Jaynes-Cummings dynamics in mesoscopic ensembles of Rydberg-blockaded atoms

We show that Jaynes-Cummings dynamics can be observed in mesoscopic atomic ensembles interacting with a classical electromagnetic field in the regime of a Rydberg blockade where the time dynamics of the average number of Rydberg excitations in mesoscopic ensembles displays collapses and revivals typical of this model. As the frequency of Rabi oscillations between collective states of Rydberg-blockaded ensembles depends on the number of interacting atoms, for randomly loaded optical dipole traps, we predict collapses and revivals of Rabi oscillations. We have studied the effects of finite interaction strengths and a finite laser linewidth on the visibility of the revivals. We have shown that observation of collapses and revivals of Rabi oscillations can be used as a signature of the Rydberg blockade without the need to measure the exact number of Rydberg atoms.

Figure 1

(a) Interaction of a single atom with the quantized electromagnetic field in a cavity is described by the Jaynes-Cummings model [1]. Two coupled atom-field states are $|g,n\rangle$ and $|e,n-1\rangle$, where $n$ is the number of photons and $|g\rangle ,|e\rangle$ are ground and excited states of the atom; (b) Rydberg blockade for two interacting atoms. The shift in the collective energy level $|rr\rangle$ caused by the interaction prohibits excitation of two Rydberg atoms by a narrow-band laser radiation; (c) scheme of the coupled states of a mesoscopic atomic ensemble with the number of ground-state atoms $N_g$ is considered as being equivalent to the number of photons in the Jaynes-Cummings model; (d) scheme of the collective states of the atomic ensemble with $N$ atoms interacting with laser radiation in the regime of the Rydberg blockade.

Figure 2

The numerically calculated dependencies of the probabilities $q_{\mathrm{Ry}}\left(n\right)$ to excite $n$ Rydberg atoms in a mesoscopic ensemble with $\overline{N}=7$ atoms, randomly distributed in the optical dipole trap with the radius (a) $r=2\mu \mathrm{m}$, (b) $r=3\mu \mathrm{m}$, (c) $r=4\mu \mathrm{m}$, and (d) $r=5\mu \mathrm{m}$. (e) The calculated density plot of the probability to excite at least one Rydberg atom $P_{\mathrm{Ry}}$ as a function of $r$ and time.

Figure 3

The numerically calculated dependencies of the probability $s_{\mathrm{Ry}}\left(1\right)$ to detect a single Rydberg atom in a mesoscopic ensemble with $\overline{N}=7$ atoms for finite detection efficiency (a) $T=0.1$ and (b) $T=0.5$ using Eq. (12); the numerically calculated probabilities to excite at least one Rydberg atom $P_{\mathrm{Ry}}$ calculated using Eqs. (13, 14, 15, 16, 17) taking into account the finite linewidth of the Rydberg Cs $80S$ state ${\gamma }_2/\left(2\pi \right)=0.8\mathrm{kHz}$ and laser linewidth (c) $\gamma /\left(2\pi \right)=0\mathrm{kHz}$; (d) $\gamma /\left(2\pi \right)=10\mathrm{kHz}$ (solid curve) and $\gamma /\left(2\pi \right)=100\mathrm{kHz}$ (dashed curve).

Figure 4

The time dependence of the average number of Rydberg atoms excited in the optical lattice containing (a) nine and (b) four sites with 50% probability to load a single atom in each site.

Figure 5

Time dependence of the average number $N_{\mathrm{Ry}}$ of Rydberg atoms excited in two spatially separated, at distance $d$, mesoscopic atomic ensembles with $\overline{N}=10$ atoms in each ensemble for (a) $d=20\mu \mathrm{m}$ and (b) $d=4\mu \mathrm{m}$; (c) density plot of the average number of Rydberg atoms calculated as a function of time and distance between ensembles; (d) the Fourier spectrum of the time dependencies of $N_{\mathrm{Ry}}$ for $d=4\mu \mathrm{m}$ (solid curve) and $d=20\mu \mathrm{m}$ (dashed curve).