Lévy statistics of interacting Rydberg gases

A statistical analysis of the laser excitation of cold and randomly distributed atoms to Rydberg states is developed. We first demonstrate with a hard-ball model that the distribution of energy level shifts in an interacting gas obeys Lévy statistics in any dimension $d$ and for any interaction $-C_p/R^p$ under the condition $d/p<1$. This result is confirmed with a Monte Carlo rate equations simulation of the actual laser excitation in the particular case $p=6$ and $d=3$. With this finding, we develop a statistical approach for the modeling of probe light transmission through a cold atom gas driven under conditions of electromagnetically induced transparency involving a Rydberg state. The simulated results are in good agreement with experiment.

Figure 1

Comparison between a histogram of the distribution of energy level shifts $f\left(\nu \right)$ computed by Markov-chain Monte Carlo (MCMC) simulation of hard balls (black solid line) and the Lévy distribution function $f_{\frac{1}{2},{\lambda }_0\times \left(1+0.26D_b^3/R_3^3\right)}\left(\nu \right)$ with ${\lambda }_0=\frac{4{\pi }^{3/2}}{3}{n}_r\sqrt{-C_6/h}$, where $C_6/h=-1\mathrm{MHz}\mu {\mathrm{m}}^6$ and $n_r=7.07\times {10}^{10}{\mathrm{cm}}^{-3}$ (red dashed line). The system is composed of a cube of $33.8\mu \mathrm{m}$ edge length, containing hard balls of diameter $D_b=1.05\mu \mathrm{m}$ with Wigner-Seitz radius $R_3={\left(\frac{1}{4/3\pi n_r}\right)}^{1/3}=D_b/0.7=1.5\mu \mathrm{m}$, and periodic boundary conditions are assumed. The histogram of $f\left(\nu \right)$ is realized with the computed data of the total shift at the center of the cube, $\sum \frac{-C_6/h}{R^6}$, accumulated over 8000 different trajectories, with a different initial state and $200000$ Markov chain steps for each trajectory.

Figure 2

Correction term $g\left(x\right)$, as a function of $x=D_b/R_3$. The analytical approximate curve (red solid line) is given by $g\left(x\right)=\varepsilon x^3$ with $\varepsilon =\frac{2-\sqrt{2}}{2}$. The data points (blue circles) are extracted from fitting of the simulated energetic distributions $f\left(\nu \right)$ to a Lévy distribution $f_{\frac{1}{2},{\lambda }_0\times \left[1+g\left(x\right)\right]}\left(\nu \right)$ of parameter ${\lambda }_0\times \left[1+g\left(x\right)\right]$. The distributions $f\left(\nu \right)$ are simulated with MCMC computations as in Fig. 1, at fixed particles density, and with various $D_b$. The fit of the data points to a model curve of the form $g\left(x\right)=\alpha x^3+\beta x^5$ gives $\alpha =0.281\approx \varepsilon$ and $\beta =-0.044$ (black dashed line).

Figure 3

Scheme for laser excitation of ground-state atoms to a Rydberg state as used in the Monte Carlo rate equations (MCRE) computation. Two laser beams with Rabi frequencies ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ are detuned ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ from the resonance of the $|g\rangle \to |e\rangle$ and $|e\rangle \to |r\rangle$ transitions, respectively.

Figure 4

Comparison between a histogram of the distribution of energy shifts $f\left(\nu \right)$ computed using the MCRE approach (black solid line) and the Lévy distribution $f_{\frac{1}{2},{\lambda }_0\times \left(1+0.31D_b^3/R_3^3\right)}$ (red dashed line). The system is composed of a cube containing 1000 atoms at a density of $n_{\mathrm{at}}={10}^{11}{\mathrm{cm}}^{-3}$ and the computation is carried out with periodic boundary conditions. $f\left(\nu \right)$ corresponds here to the distribution for an equilibrium excitation of ${}^{87}\mathrm{Rb}$ atoms, with ${\mathrm{\Omega }}_1/2\pi =1.6$ MHz, ${\mathrm{\Omega }}_2/2\pi =5.6$ MHz, ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=0$, Rydberg state $|r\rangle =|38s\rangle$, and decay rate of atomic coherences ${\gamma }_{ge}\approx {\mathrm{\Gamma }}_e/2$ and ${\gamma }_{gr}/2\pi =0.1$ MHz. The Wigner-Seitz radius is equal to $R_3={\left(\frac{1}{4/3\pi n_r}\right)}^{1/3}\sim D_b/0.67$, where $D_b=2.4\mu \mathrm{m}$. The histogram is plotted with the Rydberg energy level shifts of all the atoms excluded the ones that are in the Rydberg state, computed over 23 different trajectories, with nearly 1300 Monte Carlo time steps each. The inset shows the radial pair-correlation function $g_2\left(r\right)$ in this system (black solid line), compared to that of the ideal system of hard balls of diameter $D_b$ obtained considering the same density $n_r$ (red dashed line). Both pair-correlation functions are computed with Monte Carlo simulations.

Figure 5

Imaginary part of the susceptibility and $|38s\rangle$ Rydberg state population fraction $f_r$ for an ensemble of interacting atoms at a density of $n_{\mathrm{at}}=1.25\times {10}^{11}{\mathrm{cm}}^{-3}$, driven in configuration of EIT with probe field Rabi frequency ${\mathrm{\Omega }}_p/2\pi =1.45$ MHz and coupling field Rabi frequency ${\mathrm{\Omega }}_c/2\pi =5.6$ MHz. The plotted curves are as follows: $\mathrm{Im}\left[\chi \right]$ obtained from Eq. (18), where the Rydberg fraction $f_r$ is calculated with Eq. (19) (red dotted line); $\mathrm{Im}\left[{\chi }_{\mathrm{MC}}\right]$ computed with the MCRE simulation over 800 atoms in a square box (black thin solid line); the population fraction in the Rydberg state $f_r$ obtained directly with the MCRE simulation (black thick solid line); and the approximate Rydberg population fraction $f_r$ as given by Eq. (19) (green dashed line).

Figure 6

Comparison between the probe transmission spectra acquired in an experiment similar to that in Ref. [25] and generated by theoretical models. The experimental spectra are acquired with a coupling field Rabi frequency of ${\mathrm{\Omega }}_c/2\pi =5.6\mathrm{MHz}$, an input probe field Rabi frequency of ${\mathrm{\Omega }}_{p0}/2\pi =1.45\mathrm{MHz}$, the Rydberg state $|r\rangle =|38s\rangle$, and in the case of two different Gaussian atomic clouds. The first one is of $1/e^2$ radius $w_z=51\mu \mathrm{m}$ with peak atomic density, $n_0=0.38\times {10}^{11}{\mathrm{cm}}^{-3}$ ($•$), while the second one is of $1/e^2$ radius $w_z=24\mu \mathrm{m}$ with peak atomic density $n_0=1.76\times {10}^{11}{\mathrm{cm}}^{-3}$ ($\blacksquare$). The dashed lines are generated by the Lévy statistics model with the corresponding experimental parameters as inputs while the solid lines are obtained using the MCRE approach (see text).