Universal Nonequilibrium Properties of Dissipative Rydberg Gases

We investigate the out-of-equilibrium behavior of a dissipative gas of Rydberg atoms that features a dynamical transition between two stationary states characterized by different excitation densities. We determine the structure and properties of the phase diagram and identify the universality class of the transition, both for the statics and the dynamics. We show that the proper dynamical order parameter is in fact not the excitation density and find evidence that the dynamical transition is in the “model $A$” universality class; i.e., it features a nontrivial ${\mathbb{Z}}_2$ symmetry and a dynamics with nonconserved order parameter. This sheds light on some relevant and observable aspects of dynamical transitions in Rydberg gases. In particular it permits a quantitative understanding of a recent experiment [C. Carr, Phys. Rev. Lett. 111, 113901 (2013)] which observed bistable behavior as well as power-law scaling of the relaxation time. The latter emerges not due to critical slowing down in the vicinity of a second order transition, but from the nonequilibrium dynamics near a so-called spinodal line.

Figure 1

Phase diagram in the $a-b$ plane [as defined in Eq. (4)] for three different values of $c$. The shaded areas correspond to domains portraying three stationary real solutions. Their boundaries identify the spinodal lines. The black curve represents the path threaded by the critical point when varying $c$, corresponding to the projection of the critical line $\{a_c,b_c,c\}=\{-9/(8c),-27/(8c^3),c\}$ onto the $a-b$ plane. This line meets the vertical axis at $b_{\min }=512/27$. Panels (b) relate to $c=-0.42$ and show the excitation density $n$ taken along the three cuts shown in panel (a): in panels (b.1) and (b.2) we show $n$ as observed on the blue and red dashed lines, respectively, which correspond to the “thermal” and “magnetic” directions (see main text). The black dashed line which crosses the spinodal boundaries probes instead the stable-bistable transition, corresponding to the hysteresislike profile in panel (b.3).

Figure 2

The nonlinear transformation from the stationary basis of observables $\{S^x,S^y,n\}$ to the dynamical one ($\{{S^x}^{\prime },{S^y}^{\prime },n^{\prime }\}$) is qualitatively depicted in the main panel. The critical or off-critical behavior of the $\delta S^{\prime }$ components can be captured by superimposing an effective potential $\mathcal{V}$ which we discuss later in the text. Crucially, the double-well structure (responsible for the model $A$ physics) is only felt by the critical $n^{\prime }$, whereas along the other “massive” directions the system only probes single quadratic wells which play no role in the transition, as we sketch on the right.

Figure 3

Emergence of metastable regimes in the vicinity of the spinodal lines. In panel (a) we show the phase diagram in the $\mathrm{\Delta }/\mathrm{\Gamma }-\mathrm{\Omega }/\mathrm{\Gamma }$ plane at $K=0$ and $V$ fixed. The solid and dashed curves show the spinodal and symmetry lines of Fig. 1, respectively. On the right we display the qualitative structure of the mean-field potential $\mathcal{V}(n^{\prime })$ for parameters corresponding to inside (stars), on (squares) and outside (triangles) the spinodal lines (see text). In panel (b) we show an example for the relaxation of the excitation density $n$ (from an initial value $n_0$) towards the stationary value $n_{\mathrm{st}}$, for parameters near one of these boundaries. Here we observe a metastable plateau whose lifetime $\tau$ is determined by the first crossing time of the midpoint $\overline{n}=(n_{\mathrm{st}}+n_0)/2$. Panels (c),(d) display a power-law divergence of $\tau$ as a function of the reduced Rabi frequency $\omega =(\mathrm{\Omega }-{\mathrm{\Omega }}_c)/{\mathrm{\Omega }}_c$ varied along the red and purple curves in panel (a) (see for comparison the experimental data shown in Fig. 4 of Ref. [32]). The power changes from $\theta =1/2$ in the bistable region [diamonds in panels (a),(c),(d)] to $\theta =2/3$ when intersecting the critical point [disks in panels (a),(c),(d)].