Many-body Rabi oscillations of Rydberg excitation in small mesoscopic samples

We investigate collective aspects of Rydberg excitations in ultracold mesoscopic samples. Strong interactions between Rydberg atoms influence the excitation process and impose correlations between excited atoms. The manifestations of the collective behavior of Rydberg excitations are many-body Rabi oscillations, spatial correlations between atoms, as well as fluctuations in the number of excited atoms. We study these phenomena in detail by numerically solving the many-body Schrödinger equation, using a “superatom” approach. We find that, under certain conditions, these many-body behaviors could be observed experimentally.

Figure 1

(Color online) The number of excited $70p_{3/2}$ atoms as a function of the pulse area for square laser pulses ($N=70$ and $\rho ={10}^{11}{\text{cm}}^{-3}$). In (a) the pulse duration was varied and in (b) the single-atom Rabi frequency. In both graphs the dashed lines represent some particular random distributions of atoms, while the solid lines are the average dependencies over 100 random arrangements of atoms.

Figure 2

(Color online) Comparison of the number of excited atoms of Figs. 1a, 1b with the simple domain picture described by the expression $N_{\text{exc}}\sim N_{\text{exc}}^{\left(1\right)}{\text{sin}}^2\left(\sqrt{N_D}\Omega \tau /2\right)$, with $N_{\text{exc}}^{\left(1\right)}\sim 3.5$. Also shown is the fully blockaded and isolated atoms results with $N_{\text{exc}}={\text{sin}}^2\left(\sqrt{N}\Omega \tau /2\right)$ and $N{\text{sin}}^2\left(\Omega \tau /2\right)$, respectively.

Figure 3

(Color online) The average dependence of the number of excited $70p_{3/2}$ atoms versus the pulse area for a square (solid) and a Gaussian (dashed) laser pulse ($N=70$ and $\rho ={10}^{11}{\text{cm}}^{-3}$). In (a) the pulse duration $\tau$ was varied and in (b), the single-atom Rabi frequency $\Omega$. The excitation blockade is slightly more efficient for Gaussian pulses.

Figure 4

(Color online) Time dependence of the number of excited atoms with scaled interaction strengths $K_{ij}$ 15 times larger than in previous simulations of Figs. 1, 3. To explore the robustness of the many-body oscillations against density fluctuations, the number of atoms in the sample is varied according to a Poissonian distribution. The solid curve is the dependence for $N=70$ and the dashed curve is the average dependence over density fluctuations with $⟨N⟩=70$.

Figure 5

(Color online) The relative variance $\Delta N_{\text{exc}}/\Delta N_{\text{exc}}^{\prime }$ of the number of excited Rydberg atoms versus the square of the pulse area for the same parameters used in Figs. 1, 2, 3. The actual variance $\Delta N_{\text{exc}}$ is expressed using the calculated variance $\Delta N_{\text{exc}}^{\prime }$, obtained from $P$, assuming no correlation between excited atoms. In (a) pulse duration was varied and in (b) the single-atom Rabi frequency. Both solid lines correspond to square pulses and dashed lines to Gaussian ones.

Figure 6

(Color online) Radial excitation probabilities $P_{\text{exc}}\left(R\right)$ (a), and the correlation function $c\left(p,q\right)$ between a central atom and other atoms (b), versus the distance between them (using the parameters of Figs. 1, 2, 3, but with doubled excitation time). These functions are obtained by averaging over 150 different random distributions of 70 atoms. In (a), blue opened circles depict a stage when $P_{\text{exc}}\left(R\right)$ increases nowhere in the sample $\left(\tau =6.8\text{ns}\right)$, while black diamonds show a stage with outward propagation of $P_{\text{exc}}\left(R\right)$ $\left(\tau =10.8\text{ns}\right)$: upward green triangles describe the final profile of $P_{\text{exc}}\left(R\right)$ $\left(\tau =20,0\text{ns}\right)$. In (b), red squares show $c\left(p,q\right)$ in the very beginning of excitation $\left(\tau =1.0\text{ns}\right)$, while the other three profiles show small oscillations in $c\left(p,q\right)$ during its time evolution, with a large value at 6.8 ns (blue opened circles), a lower value at 12.0 ns (black diamonds), and again a large value at 16.4 ns (upward full green triangles).