Experimental observation of controllable kinetic constraints in a cold atomic gas

Many-body systems relaxing to equilibrium can exhibit complex dynamics even if their steady state is trivial. In situations where relaxation requires highly constrained local particle rearrangements, such as in glassy systems, this dynamics can be difficult to analyze from first principles. The essential physical ingredients, however, can be captured by idealized lattice models with so-called kinetic constraints. While so far constrained dynamics has been considered mostly as an effective and idealized theoretical description of complex relaxation, here we experimentally realize a many-body system exhibiting manifest kinetic constraints and measure its dynamical properties. In the cold Rydberg gas used in our experiments, the nature of the kinetic constraints can be tailored through the detuning of the excitation lasers from resonance. The system undergoes a dynamics which is characterized by pronounced spatial correlations or anticorrelations, depending on the detuning. Our results confirm recent theoretical predictions, and highlight the analogy between the dynamics of interacting Rydberg gases and that of certain soft-matter systems.

Figure 1

Kinetic constraints realized in a gas of Rydberg atoms. (a) Schematic representation of the laser-induced coupling between ground and Rydberg states with Rabi frequency $\mathrm{\Omega }$ and detuning $\mathrm{\Delta }$, where coherence between the atomic states is lost at a rate $\gamma$. Each atom can be effectively described as an Ising pseudospin. In the incoherent regime $\gamma \gg \mathrm{\Omega }$, the dynamics of an atom reduces to incoherent state changes (spin flips) at a rate ${\mathrm{\Gamma }}_i\left(\mathrm{\Delta }\right)$. (b) Interaction potential $V\left(\right|r_i-r_j\left|\right)$ between two excited atoms at positions $r_i$ and $r_j$ (red line) and excitation rates ${\mathrm{\Gamma }}_i\left(\mathrm{\Delta }\right)$ for $\mathrm{\Delta }>0$ (blue line) and $\mathrm{\Delta }=0$ (gray line). The excitation rate for $\mathrm{\Delta }=0$ drops to zero for interatomic distances $|r_i-r_j|$ below the blockade radius $r_b$ (blockade constraint), whereas it peaks at the facilitation radius $r_{\mathrm{fac}}$ for $\mathrm{\Delta }>0$ (facilitation constraint). This leads to blockaded dynamics, shown in (c), and to facilitated dynamics, shown in (d), respectively. In (c), excitation of individual atoms occurs at the resonant rate until the distance between adjacent excitations approaches the blockade radius. The third spin from the left is excited at a strongly reduced rate as it is located within the blockade radius. In (d), a seed excitation enables further excitations at the facilitation distance $r_{\mathrm{fac}}$.

Figure 2

Dependence of incoherent dynamics on the Rabi frequency for resonant excitation. The inset shows the mean number of excitations in a sample of $7\times {10}^5$ atoms in a MOT of width $60\mu \mathrm{m}$ in the 3D configuration as a function of time for three different Rabi frequencies: $\mathrm{\Omega }/2\pi =81$ (open diamonds), 43 (green squares), and 20 kHz (red circles). When multiplying the excitation times by ${\mathrm{\Omega }}^2/\gamma$ (where $\gamma /2\pi =0.7\mathrm{MHz}$), the three curves collapse onto each other (main figure), demonstrating the expected ${\mathrm{\Omega }}^2$ scaling in the incoherent excitation regime. Representative error bars are one standard deviation of the mean (s.d.m.).

Figure 3

Evidence for the blockade constraint in a gas of Rydberg atoms. (a) Mean number of excitations and (b) excitation fraction as a function of time for different atom numbers $N_g$ inside the interaction volume: 5600 (open diamonds), 715 (green squares), and 180 (red circles). These correspond to values of the parameter $\frac{r_b}{a}$ of 4.2, 2.1, and 1.3, respectively. The numerical simulations (solid lines) have been scaled vertically by a factor of $1.8$. (c) Normalized growth rate as a function of the mean distance $d$ between excited atoms. The solid line is obtained from the expression for ${\mathrm{\Gamma }}_i\left(\mathrm{\Delta }\right)$ in a mean-field approach. (d) Mandel $Q$ parameter as a function of mean excitation number. The vertical axis on the left corresponds to experimental data, and that on the right to the numerical simulation (solid lines). Representative error bars are one s.d.m.

Figure 4

Evidence for the facilitation constraint in a gas of Rydberg atoms. (a) Mean number of excitations as a function of time for $\mathrm{\Delta }/2\pi =+19\mathrm{MHz}$ (blue circles), $\mathrm{\Delta }/2\pi =-19\mathrm{MHz}$ (red diamonds), and $\mathrm{\Delta }=0$ (gray squares). The numerical simulations (solid lines) have been scaled vertically by a factor $0.56$. Representative error bars are one s.d.m. (b) The Mandel $Q$ parameter as a function of time. The numerical simulations (solid lines) have been scaled by a factor 2.