Antiferromagnetic long-range order in dissipative Rydberg lattices

We study the dynamics of dissipative spin lattices with power-law interactions, realized via few-level atoms driven by coherent laser-coupling and decoherence processes. Using Monte Carlo simulations, we determine the phase diagram in the steady state and analyze its generation dynamics. As opposed to mean-field predictions and nearest-neighbor models, there is no transition to long-range-ordered phases for realistic interactions and resonant driving. However, for finite laser detunings, we demonstrate the emergence of crystalline order. Although the static and dynamical critical exponents of the revealed dissipative phase transition fall into the two-dimensional Ising universality class, the found steady states differ considerably from those of an equilibrium Ising magnet. Two complementary schemes for an experimental implementation with cold Rydberg atoms are discussed.

Figure 1

(a) Schematics of a two-dimensional lattice in which ground-state atoms (small blue spheres) are laser excited to Rydberg states (large red spheres). The interplay of dissipation and Rydberg-Rydberg interactions (b) can give rise to antiferromagnetic long-range order, where excitations predominantly occupy one checkerboard sublattice. Two possible realizations of such effective two-level systems with tunable excitation rates ${\Gamma }_{\uparrow }$ and ${\Gamma }_{\downarrow }$ are illustrated in (c) and (d) (see text for details).

Figure 2

Order parameter as a function of the power-law exponent $\alpha$, for $p_0=0.95$ and $V_0=5\omega$ and resonant driving $\Delta =0$. The symbols show results for finite system sizes given in the legend. The thick solid line shows the extrapolation to the thermodynamic limit, $L\to \infty$. Finite-size scaling shows a linear increase of the order parameter in the critical regime (exponent: $1\pm 0.05$).

Figure 3

Order parameter obtained for a $30\times 30$ lattice as a function of (a) $\Delta$ and $p_0$ for $V_0=5\omega$ and (b) $\Delta$ and $V_0$ for $p_0=0.96$. The dashed line shows an estimate of the phase boundary (see text). (c) shows the order parameter after finite-size scaling for $p_0=0.96$ and $V_0=5\omega$ (solid line) compared to mean-field predictions for NN interactions (dotted line) and the full vdW interactions (dashed line).

Figure 4

(a) Finite-size extrapolation of the inverse correlation length ${\xi }^{-1}$ near the phase transition at $\Delta ={\Delta }_c\approx 0.79\omega$ for $\alpha =6,p_0=0.9,V_0/\omega =2.97$. The dashed line corresponds to ${\xi }^{-1}\propto {|\Delta -{\Delta }_c|}^1$. (b) Scaling of relaxation time with the linear size of the two-dimensional system for parameters as in (a) and $\Delta =\omega$ within the ordered phase.

Figure 5

Trace norm distance $\mathcal{F}$ of the steady state to a thermal state of an Ising model with optimized values of $\beta ,h$, and $V_{ij}$ [cf. Eq. (5)] for a $4\times 4$ lattice with periodic boundary conditions and $\alpha =6,p_0=0.9,V_0/\omega =2.97$.