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arXiv:2404.16523 · Physical Review Letters

Anomalous Directed Percolation on a Dynamic Network Using Rydberg Facilitation

Daniel Brady, Simon Ohler, Johannes Otterbach, Michael Fleischhauer

Published Oct 22, 2024• 4 citations
arXivHTMLPDF

Abstract

The facilitation of Rydberg excitations in a gas of atoms provides an ideal model system to study epidemic evolution on (dynamic) networks and self-organization of complex systems to the critical point of a nonequilibrium phase transition. Using Monte Carlo simulations and a machine learning algorithm we show that the universality class of this phase transition can be tuned but is robust against decay inherent to the self-organization process. The classes include directed percolation (DP), the most common class in short-range spreading models, and mean-field (MF) behavior, but also different types of anomalous directed percolation (ADP), characterized by rare long-range excitation processes. In a frozen gas, ground state atoms that can facilitate each other form a static network, for which we predict DP universality. With atomic motion the network becomes dynamic by long-range (Lévy-flight type) excitations. This leads to continuously varying critical exponents, varying smoothly between DP and MF values, corresponding to the ADP universality class. These findings also explain the recently observed critical exponent of Rydberg facilitation in an ultracold gas experiment [Helmrich et al., Nature (London) 577, 481 (2020)NATUAS0028-083610.1038/s41586-019-1908-6], which was in between DP and MF values.

DOI
10.1103/physrevlett.133.173401

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The facilitation of Rydberg excitations in a gas of atoms provides an ideal model system to study epidemic evolution on (dynamic) networks and self-organization of complex systems to the critical point of a nonequilibrium phase transition. Using Monte Carlo simulations and a machine learning algorithm we show that the universality class of this phase transition can be tuned but is robust against decay inherent to the self-organization process. The classes include directed percolation (DP), the most common class in short-range spreading models, and mean-field (MF) behavior, but also different types of anomalous directed percolation (ADP), characterized by rare long-range excitation processes. In a frozen gas, ground state atoms that can facilitate each other form a static network, for which we predict DP universality. With atomic motion the network becomes dynamic by long-range (Lévy-flight type) excitations. This leads to continuously varying critical exponents, varying smoothly between DP and MF values, corresponding to the ADP universality class. These findings also explain the recently observed critical exponent of Rydberg facilitation in an ultracold gas experiment [Helmrich et al., Nature (London) 577, 481 (2020)NATUAS0028-083610.1038/s41586-019-1908-6], which was in between DP and MF values.
Understanding the universal properties of nonequilibrium phase transitions of spreading processes is a challenging problem. This applies in particular to irregular and dynamically varying networks. We here investigate an experimentally accessible model system for such processes, namely, the absorbing-state phase transition (ASPT) of Rydberg-excitation spreading, known as Rydberg facilitation, in a laser-driven gas of mobile atoms. It occurs on an irregular graph, set by the random atom positions in the gas, and, depending on temperature, changes its character from static to dynamic. By studying the behavior of the order parameter in the work of Brady . [], we showed numerical evidence for a crossover from directed percolation (DP) universality through various phases of anomalous directed percolation (ADP) to mean-field (MF) behavior when the temperature of the gas is increased. As the behavior of the order parameter is not sufficient to uniquely determine the universality class, we here analyze the distribution of avalanches—characteristic of nonequilibrium critical behavior—to fully characterize the ASPT. Performing extended numerical calculations and experiments on a cold Rb 87 atom gas, we confirm our earlier numerical findings and our phenomenological model that maps the dynamic network to a static one with power-law tails of the distribution of excitation distances. Furthermore, we discuss the influence of dissipation, present in the experiment and a necessary ingredient for the self-organization of the system to the critical point. In particular, we study the potential modification of the universality class by losses as a function of dissipation strength.
From financial markets and neuronal activity in the brain to the way forest fires and diseases spread, the dynamics of complex systems are governed by critical events and emergent phenomena that are exceedingly difficult to understand or predict from underlying principles. We recently discovered that a gas of ultracold atoms continuously driven to strongly-interacting Rydberg states by an off-resonant laser field displays all the hallmarks of complex systems dynamics in a highly-controllable experimental system: (i) At early times we observe rapid growth of Rydberg excitations that has a striking correspondence with the spreading of diseases empirically observed in epidemics [1]; (ii) At later times we find that the system evolves toward a self-organised critical (SOC) state [2], a phenomenon that has been conjectured to explain the abundance of scale-invariant systems found in nature. I will discuss how these experiments can be understood in terms of an emergent atomic network that bridges the gap between mathematical models and empirical observations. This provides the opportunity to identify general principles governing non-equilibrium dynamics and to learn how seemingly universal properties emerge from microscopic physical details. References: [1] T. M. Wintermantel et al., Epidemic growth and Griffiths effects on an emergent network of excited atoms. Nature Communications 12, 103 (2021) [2] S. Helmrich et al., Signatures of self-organized criticality in an ultracold atomic gas. Nature 577, 481-486 (2020)
The spread of excitations by Rydberg facilitation bears many similarities to epidemics. Such systems can be modeled with Monte Carlo simulations of classical rate equations to great accuracy as a result of high dephasing. Motivated by experiments, we theoretically analyze the dynamics of a Rydberg many-body system in the facilitation regime in the limits of high and low temperatures. In the high-temperature limit, a homogeneous mean-field behavior is recovered, while characteristic effects of heterogeneity can be seen in a frozen gas. At high temperatures, the system displays an absorbing-state phase transition and, in the presence of an additional loss channel, self-organized criticality. In a frozen or low-temperature gas, excitations are constrained to a network resembling an Erdős-Rényi graph. We show that the absorbing-state phase transition is replaced with an extended Griffiths phase, which we accurately describe by a susceptible-infected-susceptible model on the Erdős-Rényi network taking into account Rydberg blockade. Published by the American Physical Society 2024
The spread of excitations by Rydberg facilitation bears many similarities to epidemics. Such systems can be modeled with Monte-Carlo simulations of classical rate equations to great accuracy as a result of high dephasing. In this paper, we analyze the dynamics of a Rydberg many-body system in the facilitation regime in the limits of high and low temperatures. While in the high-temperature limit a homogeneous mean-field behaviour is recovered, characteristic effects of heterogeneity can be seen in a frozen gas. At large temperatures the system displays an absorbing-state phase transition and, in the presence of an additional loss channel, self-organized criticality. In a frozen or low-temperature gas, excitations are constrained to a network resembling an Erdös-Renyi graph. We show that the absorbing-state phase transition is replaced with an extended Griffiths phase, which we accurately describe by a susceptible-infected-susceptible model on the Erdös-Renyi network taking into account Rydberg blockade. Furthermore, we expand upon an existing macroscopic Langevin equation to more accurately describe the density of Rydberg atoms in the frozen and finite temperature regimes.
The theory of continuous phase transitions predicts the universal collective properties of a physical system near a critical point, which for instance manifest in characteristic power-law behaviours of physical observables. The well-established concept at or near equilibrium, universality, can also characterize the physics of systems out of equilibrium. The most fundamental instance of a genuine non-equilibrium phase transition is the directed percolation (DP) universality class, where a system switches from an absorbing inactive to a fluctuating active phase. Despite being known for several decades it has been challenging to find experimental systems that manifest this transition. Here we show theoretically that signatures of the DP universality class can be observed in an atomic system with long-range interactions. Moreover, we demonstrate that even mesoscopic ensembles—which are currently studied experimentally—are sufficient to observe traces of this non-equilibrium phase transition in one, two and three dimensions.
At nonequilibrium phase transitions into absorbing (trapped) states, it is well known that the directed percolation (DP) critical scaling is shared by two classes of models with a single (S) absorbing state and with infinitely many (IM) absorbing states. We study the crossover behavior in one dimension, arising from a considerable reduction of the number of absorbing states (typically from the IM-type to the S -type DP models) by following two different (excitatory or inhibitory) routes which make the auxiliary field density abruptly jump at the crossover. Along the excitatory route, the system becomes overly activated even for an infinitesimal perturbation and its crossover becomes discontinuous. Along the inhibitory route, we find a continuous crossover with universal crossover exponent phi approximately=1.78(6), which is argued to be equal to nu||, the relaxation time exponent of the DP universality class on a general footing. This conjecture is also confirmed in the case of the directed Ising (parity-conserving) class. Finally, we discuss the effect of diffusion on the IM-type models and suggest an argument why diffusive models with some hybrid-type reactions should belong to the DP class.