Signatures of self-organized criticality in an ultracold atomic gas


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.

Data availability

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


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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.

Author information

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.

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.

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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).

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