Article
the onsets for critical (brown triangles) and active (red squares) initial
states both coincide at δ ≈ 0 within the experimental resolution. We
can compare these data to a prediction of the susceptibility obtained
from a derivative of the experimentally determined scaling function
using β = 0.910. The scaling-function predictions (solid lines in Extended
Data Fig. 2) are in good agreement with the data, whereas mean-field
predictions (dotted lines) systematically fail to capture the widths and
heights of the peaks. However, we note that starting from the initial
active state, the measured response is narrower and slightly stronger
than the scaling-function prediction. This is further evidence that the
system evolves towards a state that is sharply concentrated at the crit-
ical point, instead of one that is a statistical mixture of many different
accessible macro states. From these experiments we confirm that when
starting from a supercritical state (irrespective of the precise value of
Ωi > Ωc), the system self-organizes to a critical state that is characterized
by a vanishing excitation gap—underpinning SOC signature (3).
Fitting the generalized scaling function to the rescaled data yields
β = 0.95(3), where the larger statistical uncertainty reflects the fact that
the generalized model function has more parameters. This is close to
the value β = 0.910(4) determined from the density-dependent data in
the main text. Refitting the density-dependent data with the general-
ized scaling function yields β = 0.920(7). This shows that although the
full form of the scaling function is not universal, data taken under very
different conditions concerning initial densities and detuning of the
driving field do in fact share a common universal critical exponent
describing the asymptotic scaling regime.
Role of trap inhomogeneities and residual coherence
We can also rule out possible modifications to the scaling behaviour
due to other experimental details such as the inhomogeneous density
or residual effects of quantum coherence.
Inhomogeneities. In the experiment the atoms are laser-trapped in a
cylindrical geometry of finite diameter and length, causing a nearly
homogeneous density distribution in the trap centre and smooth
variation of nt at the boundaries. nt smoothly follows the Gaussian
trapping profile of the lasers. To estimate the impact of inhomogenei-
ties, on this basis we now study a local density approximation for the
Langevin equation. In this approximation, ρ(r, t) experiences a constant
background density n(,rrtn)=∼(,tI)(r) at each point in space r, which
is modulated by the trapping profile I(r), whereas n∼(,rt) only incorpo-
rates fluctuations owing to the coupling to ρt. An appropriate mean-
field theory considers ρt=(Vρ−1∫V r,)t as the spatially averaged den-
sity over the system volume V, and ∼nt=0=n 0 in the absence of fluctua-
tions. The corresponding spatially averaged SOC line is located at
Ωc^2 ∝=κΓc0(/nI)[∫V1/()r]∝n 0 −1, demonstrating that the mean-field
exponent β = 1 is not modified by the inhomogeneous geometry.
Quantum coherence. The evolution of the density averages
(Supplementary Equations (S4) and (S5)) is real and linear in time,
which maps the final Langevin equation to a stochastic differential
equation for classical processes. It incorporates strong, classical cor-
relations between different atoms but lacks the possibility for long
range coherence. Coherence between different atoms can be system-
atically built-in by replacing the adiabatic elimination (Supplemen-
tary Equation (S1)) with the exact solution, which amounts to a shift
Γ → Γ + ∂t (Supplementary Equation (S5)). To leading order it intro-
duces a coherent contribution, (+Γγde)∂−1tl^2 m, to the right-hand side
of Supplementary Equation (S5), where ml is the probability for having
an excited particle within the coarse grained region l corresponding
to the characteristic facilitation volume. Analogously to a damped
harmonic oscillator, this evolution is observable on timescales
t(Γ + γde) ≤ 1, but washed out on larger timescales, that is, on the re-
laxation towards the SOC steady state. Fast coherent processes might
modify the parameters κ, D and τ, but not the structure of the Langevin
equation.Data availability
The data that support the findings of this study are available from the
corresponding author upon reasonable request.- 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). - 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). - Bauke, H. Parameter estimation for power-law distributions by maximum likelihood
methods. Eur. Phys. J. B 58 , 167–173 (2007).
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.Author 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.Competing interests The authors declare no competing interests.Additional information
Supplementary information is available for this paper at https://doi.org/10.1038/s41586-019-
1908-6.
Correspondence and requests for materials should be addressed to S.W.
Peer review information Nature thanks Ronald Dickman and the other, anonymous, reviewer(s)
for their contribution to the peer review of this work.
Reprints and permissions information is available at http://www.nature.com/reprints.