Nature | Vol 577 | 23 January 2020 | 481Article
Signatures of self-organized criticality in an
ultracold atomic gas
S. Helmrich^1 , A. Arias1,2,3, G. Lochead1,2,3, T. M. Wintermantel1,2,3, M. Buchhold4,5,
S. Diehl^6 & S. Whitlock1,2,3*Self-organized criticality is an elegant explanation of how complex structures emerge
and persist throughout nature^1 , and why such structures often exhibit similar scale-
invariant properties^2 –^9. Although self-organized criticality is sometimes captured by
simple models that feature a critical point as an attractor for the dynamics^10 –^15 , the
connection to real-world systems is exceptionally hard to test quantitatively^16 –^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.Self-organized criticality (SOC) was conceptualized as a way to explain
the abundance of scale-invariant systems found in nature^1. It is thought
to underlie a range of complex dynamical phenomena, from activ-
ity in electrical circuits and neural networks^7 ,^9 , to the likelihood of
avalanches and earthquakes^2 as well as how forest fires^4 ,^10 , diseases^3 and
even ideas spread^8. However, despite the fundamental and practical
importance of SOC phenomena, much-needed controlled experiments
are hindered by numerous complexities concerning the relevant micro-
scopic degrees of freedom^16 –^19 and even the simplest models (beyond
mean-field approximations) present serious challenges to theory^11 –^15 ,^20.
SOC can be understood as an organizing principle that governs a
class of dissipative interacting systems that display three key signa-
tures: (1) self-organization to a stationary state (bringing observables
to values that are independent of initial conditions); (2) scale invariance
of spatio-temporal correlation functions, including bulk observables;
and (3) critical response to small perturbations, usually encountered in
the form of avalanches that have a broad range of sizes and durations
and that are described by power-law distributions. This differs from
an equilibrium phase transition, where scale invariance and a critical
response ensue only for a fine-tuned parameter set. The common root
of these emergent SOC properties is that the respective gap (that is,
the distance in parameter space from the critical state) is replaced
by a ‘dynamical gap’ that self-tunes to zero by an intrinsic feedback
mechanism. This property, and signatures (1)–(3), set SOC apart from
other occurrences of non-equilibrium scaling behaviour—such as
hydrodynamic long-time tails^22 , the Kosterlitz–Thouless critical phase
in two-dimensional quantum fluids^23 and the transient dynamics of
turbulent cascades in isolated systems^24 ,^25 —which have also been
studied with ultracold atoms^26 –^31 ; see also related experiments on
superradiance^32 ,^33 and scaling in unitary Bose gases^34.
In this work, we demonstrate signatures (1)–(3) of SOC in a micro-
scopically well-controlled physical system: a three-dimensional trapped
gas of ultracold potassium atoms driven to highly excited Rydberg
states by a laser field (Fig. 1a). The ingredient that leads to SOC is the
slow, irreversible decay of the excited population to auxiliary inac-
tive states, which has been largely disregarded in the investigation of
Rydberg many-body dynamics. This enables the observation of a phase
transition from a self-organizing active phase to an absorbing phase,
scale-invariance of the self-organized density and large fluctuations
of the active density in the form of power-law distributed avalanches.
Beyond these experimental results, we derive a Langevin equation from
the underlying microscopic many-body quantum master equation that
governs driven–dissipative Rydberg dynamics, which coincides with
one of the emblematic classes of SOC models^20. This provides the crucial
link from the microscopic atomic physics to the observed macroscopic
SOC phenomenology, and establishes ultracold Rydberg atomic gases
as a widely tunable and theoretically accessible platform for studying
self-organization and universality in non-equilibrium dynamics.Physical system
Each of the approximately 10^5 atoms held in the optical trap can be
represented by a three state system: a ground state |g⟩ = |4s1/2, F = 1⟩, an
excited Rydberg state |r⟩, and auxiliary removed states, which we refer
to collectively by |0⟩ (Fig. 1b). The laser field drives the |g⟩ → |r⟩ transi-
tion with a fixed detuning Δ from resonance and with an amplitude
parameterized by the Rabi frequency Ω. In our experiments Δ ≫ Ω, such
that single-atom excitation processes are strongly suppressed. Once
excited, however, atoms can facilitate further excitations (when the
laser detuning is compensated by the interaction energy of Rydberghttps://doi.org/10.1038/s41586-019-1908-6
Received: 25 June 2018
Accepted: 23 October 2019
Published online: 15 January 2020
(^1) Physikalisches Institut, Universität Heidelberg, Heidelberg, Germany. (^2) Institut de Science et d’Ingénierie Supramoléculaires (ISIS, UMR 7006), University of Strasbourg and CNRS, Strasbourg,
France.^3 Institut de Physique et Chimie des Matériaux de Strasbourg (IPCMS, UMR 7504), University of Strasbourg and CNRS, Strasbourg, France.^4 Department of Physics, California Institute of
Technology, Pasadena, CA, USA.^5 Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USA.^6 Institut für Theoretische Physik, Universität zu Köln,
Cologne, Germany. *e-mail: [email protected]