Semianalytical model for nonlinear absorption in strongly interacting Rydberg gases

Rate equation models are extensively used to describe the many-body states of laser driven atomic gases. We show that the properties of the rate equation model used to describe nonlinear optical effects arising in interacting Rydberg gases can be understood by considering the excitation of individual superatoms. From this we deduce a simple semianalytical model that accurately describes the Rydberg density and optical susceptibility for different dimensionalities. We identify the previously reported universal dependence of the susceptibility on the Rydberg excited fraction as an intrinsic property of the rate equation model that is rooted in one-body properties. Benchmarking against exact master equation calculations, we identify regimes in which the semianalytical model is particularly reliable. The performance of the model improves in the presence of dephasing which destroys higher-order atomic coherences.

Figure 1

Rescaled absorption for varying atomic density. The symbols show data from RE simulations while the solid lines are the result of a semianalytical model described in Sec. 4b. Parameters as in Ref. [8]: $C_6=50$ GHz $\mu$m${}^6$, ${\Omega }_p=1$ MHz, ${\Omega }_c=5.1$ MHz, $\Gamma =6.06$ MHz, dephasing of the probe transition 0.1(0.33) MHz, dephasing of the two-photon transition 0.1(1.7) MHz for the green (red) points. The inset shows the comparison between the RE model results and the modified universal relation (7).

Figure 2

Blockade fraction as a function of atomic density for different trap geometries. The density is varied by changing the edge length of the cube/square/string for a fixed atom number of $N=5\times {10}^4$. The dashed line has unit slope $f_{\mathrm{bl}}={\rho }_{RR}^{\left(0\right)}{N}_b$. The solid lines are proportional to $N_b^{\nu }$ with $\nu =12/13$, $6/7$, and $4/5$, for 1D, 2D, and 3D, respectively. The inset shows $f_{\mathrm{bl}}$ plotted as a function of ${\rho }_{RR}^{\left(0\right)}\mu$, where $\mu$ is the average superatom size including collective enhancement of the atom-light coupling (Sec. 4b2).

Figure 3

Steady-state Rydberg population ${\rho }_{RR}$ and intermediate state population ${\rho }_{ee}$ for an ensemble of four perfectly blockaded atoms as a function of the probe Rabi frequency ${\Omega }_p$. Laser dephasings and decay from the Rydberg level are set to zero (perfect EIT). In panels (a) and (c), the coherent laser driving is weak compared to the decay rate $\Gamma$ of the intermediate level. In panels (b) and (d) the coherent driving is rather strong. Solid black lines show the exact ME solution; red dashed lines show the semianalytic result. The blue dot-dashed lines show the noninteracting value of ${\rho }_{RR}$ and the two-level value (without Rydberg level) of ${\rho }_{ee}$, respectively.

Figure 4

Absolute value of the relative deviation in Rydberg and intermediate-state populations between RE and ME for a fully blockaded ensemble of $N=2$ atoms. In each figure, the parameters are varied between weak and strong coherent drive from left to right, and between weak and strong probe from bottom to top. We use perfect EIT conditions for the left column, panels (a) and (c), while in the right column, panels (b) and (d), a dephasing of the coherences with the Rydberg level ${\gamma }_R=0.3{\Omega }_c$ is included. The red cross marks (approximately) the parameters used in Fig. 2.