Non-equilibrium universality in the dynamics of dissipative cold atomic gases

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 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 directed percolation 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.

In strongly interacting systems, long-range correlations can lead to the emergence of collective behaviors which can be distinctively different from the single-body physics governing the constituents. A remarkable example can be found within the theory of continuous phase transitions [1][2][3], which indeed predicts characteristic singular behaviors of macroscopic observables, in contrast to the usually smooth and continuous microscopic description. At a so-called critical point, the correlation length of the system diverges and the overall behavior is fundamentally determined by certain properties -such as dimensionality, range of interactions and symmetries -that do not depend on the specific scale. All systems sharing these few coarse-grained features display algebraic singularities of the same kind [4] and form a universality class, which is in turn characterized by the corresponding critical exponents [1][2][3]5]. Phase transitions have been extensively studied in equilibrium -both in the classical regime [2,3,6,7] and when quantum fluctuations dominate [8,9] -within a statistical mechanics framework. Despite the lack of an analogous unified picture, critical phenomena are known to occur out of equilibrium as well [10][11][12][13][14][15][16][17]. In particular, phase transitions can take place in the properties of the steady state; this typically leads to an enrichment of the stationary phase diagram which depends upon the coarse-grained aspects of the dynamics, such as symmetries and conservation laws (see, e.g., [18,19]). Equilibrium conditions, for example, are specifically related to the microreversibility symmetry [20][21][22]. Systems in which the latter is not recovered at large length and time scales undergo genuine non-equilibrium phase transitions [23][24][25].
One of the most fundamental and well-studied universality classes of this kind is directed percolation (DP). An intuitive description can be provided in terms of the "contact process", i.e., a classical stochastic process on a chain of Ising spins in which an up spin (or excitation) can always flip down, whereas a down spin can only flip up if another up spin is present in its neighborhood. At a critical ratio between the rates at which these two processes occur the system switches from an absorbing "all-down" steady state to a fluctuating active one with FIG. 1. (a) Two-level description of Rydberg atoms driven coherently by a laser field with Rabi frequency Ω and detuning ∆. We also consider dephasing (at rate γ) of coherent superpositions between |↑ and |↓ and decay from |↑ to |↓ with rate Γ. Two atoms interact only when simultaneously excited to the Rydberg state |↑ . The laser detuning ∆ is chosen such that it is the negative of the nearest neighbor interaction V . (b) Rates (in units of Γ) of the main dynamical processes occurring in the Rydberg gas (see main text). The death and branching (first and second row, shaded in green) of an excitation are fundamental processes of DP, while the creation of an isolated excitation (fourth row, shaded in red) is a relevant perturbation away from the DP critical region. The remaining process ("coagulation") does not modify the critical properties. a finite density of excitations, the defining feature of the DP universality class. Despite its simplicity and robustness [24], it has been very difficult to identify clear signs of DP universality in physical systems [26]. Only recently an experiment focusing on two distinct topological phases of nematic liquid crystals provided the first clean realization of DP in a two-dimensional physical system [27,28].
In this work we show that strongly interacting ensembles of Rydberg atoms [29] feature a dynamics which -in a certain limit -is governed by the elementary rules of a DP process [24]. Beyond revealing insights into the outof-equilibrium behavior of this currently much studied system, it highlights an alternative approach for the experimental exploration of DP universal features not only in two but in one and three dimensions as well. Re-markably, our results show that even for relatively small (mesoscopic) system sizes which compare to those currently studied in experiment [30][31][32][33], signatures of DP are observable.
The specific setup we consider [ Fig. 1(a)] is an ensemble of N atoms trapped in a lattice with spacing a [34]. We describe the internal level structure of the atoms with two relevant levels: the ground state |↓ and a Rydberg state |↑ [35,36], coupled by a laser with Rabi frequency Ω and detuning ∆. When two atoms (at positions r k and r m ) are in the Rydberg state they experience an interaction of strength V km which is parameterized as We will focus here on Rydberg s-states that interact via van-der-Waals (vdW) force and thus α = 6. Moreover, we consider here that the system is subject to noise (or dissipation) that leads to spontaneous radiative decay from the Rydberg state to the ground state at a rate Γ and the dephasing of atomic superposition states at a rate γ [37,38].
In the limit of strong dephasing the dynamics of this system is well described by a rate equation [37][38][39][40] for the probability vector v whose components are the statistical weights of the classical spin configurations (e.g., This depicts a classical stochastic process in which the kth spin flips up with rate Γ k and down with rate Γ + Γ k [see Fig. 1 This rate, which depends on the state of all spins but the k-th one, is analogous to those that appear in dynamical facilitation models of glasses [41,42] and is recovered from a perturbative treatment (in the limit γ Γ, Ω) of the full dissipative quantum dynamics [38]. In the final section of this work we provide evidence of the validity of this approach in the current case. Emergent DP process. It has been conjectured that the defining conditions for the emergence of DP universality are: (i) a local dynamics with a unique absorbing state, (ii) a continuous phase transition with a positive, onecomponent order parameter and (iii) absence of symmetries [24,43]. In a spin chain, these can be reformulated in terms of two fundamental processes: flipping down spins and flipping them up provided there is an up spin in the neighborhood [see Fig. 1(b)]. While the former are in our system provided by the radiative decay, the latter emerge by fixing the detuning such that it cancels exactly the nearest neighbor interaction V , i.e., ∆ = −V . This facilitates the excitation of atoms which are adjacent to an already excited one [44,45].
Let us now study the dynamical processes in our system and the underlying rates in more detail. To this end we introduce the projectors P (j) k (j = 1 . . . z, with z being the coordination number of the lattice) on the subspaces where exactly j nearest neighbors of the kth atom are excited. As these projectors commute with Γ k , we can decompose Γ k = Γ k j P (j) where k denotes all sites included in the neighborhood of site k. Since the interaction is quickly decaying with the distance we have q =k,{k} V qk γ which allows us k . This represents the rate at which a spin flips up if it has a single excited neighbor [second row in Fig. 1 Correspondingly, Γ k provides an estimate of the rate at which spins flip up when far away from excitations. These processes can create isolated excitations, thereby destroying the absorbing property of the "all-down" state. They therefore constitute a relevant perturbation away from the DP critical region [fourth row in Fig.  1(b)]. Using the fact that the Rydberg interaction is quickly decaying, i.e. q =k,{k} V qk V = |∆|, the rate of these processes can be estimated as Hence, the magnitude of the "DP-breaking" processes is strongly suppressed at large laser detuning: when |∆| γ one finds that Γ where A denotes the largest eigenvalue of A in modulus.
These considerations show that the main dynamical processes of the Rydberg system (for large |∆| and ∆ = −V ) are those displayed in Fig. 1(b). Their corresponding rates depend on the two parameters λ = 4Ω 2 /γΓ and ν = 2 |∆| /γ. It is important to remark that the presence of next-to-nearest neighbor interaction of strength ηV imposes an additional constraint on the actual emergence of DP universality. The reason is that processes of the type ↑↑↓↓↓ → ↑↑↑↓↓ occur at a rate λ/ 1 + η 2 ν 2 which is smaller than the one given in the second line of Fig. 1(b). Hence, taking the limit of large detuning or, equivalently, large ν, reduces the unwanted rate Γ To prevent this one needs to impose γ η |∆| (ην 1). In 1D and for a vdW potential we have η = 1/64 which allows to make ν reasonably large. In contrast, in higher dimensions we find the more restrictive η = 1/8. Still, this issue can be overcome, as we discuss at a later stage.
Mean-field analysis. Our first step towards understanding the out-of-equilibrium dynamics is a mean-field analysis. The time evolution of the average local density of excitations n k = tr {n k µ}, after factorizing all spatial correlations n k n p = n k n p (k = p), is given by ∂ t n k = −Γ n k + 1 − 2n k Γ k . Employing the representation (3) and assuming the density to be uniform, i.e., n k = n ∀k, one finds the mean-field equation where τ = Γt. Here, the first two terms are related to spontaneous decay (death) and processes occurring where there is a single excitation around (branching and coagulation) [three first rows in Fig. 1(b)]. The j = 0 term is the one which drives the system away from criticality [last row in Fig. 1(b)]. The stationary mean-field solution as a function of λ and ν is shown in Fig. 2(a) for a 1D chain (z = 2). For finite ν we observe a smooth crossover from a low-density region to a high-density one. As we increase ν 1, λ [see Fig. 2(b)], criticality emerges and one observes a sharp transition at λ c = 1/2. In this regime, the last terms of Eq. (5) affect the evolution only on long timescalesdictated for a single spin by ν 2 /(λΓ) = ∆ 2 /(Ω 2 γ). The dynamics of intermediate times is thus given by which is similar to the mean-field equations of other known DP processes [24]. In fact, it features a DP-like phase transition: while for λz < 1 the only stationary state is the absorbing one, n ss = 0, for λz > 1 the latter becomes unstable and a new, stable one with a non-vanishing density appears, identifying λ c = 1/z as a critical point. The mean field critical exponents of DP can also be extracted from Eq. (6). Numerical results. We have performed Monte Carlo dynamic simulations of small 1D and 2D systems with open boundary conditions for Eq. (1) (see Fig. 3) [46]. We have used ν = 128 and the initial state is a random classical spin configuration with fixed density n(0) = 0.4. We first focus on the 1D case, where we have accounted for both, vdW (α = 6) and nearest neighbor (NN) interactions in a system of N = 50 atoms. As discussed earlier, the main difference between the dynamics resulting from the two interactions is the rate of growth from a two-to a three-excitation cluster (↑↑↓↓↓ → ↑↑↑↓↓) which, with our choice of ν = 128, is 5 times smaller in the vdW case. Correspondingly, we expect a shift in the critical point by approximately the same factor, as it is indeed observed in Fig. 3(b). The data displayed in Figs. 3(a) and (c) is for the NN case only, as the vdW behavior is qualitatively identical. Signatures of the transition can also be observed in the dynamics when increasing λ by increasing the Rabi frequency Ω [see Fig. 3(a)]. One then finds a qualitative change of the curves from an exponential decay towards 0 (lower curves) to an exponential relaxation to a finite value (upper curves) with a characteristic bending down and up, respectively. The intermediate, critical regime (λ = λ c ) displays instead an algebraic decay in reasonable agreement with the DP one [24] (t −δ with δ (1D) = 0.159 is shown for comparison). We then analyze the behavior of the stationary density in Fig. 3(b). This yields again an algebraic law close to λ c which resembles the DP one [the expected value β (1D) = 0.276 is again shown for comparison in panel (c)]. As a further check, we have assumed scaling with the DP critical exponents [24] and plotted t δ (1D) n(t) as a function of t|λ − λ c | β (1D) /δ (1D) for different values of λ close to the critical point as an inset in panel (a). This shows that the curves tend to collapse, as expected in the presence of a phase transition. This collapse is not present at short times, since we have not accounted for initial slip effects, and at long times t ∆ 2 /(Ω 2 γ), where the relevant processes perturbing DP become non-negligible.
The dynamics in 2D with vdW interactions gives rise to 1D, rather than 2D critical behaviour. This can be understood by analyzing once again the growth from a two-to a three-excitation cluster: The growth rate along the direction identified by the original two excitations is λ/ 1 + (ν/64) 2 . In contrast, the rate for growth into the orthogonal direction is λ/ 1 + (ν/8) 2 , since in this case the next-to-nearest excitation lies at distance √ 2a to the first excitation. Hence, it is much more likely to grow linear structures than it is to percolate in both directions. As mentioned above, in order to reduce this effect the Potential shaping and effectively classical dynamics. In order to experimentally observe DP in Rydberg gases in dimension two and three, Rydberg states displaying vdW interactions are not suitable. In fact one requires a potential close to a NN interaction. This can be achieved through potential shaping via microwave-dressing of Rydberg states [47][48][49]. To illustrate this we consider a Rydberg s-state (orbital quantum number l = 0) with principal quantum number n coupled to a n p state via a linearly polarized microwave field [see Fig. 4(a)]. This results in a "hybridization" of the respective interactions, which consequently modifies the bare ns−ns purely algebraic decay of vdW potential. For example, in Fig. 4(a) we show the result of coupling the 50s and 60p states of Rubidium with Rabi frequency 50 MHz and detun- ing −30 MHz. Considering the lattice constant to be a = 2 µm, the ratio between nearest (distance a) and next nearest neighbor ( √ 2a in 2D and 3D) interaction increases from η −1 = 8 (in the vdW case) to η −1 ≈ 55.
Finally, we would like to briefly provide evidence for the validity of the effectively classical equation of motion (1) in the specific parameter regime considered in this work. A numerically exact solution of the full quantum Master equation underlying this problem is only possible for small systems sizes. Here we use N = 9 atoms in a 1D chain with NN interactions. Initially there is a single excitation in the center of the lattice and the corresponding dissipative quantum dynamics is simulated with Quantum Jump Monte Carlo. A comparison to the dynamics obtained from (1) is displayed in Fig. 4(b) for ∆ = −V = −10γ, Ω/γ = 0.1 and Γ/γ = 1/100, 1/200 and 1/300. Indeed an excellent agreement between the full quantum and the approximate classical dynamics is found [38]. Conclusions. We have shown that interacting gases of Rydberg atoms feature a microscopic dynamics that leads to an emergent out-of-equilibrium behavior displaying features of DP. Surprisingly this is the case already for rather small mesoscopic system sizes which are currently studied experimentally. This renders these cold atomic systems into a viable candidate for the experimental exploration of the simple -yet intriguingly hard to observe -DP non-equilibrium universality class. Such experimental studies could furthermore perform a systematic exploration of the role of quantum effects on the dynamics (which is challenging to do theoretically) and moreover study the case of continuous gases instead of lattice systems considered here.