Two-dimensional crystals of Rydberg excitations in a resonantly driven lattice gas

The competition between resonant optical excitation of Rydberg states of atoms and their strong, long-range van der Waals interaction results in spatial ordering of Rydberg excitations in a two-dimensional lattice gas, as observed in a recent experiment of Schauß et al. [Nature (London) 491, 87 (2012)]. Here we use semiclassical Monte Carlo simulations to obtain stationary states for hundreds of atoms in finite-size lattices. We show the formation of regular spatial structures of Rydberg excitations in a system of increasing size, and find highly sub-Poissonian distribution of the number of Rydberg excitations characterized by a large negative value of the Mandel $Q$ parameter which is nearly independent of the system size.

Figure 1

Rydberg excitations of a resonantly driven ensemble of $N=(44,188,401,688)$ atoms in a 2D lattice of diameter $d=(1,2,3,4)d_{\mathrm{b}}$, respectively. (a1)–(a4) Average spatial distribution of Rydberg excitation probabilities ${\overline{\sigma }}_{rr}$; each pixel corresponds to a single atom per lattice site. (b1)–(b4) Probabilities $p_{\mathrm{R}}\left(n\right)$ of $n$ Rydberg excitations (open bars) and the mean number of excitations $\langle n_{\mathrm{R}}\rangle$. Also shown is the Poisson distribution $p_{\mathrm{Poisson}}\left(n\right)$ for the same $\langle n_{\mathrm{R}}\rangle$ (solid brown circles). (c1)–(c4) Average spatial distributions of $n$ Rydberg excitations, corresponding to two largest $p_{\mathrm{R}}\left(n\right)$ in (b1)–(b4), after centering and aligning each configuration ${\left\{{\widehat{\sigma }}_{rr}^j\right\}}_n$, and the corresponding axial $P\left(\rho \right)$ and angular $P\left(\varphi \right)$ probability distributions [$P\left(\rho \right)$ are binned over the lattice spacing $a$ and $P\left(\varphi \right)$ are binned over the interval $\pi a/d$].

Figure 2

(a) Mean number of Rydberg excitations $\langle n_{\mathrm{R}}\rangle$, and (b) the corresponding $Q$ parameter vs the system size $d/d_{\mathrm{b}}$. In (a) the black dashed line for $d<d_{\mathrm{b}}$ is $\langle n_{\mathrm{R}}\rangle =1$, and the dotted line for $d\ge d_{\mathrm{b}}$ is $\langle n_{\mathrm{R}}\rangle =0.2+{(d/d_{\mathrm{b}})}^{1.66}$.

Figure 3

Correlations of Rydberg excitations $g^{\left(2\right)}\left(z\right)$ vs distance $z$; the maximum is at $z_{\max }=1.15d_{\mathrm{b}}$.