Experimental investigation of Lévy flights for step-length distributions with a length-dependent local power exponent

Abstract

We experimentally investigate the transmission of light by dense atomic vapor. The light propagating in dense atomic vapor can be modeled as a Lévy flight random walk. For such system, the step-length distribution can be modeled as $P(\ell)\sim \ell^{-1-α(\ell)}$, with the Lévy index $α(\ell)$ varying smoothly with the step length. Moreover, the walkers alternate between two distinct distributions depending on the occurrence of collisions between atoms in the light scattering. We obtain the Lévy index from transmission measurements for different system sizes and atomic densities. The measured Lévy index is determined by the system size $α=α(\ell=L)$. Simulations are made for walkers alternating between two Lévy like step-length

Figures (9)

• Figure 1: Measured Lévy index for different atomic densities (diamonds). Lines are step-length-dependent Lévy index $\alpha(\ell)$ calculated from $P_{II}(\ell)$ (black lines),$P_{III}(\ell)$ (red lines) and $P(\ell)$ (green lines). (a) For density $N=3.5\times 10^{12}$ atoms/cm$^3$ corresponding to $P_C=6\%$; (b) for $N=10\times 10^{12}$ atoms/cm$^3$ and $P_C=15\%$; (c) for $N=25\times 10^{12}$ atoms/cm$^3$ and $P_C=30\%$; (d) for $N=44\times 10^{12}$ atoms/cm$^3$ and $P_C=43\%$. For density (a) measurement were made for two different cells of thickness $L=1$ cm and $L=2$ cm. For densities (b-d) only the cell of thickness $L=2$ cm were used.
• Figure 2: Scheme of the experimental set-up. A low power laser beam is incident on the atomic vapor and diffuse transmission if collected by photodetector PD2 at 10$^\circ$ relative to laser beam.
• Figure 3: (a) Diffuse transmission spectrum for a cell of thickness $L=2$ cm and atomic density of $N=3.5\times10^{12}$ atoms/cm$^3$. The zero detuning corresponds to the $6S_{1/2}(F=4)\rightarrow 6P_{3/2}(F'=5)$ transition. (b) Power law dependence of diffuse transmission as a function of laser penetration depth $z_0$. The range of frequencies analysed in (b) corresponds to red-solid line in (a).
• Figure 4: (a) Transmission as a function of starting point $z_0$ for two different) sizes of system: $L=5\times 10^{3}$ (black-solid line) and $L=10^5$ (red-solid line). (b) Local Lévy exponent $\alpha$ obtained by fitting locally Eq. \ref{['Eq.Klinger']} for $L=5\times 10^{3}$ (black-solid line) and $L=10^5$ (red-solid line). In blue-solid line we plot expected $\alpha$ value calculated from the step-length distribution: $1+\alpha=-\frac{d log(P^{(DP)})}{dlog(z_0)}$. In green-solid line we plot expected $\alpha^{(w)}$ obtained from Eq. \ref{['Eq:alfa_Weigthed']}
• Figure 5: Diffuse transmission as a function of $\ell_0$ for different sizes of systems $L$: $L=100$ (black circles), $L=280$ (red triangle), $L=780$ (blue triangle), $L=2150$ (magenta squares) and $L=6000$ (cyan diamonds). (b) Extracted Lévy index $\alpha$ by fitting Eq. \ref{['Eq.TD_r0']} to transmission as a function of $L$ (blue circles). Black-solid line: expected $\alpha$ values calculated from $1+\alpha(L)=-\frac{dlog(P(L))}{dlog(L)}$. Red-diamonds: Expected $\alpha^{(w)}$ calculated using Eq. \ref{['Eq:Weigthed2']}.