Many-body theory of excitation dynamics in an ultracold Rydberg gas

We develop a theoretical approach for the dynamics of Rydberg excitations in ultracold gases,with a realistically large number of atoms. We rely on the reduction of the single-atom Bloch equations to rate equations, which is possible under various experimentally relevant conditions. Here, we explicitly refer to a two-step excitation scheme. We discuss the conditions under which our approach is valid by comparing the results with the solution of the exact quantum master equation for two interacting atoms. Concerning the emergence of an excitation blockade in a Rydberg gas, our results are in qualitative agreement with experiment. Possible sources of quantitative discrepancy are carefully examined. Based on the two-step excitation scheme, we predict the occurrence of an antiblockade effect and propose possible ways to detect this excitation enhancement experimentally in an optical lattice, as well as in the gas phase.

Figure 1

Sketch of the two-step excitation scheme for rubidium.

Figure 2

(Color online) Population of the Rydberg level for the three-level system of Fig. 1 according to the RE (14) (solid lines) and OBE (6) (dashed lines) for different pulse lengths: $0.3\mathrm{\mu }\mathrm{s}$ (lowest pair of curves), $1.0\mathrm{\mu }\mathrm{s}$ (middle pair), and $2.0\mathrm{\mu }\mathrm{s}$. The parameters in MHz are $(\Omega ,\omega ,\Gamma )=(4,0.2,6)$ in (a) and $(22.1,0.8,6)$ in (b).

Figure 3

(Color online) Comparison of the solutions of the master equation (22) (dashed lines) and the rate equation (28) (solid lines) for two interacting atoms at distance $r=5\mathrm{\mu }\mathrm{m}$. Upper graphs [(a) and (b)] show the fraction of excited atoms $f_e$, lower graphs [(c) and (d)] the probability ${\rho }_{ee;ee}$ that both atoms are in the Rydberg state as a function of the principal quantum number $n$ for a pulse length $\tau =2\mathrm{\mu }\mathrm{s}$. The parameters of [(a) and (c)] and [(b) and (d)] are those of Figs. 2a, 2b, respectively.

Figure 4

(Color online) Probability ${\rho }_{ee;ee}$ that both atoms are in the Rydberg state $82S$ as a function of the laser detuning $\Delta$ after an excitation time of $\tau =2\mathrm{\mu }\mathrm{s}$ at an interatomic distance of $r=5\mathrm{\mu }\mathrm{m}$ (a) and $r=7\mathrm{\mu }\mathrm{m}$ (b). Solid lines are the solutions of Eq. (28), the dashed lines of Eq. (22). The excitation parameters are those of Fig. 2b. The insets show the corresponding fraction of excited atoms $f_e$.

Figure 5

Density of Rydberg atoms as a function of the peak density in the MOT for a pulse length of $\tau =20\mathrm{\mu }\mathrm{s}$ for the $82S$ (black) and the $62S$ state (gray) of Rb. Circles: experimental data taken from [8]. Lines: Calculations using different models for the pair interactions potential: two-state model of Ref. 19 (solid) and pure van der Waals interaction from perturbative treatment [16] (dashed).

Figure 6

Estimated average number $N_p$ of $n=82$ Rydberg pairs excited by multiphoton transitions as a function of the laser detuning $\Delta$ after $\tau =20\mathrm{\mu }\mathrm{s}$ for a ground state peak density ${\rho }_0={10}^{10}{\mathrm{cm}}^{-3}$.

Figure 7

(Color online) (a) Fraction of excited atoms for atoms on a simple cubic lattice with 20% unoccupied sites as a function of the principal quantum number $n$. The lattice constant is $a=5\mathrm{\mu }\mathrm{m}$; all other parameters are those of Fig. 2b. [(b)–(d)] Corresponding number of “Rydberg clusters” per lattice site $n_s$ normalized to the number of 1-clusters (i.e., isolated Rydberg atoms) $n_1$ as a function of the cluster size $s$ for principal quantum number $n=40$ (b), $n=65$ (c), and $n=68$ (d). The shaded areas represent predictions from percolation theory [24] for a system with the same number of isolated Rydberg atoms (1-clusters) per lattice site.

Figure 8

(Color online) Comparison of the $Q$ parameter in the blockade (squares) and antiblockade (circles) configuration as a function of the principal quantum number $n$ for a sample with a homogeneous atomic density ${\rho }_0=8\times {10}^9{\mathrm{cm}}^{-3}$ and for an excitation pulse length $\tau =2\mathrm{\mu }\mathrm{s}$. $Q$ was determined by ${10}^5$ successive measurements of $⟨N_{\mathrm{Ry}}⟩$ and $⟨N_{\mathrm{Ry}}^2⟩$. The Rabi frequencies $(\Omega ,\omega )$ are $(4.0,0.24)\mathrm{MHz}$ (squares) and $(22.1,0.8)\mathrm{MHz}$ (circles).