Self-similar nonequilibrium dynamics of a many-body system with power-law interactions

The influence of power-law interactions on the dynamics of many-body systems far from equilibrium is much less explored than their effect on static and thermodynamic properties. To gain insight into this problem we introduce and analyze here an out-of-equilibrium deposition process in which the deposition rate of a given particle depends as a power law on the distance to previously deposited particles. This model draws its relevance from recent experimental progress in the domain of cold atomic gases, which are studied in a setting where atoms that are excited to high-lying Rydberg states interact through power-law potentials that translate into power-law excitation rates. The out-of-equilibrium dynamics of this system turns out to be surprisingly rich. It features a self-similar evolution which leads to a characteristic power-law time dependence of observables such as the particle concentration, and results in a scale invariance of the structure factor. Our findings show that in dissipative Rydberg gases out of equilibrium the characteristic distance among excitations—often referred to as the blockade radius—is not a static but rather a dynamic quantity.

Figure 1

Deposition process vs full Rydberg gas dynamics. (a) Concentration $c\left(t\right)$ of excitations as a function of time in a 1D chain of $N={10}^4$ atoms interacting through a power-law potential $\left(\alpha =3,R=15\right)$ for the full process (deposition and removal; blue dashed line) and for pure deposition (red continuous line). The black dashed segment corresponds to a power law with exponent $d/(2\alpha +d)=1/7$. See text for details. (b) Same results for an $N=100\times 1002\text{D}$ square lattice. The black dashed segment corresponds to a power law of exponent $1/4$. (c) Natural logarithm of the deposition rate ${\mathrm{\Gamma }}_k$ [Eq. (1)] as a function of ${\widehat{r}}_k$ in a 1D lattice $\left(R=15\right)$ with excitations being present on sites 1 and 20. (The most relevant cases, $\alpha =3$ and 6, are accentuated, and the order in which the lines appear starting from the top, $\alpha =1$, and moving down to the bottom line, $\alpha =6$, is that of increasing $\alpha$ values.) (d) Sketch of the deposition model in a 2D lattice. Red (blue) discs represent excited (ground state) atoms. The darker the blue, the higher the deposition rate for subsequent excitations. Encircled discs indicate the most likely positions for deposition.

Figure 2

Scale invariant evolution. (a) Excitations (blue dots) in a 2D square lattice of $N=256\times 256$ atoms, $\alpha =3$ and $R=15$. The first column shows a subsystem of size $l_1\times l_1$ sites for $l_1=128$ at three different times: $t_1=1,t_2={10}^3,t_3={10}^6$. The second column shows the configuration within a box of linear size $l_2={(t_1/t_2)}^{1/(2\alpha +2)}{l}_1\simeq 54$ at times $t_2$ and $t_3$. The third column shows the configuration within a box of linear size $l_3={(t_1/t_3)}^{1/(2\alpha +2)}\simeq 23$ at time $t_3$. (b) Snapshots of the structure factor $S\left(k\right)$ taken at different times. (c) $S\left(k\right)$ as a function of the scaled wave number $k/\sqrt{c\left(t\right)}$. All the lines shown in panel (b) collapse onto a single curve.

Figure 3

Scale invariance of $\pi (l,t)$ in 1D and 2D for $\alpha =3$. (a) Distribution of distances between nearest excitations $\pi (l,t)$ and rescaling according to Eq. (11) for $\alpha =3$ in a 1D chain $\left(N=2^{16},R=15\right)$. Lines of different color correspond to different times. The inset shows the collapse of the distribution functions under Eq. (11) [distributions are shown as a function of $l{t}^{1/(2\alpha +d)}$, and multiplied by $t^{-1/(2\alpha +d)}$ to account for the change in the measure of the distribution]. (b) Same as panel (a) for 2D $\left(N=256\times 256,R=15\right)$.

Figure 4

Excitation number variance as a function of the system size. Variance in a chain of $2^{16}$ atoms $\left(\alpha =3,R=15\right)$ for (sub)system sizes $N=2,4,...,2^{11}$ (averaged over 1000 realizations). The dashed curves correspond to the expression in Eq. (13) applied to each $c$, where $R_{\text{B}}$ is extracted from the largest $N$ considered. Different lines correspond to concentrations starting from approximately $c=0.09$ (light green line at the bottom) to $c=0.30$ (pink line on top) [lines have been displaced vertically to improve the visibility]. Inset: Number variance curves normalized by $Nc(1-c)\left\{1-(2R_{\text{B}}-1)c\right\}$ and plotted as functions of $N/R_{\text{B}}$.

Figure 5

Time evolution of the concentration for different power-law exponents $\alpha$. (a) Concentration $c\left(t\right)$ in a square lattice $\left(N=317\times 317,R=15\right)$ with long-range interactions, $\alpha \le d=2$ (averages of 30 realizations). The larger the value of $\alpha$, the faster the dynamics (and therefore the higher the position at which the line appears in the plot). The dashed segments superimposed to each line show the result of a power-law fit $c\left(t\right)\sim t^{\gamma }$. (b) Same as in (a) for $\alpha \ge d$. In this case, the larger the value of $\alpha$, the slower the dynamics (and therefore the lower the position at which the line appears in the plot). (c) Growth exponent $\gamma$ vs. interaction exponent $\alpha$ in the same 2D square lattice (red squares) and a 1D chain (black dots; $N={10}^5,R=15)$. Dashed lines correspond to $\gamma =1/3$ (mean-field exponent, for $\alpha <d)$ and $\gamma =d/(2\alpha +d)$ (valid for $\alpha >d)$.

Figure 6

Scale invariance of $\pi (l,t)$ in 1D and 2D for $\alpha =0.8$. (a) Distribution of distances between nearest excitations $\pi (l,t)$ and rescaling according to Eq. (20) for $\alpha =0.8$ in a 1D chain $\left(N=2^{16},R=15\right)$. Lines of different color correspond to different times. The inset shows the collapse of the distribution functions under Eq. (20). [Distributions are shown as a function of $l{t}^{1/3d}$, and multiplied by $t^{-1/3d}$ to account for the change in the measure of the distribution. The black line corresponds to the analytical prediction $\pi {(x,t)}^{\left(d\right)}=d(A/2^d)B{x}^{d-1}{e}^{-(A/2^d)B{x}^d}$ (see text for an explanation).] (b) Same as panel (a) for 2D $\left(N=256\times 256,R=15\right)$.