar 

  5. de Arcangelis, L., Godano, C., Lippiello, E. & Nicodemi, M. Universality in solar flare and earthquake occurrence. Phys. Rev. Lett. 96, 051102 (2006).

    ADS  PubMed  Google Scholar 

  6. Klaus, A., Yu, S. & Plenz, D. Statistical analyses support power law distributions found in neuronal avalanches. PLoS ONE 6, e19779 (2011).

    ADS  CAS  PubMed  PubMed Central  Google Scholar 

  7. Hesse, J. & Gross, T. Self-organized criticality as a fundamental property of neural systems. Front. Syst. Neurosci. 8, 166 (2014).

    PubMed  PubMed Central  Google Scholar 

  8. Gleeson, J. P., Ward, J. A., O’Sullivan, K. P. & Lee, W. T. Competition-induced criticality in a model of meme popularity. Phys. Rev. Lett. 112, 048701 (2014).

    ADS  PubMed  Google Scholar 

  9. Shew, W. L. et al. Adaptation to sensory input tunes visual cortex to criticality. Nat. Phys. 11, 659–663 (2015).

    CAS  Google Scholar 

  10. Drossel, B. & Schwabl, F. Self-organized critical forest-fire model. Phys. Rev. Lett. 69, 1629–1632 (1992).

    ADS  CAS  PubMed  Google Scholar 

  11. Dickman, R. Numerical study of a field theory for directed percolation. Phys. Rev. E 50, 4404–4409 (1994).

    ADS  CAS  Google Scholar 

  12. Muñoz, M. A., Grinstein, G., Dickman, R. & Livi, R. Critical behavior of systems with many absorbing states. Phys. Rev. Lett. 76, 451–454 (1996).

    ADS  PubMed  MATH  Google Scholar 

  13. Vespignani, A. & Zapperi, S. Order parameter and scaling fields in self-organized criticality. Phys. Rev. Lett. 78, 4793–4796 (1997).

    ADS  CAS  Google Scholar 

  14. Dornic, I., Chaté, H. & Muñoz, M. A. Integration of Langevin equations with multiplicative noise and the viability of field theories for absorbing phase transitions. Phys. Rev. Lett. 94, 100601 (2005).

    ADS  PubMed  Google Scholar 

  15. Henkel, M., Hinrichsen, H. & Lübeck, S. Non-equilibrium Phase Transitions: Absorbing Phase Transitions Vol. 1 (Springer, 2008).

  16. Field, S., Witt, J., Nori, F. & Ling, X. Superconducting vortex avalanches. Phys. Rev. Lett. 74, 1206–1209 (1995).

    ADS  CAS  PubMed  Google Scholar 

  17. Frette, V. et al. Avalanche dynamics in a pile of rice. Nature 379, 49 (1996).

    ADS  CAS  Google Scholar 

  18. Dickman, R., Muñoz, M. A., Vespignani, A. & Zapperi, S. Paths to self-organized criticality. Braz. J. Phys. 30, 27–41 (2000).

    ADS  Google Scholar 

  19. Altshuler, E. & Johansen, T. H. Experiments in vortex avalanches. Rev. Mod. Phys. 76, 471–487 (2004).

    ADS  CAS  Google Scholar 

  20. Bonachela, J. A. & Muñoz, M. A. Self-organization without conservation: true or just apparent scale-invariance? J. Stat. Mech. 2009, P09009 (2009).

  21. Watkins, N. W., Pruessner, G., Chapman, S. C., Crosby, N. B. & Jensen, H. J. 25 years of self-organized criticality: concepts and controversies. Space Sci. Rev. 198, 3–44 (2016).

    ADS  Google Scholar 

  22. Forster, D. Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions (Taylor & Francis, 1990).

  23. Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in two- dimensional systems. J. Phys. C 6, 1181 (1973).

    ADS  CAS  Google Scholar 

  24. Frisch, U. Turbulence: The Legacy of A. N. Kolmogorov (Cambridge Univ. Press, 1995).

  25. Berges, J., Rothkopf, A. & Schmidt, J. Nonthermal fixed points: effective weak coupling for strongly correlated systems far from equilibrium. Phys. Rev. Lett. 101, 041603 (2008).

    ADS  PubMed  Google Scholar 

  26. Hadzibabic, Z., Krüger, P., Cheneau, M., Battelier, B. & Dalibard, J. Berezinskii–Kosterlitz–Thouless crossover in a trapped atomic gas. Nature 441, 1118–1121 (2006).

    ADS  CAS  PubMed  Google Scholar 

  27. Murthy, P. A. et al. Observation of the Berezinskii–Kosterlitz–Thouless phase transition in an ultracold Fermi gas. Phys. Rev. Lett. 115, 010401 (2015).

    ADS  CAS  PubMed  Google Scholar 

  28. Tsatsos, M. C. et al. Quantum turbulence in trapped atomic Bose–Einstein condensates. Phys. Rep. 622, 1–52 (2016).

    ADS  MathSciNet  CAS  Google Scholar 

  29. Navon, N., Gaunt, A. L., Smith, R. P. & Hadzibabic, Z. Emergence of a turbulent cascade in a quantum gas. Nature 539, 72–75 (2016).

    ADS  CAS  PubMed  Google Scholar 

  30. Prüfer, M. et al. Observation of universal quantum dynamics in a spinor Bose gas far from equilibrium. Nature 563, 217–220 (2018).

    ADS  PubMed  Google Scholar 

  31. Erne, S., Bücker, R., Gasenzer, T., Berges, J. & Schmiedmayer, J. Observation of universal dynamics in an isolated one-dimensional Bose gas far from equilibrium. Nature 563, 225–229 (2018).

    ADS  CAS  PubMed  Google Scholar 

  32. Inouye, S. et al. Superradiant Rayleigh scattering from a Bose–Einstein condensate. Science 285, 571–574 (1999).

    CAS  PubMed  Google Scholar 

  33. Clark, L. W., Gaj, A., Feng, L. & Chin, C. Collective emission of matter-wave jets from driven Bose–Einstein condensates. Nature 551, 356 (2017).

    ADS  CAS  PubMed  Google Scholar 

  34. Ho, T.-L. Universal thermodynamics of degenerate quantum gases in the unitarity limit. Phys. Rev. Lett. 92, 090402 (2004).

    ADS  PubMed  Google Scholar 

  35. Lesanovsky, I. & Garrahan, J. P. Kinetic constraints, hierarchical relaxation, and onset of glassiness in strongly interacting and dissipative Rydberg gases. Phys. Rev. Lett. 111, 215305 (2013).

    ADS  PubMed  Google Scholar 

  36. Carr, C., Ritter, R., Wade, C. G., Adams, C. S. & Weatherill, K. J. Nonequilibrium phase transition in a dilute Rydberg ensemble. Phys. Rev. Lett. 111, 113901 (2013).

    ADS  CAS  PubMed  Google Scholar 

  37. Schempp, H. et al. Full counting statistics of laser excited Rydberg aggregates in a one- dimensional geometry. Phys. Rev. Lett. 112, 013002 (2014).

    ADS  CAS  PubMed  Google Scholar 

  38. Malossi, N. et al. Full counting statistics and phase diagram of a dissipative Rydberg gas. Phys. Rev. Lett. 113, 023006 (2014).

    ADS  CAS  PubMed  Google Scholar 

  39. Urvoy, A. et al. Strongly correlated growth of Rydberg aggregates in a vapor cell. Phys. Rev. Lett. 114, 203002 (2015).

    ADS  CAS  PubMed  Google Scholar 

  40. Valado, M. M. et al. Experimental observation of controllable kinetic constraints in a cold atomic gas. Phys. Rev. A 93, 040701 (2016).

    ADS  Google Scholar 

  41. Goldschmidt, E. A. et al. Anomalous broadening in driven dissipative Rydberg systems. Phys. Rev. Lett. 116, 113001 (2016).

    ADS  CAS  PubMed  Google Scholar 

  42. Simonelli, C. et al. Seeded excitation avalanches in off-resonantly driven Rydberg gases. J. Phys. B 49, 154002 (2016).

    ADS  Google Scholar 

  43. Letscher, F., Thomas, O., Niederprüm, T., Fleischhauer, M. & Ott, H. Bistability versus metastability in driven dissipative Rydberg gases. Phys. Rev. X 7, 021020 (2017).

    Google Scholar 

  44. Gutiérrez, R. et al. Experimental signatures of an absorbing-state phase transition in an open driven many-body quantum system. Phys. Rev. A 96, 041602 (2017).

    ADS  Google Scholar 

  45. Ates, C., Pohl, T., Pattard, T. & Rost, J. M. Many-body theory of excitation dynamics in an ultracold Rydberg gas. Phys. Rev. A 76, 013413 (2007).

    ADS  Google Scholar 

  46. Marcuzzi, M., Schick, J., Olmos, B. & Lesanovsky, I. Effective dynamics of strongly dissipative Rydberg gases. J. Phys. A 47, 482001 (2014).

    MathSciNet  MATH  Google Scholar 

  47. Klocke, K. & Buchhold, M. Controlling excitation avalanches in driven Rydberg gases. Phys. Rev. A 99, 053616 (2019).

    ADS  CAS  Google Scholar 

  48. Marcuzzi, M., Buchhold, M., Diehl, S. & Lesanovsky, I. Absorbing state phase transition with competing quantum and classical fluctuations. Phys. Rev. Lett. 116, 245701 (2016).

    ADS  PubMed  Google Scholar 

  49. Buchhold, M., Everest, B., Marcuzzi, M., Lesanovsky, I. & Diehl, S. Nonequilibrium effective field theory for absorbing state phase transitions in driven open quantum spin systems. Phys. Rev. B 95, 014308 (2017).

    ADS  Google Scholar 

  50. Pérez-Espigares, C., Marcuzzi, M., Gutiérrez, R. & Lesanovsky, I. Epidemic dynamics in open quantum spin systems. Phys. Rev. Lett. 119, 140401 (2017).

    ADS  PubMed  Google Scholar 

  51. Arias, A., Lochead, G., Wintermantel, T. M., Helmrich, S. & Whitlock, S. Realization of a Rydberg-dressed Ramsey interferometer and electrometer. Phys. Rev. Lett. 122, 053601 (2019).

    ADS  CAS  PubMed  Google Scholar 

  52. Dennis, G. R., Hope, J. J. & Johnsson, M. T. XMDS2: fast, scalable simulation of coupled stochastic partial differential equations. Comput. Phys. Commun. 184, 201–208 (2013).

    ADS  MathSciNet  CAS  Google Scholar 

  53. Bauke, H. Parameter estimation for power-law distributions by maximum likelihood methods. Eur. Phys. J. B 58, 167–173 (2007).

    ADS  CAS  Google Scholar 

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Acknowledgements

We acknowledge T. Ebbesen, G. Pupillo and M. Weidemüller for discussions. This work is supported by the Deutsche Forschungsgemeinschaft under WH141/1-1 and is part of and supported by the DFG Collaborative Research Centre ‘SFB 1225 (ISOQUANT)’, the Heidelberg Center for Quantum Dynamics, the European Union H2020 FET Proactive project RySQ (grant number 640378) and the ‘Investissements d’Avenir’ programme through the Excellence Initiative of the University of Strasbourg (IdEx). M.B. acknowledges support from the Alexander von Humboldt Foundation. S.D. acknowledges support by the German Research Foundation (DFG) through the Institutional Strategy of the University of Cologne within the German Excellence Initiative (ZUK 81) and the European Research Council via ERC grant agreement number 647434 (DOQS). S.W. was partially supported by the University of Strasbourg Institute for Advanced Study (USIAS), S.H. acknowledges support by the Carl Zeiss Foundation, A.A. and S.H. acknowledge support by the Heidelberg Graduate School for Fundamental Physics.

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Contributions

S.H., G.L. and S.W. devised the experiments; S.H., A.A., G.L. and T.M.W. acquired and analysed the data; M.B. and S.D. developed the theoretical description; all authors contributed to interpreting the results and writing of the manuscript.

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Correspondence to S. Whitlock.

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Peer review information Nature thanks Ronald Dickman 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 Further evidence for non-equilibrium universality.

a, Stationary-state density nf measured at t = 10 ms as a function of Ω2 and for different detunings Δ. b, The same data with rescaled axes to achieve full data collapse, revealing the scaling function (with fit shown by the dashed blue line) for the stationary density nf. Inset, normalized residuals between the rescaled data and the fitted scaling function. The dashed blue line corresponds to the simple scaling function used in the main text, and the solid orange line is a generalized scaling function that reproduces the asymptotic scaling form more accurately. Each data point corresponds to a single measurement.

Extended Data Fig. 2 Response of the SOC state to external perturbations.

a, Sketch of the experimental procedure used to measure the susceptibility \(\chi ={\rm{d}}{n}_{{\rm{f}}}/{\rm{d}}{\varOmega }_{{\rm{f}}}^{2}\) by quenching the spreading parameter κ Ω2 across the absorbing state phase transition. b, Experimental data corresponding to three different initial conditions corresponding to the absorbing phase (black circles), critical phase (brown triangles) and active phase (red squares). The solid lines correspond to predictions based on the experimentally determined scaling function, and the dotted lines correspond to mean-field predictions. Each data point corresponds to the average of eight measurements. For reference we show two representative error bars, corresponding to the standard error of the mean.

Supplementary information

Supplementary Information

This file contains the Derivation of the effective Langevin equation.

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Helmrich, S., Arias, A., Lochead, G. et al. Signatures of self-organized criticality in an ultracold atomic gas. Nature 577, 481–486 (2020). https://doi.org/10.1038/s41586-019-1908-6

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