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.
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.
The directed percolation universality class possibly represents the simplest instance of a genuine non-equilibrium phase transition from an absorbing state to a fluctuating active phase. It is often thought to be as fundamental as the Ising universality class is for equilibrium. However, 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 directed percolation universality class emerge in ensembles of strongly interacting Rydberg atoms. 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.
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.
Summary form only given. Understanding and probing phase transitions in non-equilibrium systems is an ongoing challenge in physics. A particular instance are phase transitions that occur between a non-fluctuating absorbing phase [1], e.g., an extinct population, and one in which the relevant order parameter, such as the population density, assumes a finite value. Here we report the observation of signatures of such a non-equilibrium phase transition in an open driven quantum system. In our experiment rubidium atoms in a quasi one-dimensional cold disordered gas are laser-excited to Rydberg states under so-called facilitation conditions [2]. This conditional excitation process competes with spontaneous decay (see Fig. 1a) and leads to a crossover between a stationary state with no excitations and one with a finite number of Rydberg excitations (see Fig. 1b, where the phase diagram is plotted as a function of the driving strength and detuning from resonance) [3]. We relate the underlying physics to that of an absorbing state phase transition in the presence of a field which slightly offsets the system from criticality. We observe a characteristic power-law scaling of the Rydberg excitation density as well as increased fluctuations close to the transition point. Furthermore, we argue that the observed transition relies on the presence of atomic motion which introduces annealed disorder into the system and enables the formation of long-ranged correlations. Our study paves the road for future investigations into the largely unexplored physics of non-equilibrium phase transitions in open many-body quantum systems.
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 and probing phase transitions in nonequilibrium systems is an ongoing challenge in physics. A particular instance are phase transitions that occur between a nonfluctuating absorbing phase, e.g., an extinct population, and one in which the relevant order parameter, such as the population density, assumes a finite value. Here, we report the observation of signatures of such a nonequilibrium phase transition in an open driven quantum system. In our experiment, rubidium atoms in a quasi-one-dimensional cold disordered gas are laser excited to Rydberg states under so-called facilitation conditions. This conditional excitation process competes with spontaneous decay and leads to a crossover between a stationary state with no excitations and one with a finite number of excitations. We relate the underlying physics to that of an absorbing-state phase transition in the presence of a field (i.e., off-resonant excitation processes) which slightly offsets the system from criticality. We observe a characteristic power-law scaling of the Rydberg excitation density as well as increased fluctuations close to the transition point. Furthermore, we argue that the observed transition relies on the presence of atomic motion which introduces annealed disorder into the system and enables the formation of long-ranged correlations. Our study paves the road for future investigations into the largely unexplored physics of nonequilibrium phase transitions in open many-body quantum systems.
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.