Hybrid model for Rydberg gases including exact two-body correlations

A model for the simulation of ensembles of laser-driven Rydberg-Rydberg interacting multilevel atoms is discussed. Our hybrid approach combines an exact two-body treatment of nearby atom pairs with an effective approximate treatment for spatially separated pairs. We propose an optimized evolution equation based only on the system steady state, and a time-independent Monte Carlo technique is used to efficiently determine this steady state. The hybrid model predicts features in the pair-correlation function arising from multiatom processes which existing models can only partially reproduce. Our interpretation of these features shows that higher-order correlations are relevant already at low densities. Finally, we analyze the performance of our model in the high-density case.

Figure 1

(a) Our model divides atoms into pairs and single atoms, allowing an exact treatment of the two-body interaction up to an interatomic separation $L_{Cr}$. Encircled subsets interact with each other via an effective detuning. (b) On a lattice structure, overlapping pairs have to be used.

Figure 2

Three models consisting of lower-order subsets are investigated as simplified models for the $N=3$ case. The results are compared to exact three-atom calculations. The relative deviation $|{\rho }_{33}^{\mathrm{simpl}.}-{\rho }_{33}^{\mathrm{exact}}|/{\rho }_{33}^{\mathrm{exact}}$ from the exact solution is shown for varying detuning $\Delta$ and interaction $V$ with the third atom. Models including the exact treatment of pairs give more precise results than the rate equation. The dashed line depicts the models which are suited best for a given $V$. Parameters: ${\Omega }_{12}=3\mathrm{MHz},{\Omega }_{23}=2\mathrm{MHz},{\gamma }_{21}=6\mathrm{MHz},{\gamma }_{32}=25\mathrm{kHz},\mathrm{and}{V}_{12}=2\mathrm{MHz}$.

Figure 3

Schematic algorithm of the Monte Carlo method.

Figure 4

(a) Rydberg excitation for a 3D sample of 500 atoms and $C_6=50000\mu {\mathrm{m}}^6\mathrm{MHz}$. The hybrid model (dashed lines) and the single-atom rate equation (solid lines) yield the same results. (b) Rydberg excitation on a 1D lattice for different models with interaction $V_{\mathrm{NN}}=2.5\mathrm{MHz}$ between adjacent atoms. The resonances at $\Delta =V_{\mathrm{NN}}$ and $\Delta =V_{\mathrm{NN}}/2$ correspond to single-atom and two-atom excitations. Other parameters are ${\Omega }_{12}=2\mathrm{MHz},{\Omega }_{23}=1\mathrm{MHz},{\gamma }_{21}=6\mathrm{MHz},{\gamma }_{32}=25\mathrm{kHz},$ and ${\Gamma }_{32}={\Gamma }_{21}=100\mathrm{kHz}$.

Figure 5

Pair-correlation function $g^{\left(2\right)}\left(r\right)$ for different laser detunings. (a) The models reproduce the Rydberg blockade and the uncorrelated regime. In the intermediate region resonances which indicate an excitation enhancement occur. The CE does not yield the line at $2R_1$. The hybrid model data was obtained from averaging 200$$000 Monte Carlo runs, and the noise is due to statistics. (b) The strong resonance lines are shown in detail and the process $|23\rangle \to |33\rangle$ at $R_1$ can be seen. The hybrid model shows $|22\rangle \to |33\rangle$ at $R_2$ in addition. Parameters as in Fig. 4, except $C_6=900/2\pi \mu {\mathrm{m}}^6\mathrm{MHz}$ and $n_{1\mathrm{D}}=0.1\mu {\mathrm{m}}^{-1}$.

Figure 6

Mandel $Q$ parameter for a 1D lattice with 50 atoms. The interaction between adjacent atoms is $V=100$ MHz. Next to the expected sub-Poissonian behavior at $\Delta =0$ we find super-Poissonian statistics at $\Delta =V$ and $\Delta =V/2$. Other parameters are ${\Omega }_{12}=3\mathrm{MHz},{\Omega }_{23}=3\mathrm{MHz},{\gamma }_{21}=6\mathrm{MHz},$ and ${\gamma }_{32}=25\mathrm{kHz}$.

Figure 7

Comparison of computational approaches in the high-density case. The hybrid model is extended to use both a lower and an upper bound to include pairs as explained in the text. Parameters: $\Omega =1\mathrm{MHz},\Delta =7\mathrm{MHz},C_6=900\mu {\mathrm{m}}^6\mathrm{MHz},n_{1\mathrm{D}}=3\mu {\mathrm{m}}^{-1},$ and ${\gamma }_{21}=1$ kHz (SARE and HM only).

Figure 8

Probability of excited pairs for three-level atoms in the high-density regime. Simulation results for HM and SARE with 1000 atoms are shown for parameters of Fig. 6 (solid and dashed) and Fig. 4 (dotted and dash-dotted), except $\Delta =7\mathrm{MHz}\mathrm{and}{n}_{1\mathrm{D}}=3\mu {\mathrm{m}}^{-1}$.