Extended Coherently Delocalized States in a Frozen Rydberg Gas

The long-range dipole-dipole interaction can create delocalized states due to the exchange of excitation between Rydberg atoms. We show that even in a random gas many of the single-exciton eigenstates are surprisingly delocalized, composed of roughly one quarter of the participating atoms. We identify two different types of eigenstates: one which stems from strongly-interacting clusters, resulting in localized states, and one which extends over large delocalized networks of atoms. These two types of states can be excited and distinguished by appropriately tuned microwave pulses, and their relative contributions can be modified by the Rydberg blockade and the choice of microwave parameters.

Figure 1

(a),(b) Two different eigenstates of the same 2D realization. The circle size and color represent the $\uparrow$ amplitude at each site. The magnetic field axis (blue arrow) and magic angle of the dipole-dipole interaction (black arrows) are discussed in the text. For each state $c_M={\max }_n{c}_n^{(\ell )}$. (c),(d) The probability to find states with coherence $\mathcal{C}$ in a 3D random Rydberg gas with varying number of Rydberg atoms $N$. Panel (c) highlights the low coherence and (d) the high coherence regions. We averaged over ${10}^4$ random realizations. Note the different scales of the $y$ axes.

Figure 2

The distribution (probability density function) of states having coherence $\mathcal{C}$ and eigenenergy $E$ for $N=1000$. The zero of energy is the energy ${\epsilon }_n$ of noninteracting atoms. As the unit of energy we use $V_0\equiv (4\pi /9){\mu }^{2}n$, which comes from evaluating Eq. (4) for $\theta =0$ at the Wigner-Seitz radius $a=(3/4\pi n{)}^{1/3}$. It corresponds to the typical energy scale of a Rydberg gas with density $n$. The marginal distributions are plotted on the top (corresponds to the density of states) and side [cf. Figs. 1 and 1].

Figure 3

Coherence probability for several Rydberg blockade radii $R_B$ given in units of the Wigner-Seitz radius $a$. The two panels highlight different regions and use different $y$ scales. For $R_B=0$ the number of Rydberg atoms is $N=1000$; as a result of the blockade this reduces to 992 for $R_B=0.25$, 943 for $R_B=0.5$, 834 for $R_B=0.75$, and 686 for $R_B=1$. Here $5\times {10}^5$ realizations were used. The resonant and nonresonant interactions scale rather differently with $\nu$ and density $n$: $V_R\sim {\nu }^{4}n$, while $V_{\mathrm{NR}}\sim {\nu }^4{n}^2$. A scenario with dipole-dipole interactions on the order of $V_0\approx 10\mathrm{MHz}$ and a large blockade radius $R_B=1a$ can be realized for example with $\nu \sim 100$ and density $\sim {10}^7{\mathrm{cm}}^{-3}$.

Figure 4

Probability to find states with coherence values $\mathcal{C}$ after exciting a gas of $N=1000$ Rydberg atoms in the $\downarrow$ state with a microwave pulse (no blockade). The following parameters are used: we choose the $50s$ and $50p$ states of rubidium at a density around ${10}^8{\mathrm{cm}}^{-3}$ so that the relevant interactions are on the order of $V_0\approx 10\mathrm{MHz}$. We use a rectangular microwave pulse with duration $t_{\mathrm{mw}}=500\mathrm{ns}$, a carrier frequency resonant with the transition frequency ${\omega }_{ps}$, and an electric field strength of $F_{\mathrm{mw}}\approx 5\times {10}^{-16}\mathrm{au}$. The curve shown in Fig. 4 stems from averaging $\sim 1000$ realizations. The inset shows marginal coherence distributions obtained from Fig. 2 where only energy intervals [$-\mathrm{\Delta }E$, $+\mathrm{\Delta }E$] around $E=0$, rather than the entire range as in Fig. 2, are considered. Approximately $0.5N$ states lie in this energy interval when $\mathrm{\Delta }E=1$; this fraction decreases to $0.32N$ for $\mathrm{\Delta }E=0.5$ and $0.084N$ for $\mathrm{\Delta }E=0.1$.