Sub-Poissonian Statistics of Jamming Limits in Ultracold Rydberg Gases

Several recent experiments have established by measuring the Mandel $Q$ parameter that the number of Rydberg excitations in ultracold gases exhibits sub-Poissonian statistics. This effect is attributed to the Rydberg blockade that occurs due to the strong interatomic interactions between highly excited atoms. Because of this blockade effect, the system can end up in a state in which all particles are either excited or blocked: a jamming limit. We analyze appropriately constructed random-graph models that capture the blockade effect, and derive formulae for the mean and variance of the number of Rydberg excitations in jamming limits. This yields an explicit relationship between the Mandel $Q$ parameter and the blockade effect, and comparison to measurement data shows strong agreement between theory and experiment.

Figure 1

(left) A spatial Poisson point process in which neighbors within radius $r$ block each other is used to choose appropriate parameters for (right) an ER random graph so that the particles have the same distribution of number of neighbors as $N\to \infty$. Note that this identification procedure only matches the distribution of the number of neighbors, and does not entail a mapping between the specific particles in both models.

Figure 2

(left) A random first particle excites. (middle) Subsequently, random second and third particles excite. (right) The process continues until all particles are either blocked or excited, and the resulting state is a jamming limit.

Figure 3

(left) The fluid limit $u(f)$ together with a sample path of $U_m/n$ for $n=50$. The dashed line indicates the tangent at $f^{*}$, and the arrows indicate the typical fluctuations. (right) Histogram of $X(\infty )$ in a two-dimensional random sequential adsorption problem with precisely $n=800$ particles, together with the probability density function of a normal distribution with mean $n\ln (1+c)/c$ and variance $nc/[2(1+c{)}^2]$.

Figure 4

The average number of detected excitations as a function of time, $\mathbb{E}[X_D(t)]$, fitted to the measurement data in Ref. [6], Fig. 1(a). The fit results in an excitation rate of $\lambda =14\mathrm{kHz}$, and average number of neighbors of $c=2.7\times {10}^2$.

Figure 5

Histogram (Fig. 4, Ref. [7]) of the number of detected dark-state polaritons, together with the appropriately scaled probability density function of a normal distribution with mean $\mathbb{E}[X_D(\infty )]$ and variance $\mathrm{Var}[X_D(\infty )]$. Here, $Q_D(\infty )\approx -0.36$, and the dashed line indicates the Poisson distribution.