Abstract
Whether it be physical, biological or social processes, complex systems exhibit dynamics that are exceedingly difficult to understand or predict from underlying principles. Here we report a striking correspondence between the excitation dynamics of a laser driven gas of Rydberg atoms and the spreading of diseases, which in turn opens up a controllable platform for studying nonequilibrium dynamics on complex networks. The competition between facilitated excitation and spontaneous decay results in subexponential growth of the excitation number, which is empirically observed in real epidemics. Based on this we develop a quantitative microscopic susceptibleinfectedsusceptible model which links the growth and final excitation density to the dynamics of an emergent heterogeneous network and rare active region effects associated to an extended Griffiths phase. This provides physical insights into the nature of nonequilibrium criticality in driven manybody systems and the mechanisms leading to nonuniversal powerlaws in the dynamics of complex systems.
Introduction
The dynamical behavior of an exceptionally diverse spectrum of realworld systems is governed by critical events and phenomena occurring on vastly different spatial and temporal scales. A disease outbreak, for example, can be very sensitive to the type of disease and the behavior of individuals, yet epidemics generically feature a characteristic time dependence^{1} that emerges from the connections within and between communities^{2,3}. In studying these systems, complex networks provide a crucial layer of abstraction to bridge the behavior of individuals and the macroscopic consequences^{4}. Accordingly, they have found applications not only in biology and the study of epidemics^{2}, but also in informatics^{5}, marketing^{6}, finance^{7}, and traffic flow^{8}. An overarching challenge in these fields is to find general principles governing complex system dynamics and to pinpoint how apparent universal characteristics emerge from the underlying network structure.
In this work, we address this challenge using a highly controllable complex system that consists of a trapped ultracold atomic gas continuously driven to strongly interacting Rydberg states by an offresonant laser field (Fig. 1). Our main findings include: first, the rapid growth of excitations driven by a competition between microscopic facilitated excitation and decay processes (playing the role of the transmission of an infection and recovery, respectively). The observed dynamics follow a powerlaw time dependence that parallels that which is empirically observed in realworld epidemics, providing a powerful demonstration of universality reaching beyond physics. Second, a full description and interpretation of the experiment in terms of an emergent susceptibleinfectedsusceptible (SIS) network linking the observed macroscopic dynamics to the microscopic physics. Third, the unexpected presence of rare region effects and a dynamical Griffiths phase^{9,10,11} associated with the emergent network structure, which gives rise to critical dynamics over an extended parameter regime and explains the appearance of powerlaw growth and relaxation, but with nonuniversal exponents.
Results
Microscopic ingredients for an epidemic
The microscopic processes governing the dynamics of ultracold atoms driven to Rydberg states by an offresonant laser field, shown in Fig. 1a, b, bear close similarities to those in epidemics^{12}. Each atom can be considered as a twolevel system consisting of the atomic ground state (gray disks, healthy) and an excited Rydberg state (blue disks, infected). An excited atom can spontaneously decay (recovery, with rate Γ), or it can facilitate the excitation of other atoms (transmission of the infection, with rate κ) that satisfy certain constraints linked to their positions and velocities. This results in rapid spreading of the excitations through the gas (depicted by growing excitation clusters in Fig. 1a)^{13,14,15,16,17}.
Our experimental studies start from an ultracold thermal gas of 3 × 10^{4} potassium39 atoms in their ground state \(\leftg\right\rangle =4{s}_{1/2}\), which are held in a twodimensional optical trap with a peak atomic density n_{2D}(x = y = 0) = 0.76 μm^{−2} (Fig. 1a) and e^{−1/2} Gaussian widths of the atomic distribution of σ_{x} = 125 μm and σ_{y} = 50 μm. To trigger the dynamics at t = 0 we apply a lowintensity laser pulse tuned in resonance with the \(\leftg\right\rangle \to \leftr\right\rangle =66{p}_{1/2}\) transition for a duration of 4 μs. This produces around eight seed excitations at random positions within the gas. The laser is then suddenly detuned from resonance by Δ = −30 MHz and adjusted in intensity. This makes it possible to facilitate secondary excitations at an average distance R_{fac} = 3.5 μm (illustrated by red circles), corresponding to the distance where the dominant Rydberg pairstate energy compensates the laser detuning. The twobody facilitation rate κ is proportional to the laser intensity which can be tuned over a wide range. This can be understood as an effective rate averaged over the different excitation channels corresponding to many Rydberg pairstate interaction potentials. We therefore experimentally calibrate κ based on the characteristic doubling time measured at very early times (see Methods). For the following measurements we choose different values of κ ranging from 3.3 to 10 kHz. Additionally, Rydberg excitations spontaneously decay after a characteristic lifetime τ = (2πΓ)^{−1} with a calculated rate Γ = 0.84 kHz (including blackbody decay). There is also a strong dephasing of the groundRydberg transition with estimated rate γ_{de} ≳ 200 kHz. This implies that in the experimentally relevant regime t ≳ 1 μs, the dynamics can be well described by classical transition rates. Spontaneous (offresonant) excitation events are very rare, observed in an independent experiment without triggered seed excitations to be ≲1 kHz integrated over the whole cloud. To observe the system we measure the total number of Rydberg excitations present in the gas using field ionization and a microchannel plate (MCP) detector for different exposure times t up to 2 ms. Although each atom is identical and its evolution is captured by these simple excitation rules, the competition between facilitated excitation and decay gives rise to complex dynamical phases and evolution^{18,19,20,21,22,23}. However, the full manybody system is even more complex: 3 × 10^{4} multilevel atoms moving in space with random positions and velocities while interacting with the laser field and each other, which makes it challenging to connect the microscopic physics to the macroscopic excitation dynamics^{22,24,25}.
Observation of epidemic growth
To exemplify the analogy to epidemics, in Fig. 1d we present data for κ = 10 kHz showing different stages of the dynamics. Immediately following the seed excitation pulse we observe a period of very fast growth of the Rydberg excitation number, that is, within the Rydberg state lifetime the excitation number increases from its initial value to more than 400, corresponding to more than five doublings in 0.19 ms. At around t ≈ 0.5 ms, after the initial growth stage, the system saturates with a high constant excitation number (i.e., an endemic state). However, the saturation value is still significantly lower than the estimated maximum number of excitations that can fit in the system ≳2000 assuming an interRydberg spacing of ~R_{fac}. On even longer timescales than those studied here (≳10 ms), the system should eventually relax back to an absorbing or selforganized critical state due to the gradual depletion of particles^{23,26}.
The growth phase of many real epidemics is observed to follow a characteristic powerlaw dependence described by the phenomenological generalizedgrowth model (GGM)^{1},
This describes a relation between incidence rate \(C^{\prime}\) and cumulative number of infections \(C=\mathop{\int}\nolimits_{0}^{t}C^{\prime} (t^{\prime} )dt^{\prime}\), where r is the growth rate at early times and p is the “deceleration of growth” which is an important parameter in classifying epidemics^{1}. Exponential growth in time is characterized by p = 1, while p < 1 corresponds to powerlaw growth ∝ t^{η} with η = p/(1 − p).
In Fig. 2a we represent the data from Fig. 1d in terms of \(C^{\prime}\) (instantaneous number of excitations divided by their lifetime τ = (2πΓ)^{−1}) against its time integral C, shown by the darkest green data points. This clearly shows that the incidence rate follows the GGM over several decades (evidenced by a straight line on a double logarithmic scale) with a deceleration of growth parameter p = 0.59(1) that is comparable to empirical observations of real epidemics^{1}. In fact powerlaw growth with varying exponents p < 0.6 is a general feature of the system dynamics, as seen in Fig. 2a for different κ values from 3.3 to 10 kHz (depicted with different colors), together with the corresponding p values plotted in Fig. 2c as determined from fits to the initial growth stage (see Supplementary Note 1). This is to be contrasted with exponential growth (p = 1, solid black line with a steeper slope). We also point out that each curve saturates at a different κdependent value, with some curves showing evidence for slow relaxation back toward zero incidences (the lowest three curves in Fig. 2a). In the study of epidemics, powerlaw growth with p < 1 is commonly associated with a few underlying mechanisms, most prominently spatial constraints and heterogeneity in the underlying network structure^{1}. In the following we use this insight to develop a spatial network model which quantitatively describes the experimental observations and can be directly linked to the microscopic details of the system, something that is rarely possible for real epidemics.
Emergent heterogeneous network
To explain the experimental observations we develop a physically motivated SIS network model. We assume that the twodimensional gas can be subdivided into cells that represent nodes of a network (Fig. 1c). Each cell i can either be in a susceptible state (absence of Rydberg excitation, I_{i} = 0) or infected (one Rydberg excitation, I_{i} = 1), and contains a certain number of particles N_{i} that can be excited. Vacant cells with N_{i} = 0 (and hence also I_{i} = 0) translate to unconnected, missing nodes. The probability for a given node i to become infected is described by the following stochastic master equation^{2}
where E[⋅] denotes the expectation value. The node weights N_{i} and the adjacency matrix a_{ij} together determine the probability for transmission of an infection from cell j to i. In the special case \({N}_{i}={\rm{const.}},{a}_{ij}=1\), this reduces to the wellstudied homogeneous compartmental model^{2} that exhibits exponential growth. For \({N}_{i}={\rm{const.}}\) and assuming a regular lattice with nearestneighbor transmission, this model is equivalent to directed percolation concerning its universal properties^{27}. However, spatially structured adjacency matrices can give rise to more complex spatiotemporal evolution^{28}.
To define the adjacency matrix entries a_{ij} we coarse grain our system into hexagonal cells (each with area \(\sim \!{R}_{{\rm{fac}}}^{2}\)), corresponding to a triangular network of nodes, where a_{ij} = 1 for each of the six nearest neighbors j to each node i and a_{ij} = 0 for all other nodes j. This is motivated by the fact that hexagonal packing provides the densest possible tiling of strongly interacting Rydberg excitations in twodimensional space^{29}, although the underlying atomic gas has no such apparent structure. The N_{i} are sampled from a Poissonian distribution with a spatially dependent mean \({\mu }_{i}=\epsilon (\kappa ){n}_{2{\rm{d}}}({x}_{i},{y}_{i}){R}_{{\rm{fac}}}^{2}\), where ϵ(κ) < 1 is the accessible phase space fraction for facilitated excitation (a free parameter, elaborated on below) and n_{2d}(x_{i}, y_{i}) is the twodimensional Gaussian density distribution of atoms in the trap. Thus, Eq. (2) describes a heterogeneous network where each node has a (spatially) fluctuating weighted degree s_{i} = ∑_{j}a_{ij}N_{j} with a mean and variance approximately equal to 6μ_{i}.
To numerically simulate this model we solve Eq. (2) using a MonteCarlo approach^{30}. In each time step we compute the transition rate for each node R_{i} = κN_{i}(1 − I_{i})∑_{j}a_{ij}I_{j} + ΓI_{i}. One node m is then picked at random according to the weights R_{i} and its state is flipped I_{m} → 1 − I_{m}. The timestep is computed according to \(dt={\mathrm{ln}}\,(X)/(2\pi {\sum }_{i}R_{i})\), where \(\mathrm{ln}\,\) is the natural logarithm and X is sampled from a uniform distribution on [0, 1). For the initial state we consider a fixed number of ∑_{i}I_{i}(t = 0) = 8 seed excitations randomly distributed among the nodes according to their weights N_{i}.
The numerical simulations, shown as solid curves in Figs. 1d and 2, are in excellent agreement with the experimental observations. Importantly, they fully reproduce the fast powerlaw growth with p < 0.6, the different plateau heights, and even the latetime relaxation as a function of κ. The only free parameter in the model is ϵ(κ) which is adjusted for each curve and is found to be a monotonically increasing function of κ with 0.02 < ϵ(κ) < 0.1 over the explored parameter range. This parameter directly controls the network structure, that is, for κ = 10 kHz the network consists of M ≈ 2300 nodes with N_{i} > 0 and the local s_{i} follow approximately Poissonian distributions with 〈s_{i}〉 = var(s_{i}) ≤ 5.3 (maximal at trap center) while for κ = 3.3 kHz, M ≈ 660, and 〈s_{i}〉 = var(s_{i}) ≤ 1.3 (see Supplementary Note 2). For comparison, the dashed blue lines in Fig. 2 show comparable simulations with N_{i} = μ_{i}, that is, corresponding to a locally homogeneous network with the same average node degree. These homogeneous network simulations show faster initial growth, constant p values ≈ 0.7, higher plateaus saturating at the system size limit, and a dramatic shift of the critical point to lower κ values, which are inconsistent with the experimental data. The good agreement between experiment and heterogeneous network simulations demonstrates that the emergent macroscopic dynamics of the system crucially depend on the weighted node degree distributions and heterogeneity controlled by the atomic density and the parameter ϵ(κ).
The heterogeneous spatial network model described by Eq. (2) provides an accurate and computationally efficient coarsegrained description of the physical system and its dynamics involving just a few microscopically controlled parameters. The importance of heterogeneity is particularly surprising since atomic motion could be expected to quickly wash out the effects of spatial disorder (the characteristic thermal velocity corresponding to the gas temperature at 20 μK is v_{th} = 65 μm/ms ≈3.5R_{fac}/τ). Our findings can be explained by assuming that the facilitation constraint depends on both the relative positions and velocities of the atoms. Taking into account the Landau–Zener transition probability for moving atoms confirms that only atom pairs with small relative velocities v_{LZ} ≲ 1 μm/ms contribute to the spreading of facilitated excitations^{22}. This provides a qualitative explanation for the inferred ϵ(κ) ≪ 1 and its approximate κ dependence (due to the intensity dependence of v_{LZ}) (see Methods section). It also sets the timescale for diffusion in phase space longer than the duration of our observations ≳ 2 ms. Thus, spatial constraints and (effectively static) heterogeneity can be understood as properties of an emergent network structure that is dynamically formed while the laser coupling is on (see also ref. ^{31} for a related interpretation of excitation dynamics on smaller preformed emergent lattices).
Spatial disorder is known to play a very important role in condensed matter systems, giving rise to new manybody phases, localization effects, and glassy behavior^{32}. There is still much to be explored concerning analogous effects of disorder and heterogeneity on nonequilibrium processes on networks. One key theoretical finding however is the emergence of an exotic Griffiths phase^{9,10,11}, expected to replace the singular critical point between the subcritical and active phase by an extended critical phase. The dynamics in the Griffiths phase can be understood in terms of the dynamics of rare supercritical clusters (κN_{i} ≳ Γ for all sites i of the cluster), surrounded by subcritical regions. For a Poissonian degree distribution the probability for a seed to land on a supercritical cluster of M_{clust} nodes is \(p({M}_{{\rm{clust}}}) \sim \exp (x{M}_{{\rm{clust}}})\), where x is a positive function of ϵ_{i}. The excitation number in such clusters will grow initially up to a typical lifetime \(\tau ({M}_{{\rm{clust}}}) \sim \exp (y{M}_{{\rm{clust}}})\), for some positive constant y, after which it will decay due to rare fluctuations^{11}. This yields an incidence \(C^{\prime} (t)={\sum }_{{M}_{{\rm{clust}}}}p({M}_{{\rm{clust}}})\exp (t/\tau ({M}_{{\rm{clust}}})) \sim {t}^{\alpha }\) with a nonuniversal decay exponent α = −x/y (dependent on ϵ_{i}). This can lead to slow relaxation and strong modifications to the nonequilibrium critical properties (e.g., powerlaw correlations with continuously varying exponents^{10}).
Such Griffiths effects provide a natural explanation for several of our experimental observations. First of all, the relatively short time for each curve to reach the plateau and the strong κ dependence of the plateau heights are compatible with the presence of rare regions with an aboveaverage infection rate that span only a fraction of the entire system, controlled by the disorder strength entering via ϵ(κ). This also explains the sizable shift of the critical point between subcritical (\({C}^{\prime}\approx 0\)) and active (\({C}^{\prime}\,> \, 0\)) phases to higher values of κ as compared to the expectation for a locally homogeneous system seen in Fig. 2b. Finally, we point out the slow relaxation of the subcritical curves in Fig. 2a. These curves are compatible with powerlaw decays with disorder dependent relaxation exponents α < 0, which is the defining characteristic of the Griffiths phase^{11}. While these experiments were limited to relatively short times <2 ms (to minmize the impact of particle loss), the numerical simulations confirm powerlaw relaxation over two orders of magnitude in time, depicted by solid lines in Fig. 2a. These relaxation exponents α were obtained from fitting an extended GGM (see Methods Eq. (3)) to the data, with corresponding α ≤ 0 values shown in Fig. 2c. On this basis we find that powerlaw growth (with 0.5 ≤ p ≤ 0.6) is associated with the transition from a Griffiths phase to an active phase (for κ > 6 kHz coinciding with α ≈ 0), whereas the absorbing state phase transition without disorder should occur for κ ≲ Γ^{11}.
Discussion
This work highlights a controllable physical platform for experimental network science situated at the interface between simplified numerical models and empirical observations of realworld complex dynamical phenomena. Ultracold atoms provide the means to introduce and control different types of reactiondiffusion processes as studied here, but also to realize different types of spatial networks by structuring the trapping fields^{33} and to access the full spatiotemporal evolution of the system^{29}. This would allow for indepth investigations of the phase structure and critical properties with varying disorder strength and different network geometries. Our discovery that the growth dynamics of a drivendissipative atomic gas is described by an emergent heterogeneous network that is relatively robust to particle motion suggests that similar effects could also be observable in noisy room temperature environments^{16,26}. Thus, heterogeneous network dynamics and Griffiths effects may arise naturally in very different nonequilibrium systems, having important implications, for example, in understanding nonequilibrium criticality without fine tuning^{23,26,34} and for finding effective strategies for controlling dynamics on complex networks^{35}. Future experiments could also investigate the quantum contact process^{12,36,37,38} and quantum analogs of the Griffiths phase on heterogeneous networks^{39}.
Methods
Experimental sequence and calibration of parameters
The experimental procedure to observe Rydberg excitation growth consists of three main steps, during which we keep the optical trap on. Initially a small number of seed excitations are prepared at random positions in the gas. For this we keep the laser frequency fixed at Δ = −30 MHz below the zerofield resonance and briefly applying an electric field of 0.28 V/cm for 4 μs, exploiting the DC Stark effect to tune the atoms into resonance. The laser is then momentarily switched off for 6 μs to ensure the electric field is fully off before starting the offresonant driving. Next we apply the offresonant laser field which causes rapid growth of the number of excitations in the gas. We calibrate the singleatom facilitation rate κ against a measurement of the initial growth rate r = 27(8) kHz, measured for high intensity and very short times t ≪ τ where manybody effects can be safely neglected. This is then divided by an estimate of the (cloud averaged) mean number of particles that meet the facilitation condition \(\bar{\mu }=2.7\) assuming each seed excitation is isolated. The latter is estimated from the detailed experimenttheory comparison to the spatial SIS model presented in the manuscript. After a variable exposure time t we measure the total number of excitations in the gas. For this we switch on a large electric field to ionize the Rydberg states and guide the ions onto an MCP detector. The conversion factor from integrated MCP voltage to the number of Rydberg excitations is calibrated against an independent absorption measurement of the number of particles removed from the gas after a long exposure assuming each Rydberg excited atom is eventually lost from the trap with rate Γ.
Extended generalizedgrowth model (GGM)
To extract both the growth and relaxation parameters from these data, we extend the GGM to allow for different exponents in the growth and relaxation phases
In this equation p is the deceleration of growth parameter, α is the powerlaw exponent for the latetime recovery phase. K and β determine the location and sharpness of the crossover. Curves with α < 0 will eventually recover (i.e., number of excitations decreases to zero) while α = 0 describes an endemic state. The endemic state is characterized by a constant number of excitations \(C^{\prime} =r{K}^{p}\). The number of excitations at the crossover point (C = K) is \(C^{\prime} =r{K}^{p}/{2}^{\beta }\).
Landau–Zener probability for facilitated excitation
By comparing the SIS network simulations to the data, we infer that the fraction of atoms that participate in the excitation dynamics is relatively small. This is quantified by the fitted ϵ(κ) values that vary between 0.023 and 0.094 (for κ = 3.3 kHz and κ = 10 kHz, respectively). These small values of ϵ and the approximate κ dependence can be explained by the velocity dependence of the Landau–Zener transition probability, which restricts facilitation to atoms with small relative velocities v ≲ v_{LZ} ≪ v_{th} (for a related calculations see Appendix E in ref. ^{22}). The Landau–Zener velocity can be expressed as \({v}_{{\rm{LZ}}}={\pi }^{2}{\Omega }^{2}/\dot{V}\), where Ω is the lightmatter coupling strength and \(\dot{V}\) is the slope of the RydbergRydberg interaction potential evaluated at the facilitation radius^{22}. ϵ can be understood as the number of atoms that can be facilitated in a neighboring cell within the Rydberg state lifetime divided by the mean number of atoms in each cell \({n}_{2d}{R}_{fac}^{2}\). The flux of atoms passing through a 1/6 segment of the facilitation shell is Φ = πR_{fac}n_{2d}v_{th}/3. However, only a fraction of these atoms \({f}_{v}\approx {v}_{{\rm{LZ}}}/\sqrt{\pi }{v}_{{\rm{th}}}\) fulfill the Landau–Zener condition with relative velocity ∣v∣ < v_{LZ}. Combining the above gives \(\epsilon =\Phi \tau {f}_{v}/{n}_{2d}{R}_{fac}^{2}\approx \sqrt{\pi }\tau {v}_{{\rm{LZ}}}/(3{R}_{fac})\).
For realistic experimental parameters R_{fac} = 3.5 μm, \(\dot{V}=1\times 1{0}^{5}\) kHz μm^{−1} (ref. ^{40}) and Ω ~100 kHz, we find v_{LZ} = 1 μm/ms. This is small compared to the thermal velocity v_{th} = 65 μm/ms for T = 20 μK. Inserting these parameters into the expression above for the phase space fraction yield \(\epsilon =0.03{\left(\frac{\Omega }{100{\rm{kHz}}}\right)}^{2}\). This simple estimate falls within the range of values inferred from the experimenttheory comparison, even though it still does not account for all the microscopic experimental details, for example, multiple excitation resonances associated with Zeeman substructure or possible mechanical forces between the atoms.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The code to produce the simulation data that support the findings of this study is available from the corresponding author upon reasonable request.
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Acknowledgements
We acknowledge valuable discussions with Cédric Sueur. This work is supported by the “Investissements d’Avenir” program through the Excellence Initiative of the University of Strasbourg (IdEx), the University of Strasbourg Institute for Advanced Study (USIAS), and is part of and supported by the DFG SPP 1929 GiRyd and the DFG Collaborative Research Center “SFB 1225 (ISOQUANT).” T.M.W., S.S., and M.M. acknowledge the French National Research Agency (ANR) through the Programme d’Investissement d’Avenir under contract ANR17EURE0024. M.M. acknowledges QUSTEC funding from the European Union’s Horizon 2020 research and innovation program under the Marie SkłdowskaCurie Grant Agreement No. 847471. S.D. acknowledges support by the European Research Council (ERC) under the Horizon 2020 research and innovation program, Grant Agreement No. 647434 (DOQS).
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T.M.W. and S.W. devised and performed the experiments and analyzed the data. T.M.W., M.B., S.D., and S.W. contributed to the theoretical understanding. S.S., M.M., Y.W., and G.L. made contributions to the experimental setup. All authors contributed to interpreting the results and writing of the manuscript.
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Wintermantel, T.M., Buchhold, M., Shevate, S. et al. Epidemic growth and Griffiths effects on an emergent network of excited atoms. Nat Commun 12, 103 (2021). https://doi.org/10.1038/s41467020203337
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DOI: https://doi.org/10.1038/s41467020203337
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Nature Physics (2022)
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