Signatures of Lévy flights with annealed disorder

We present theoretical and experimental results of Lévy flights of light originating from a random walk of photons in a hot atomic vapor. In contrast to systems with quenched disorder, this system does not present any correlations between the position and the step length of the random walk. In an analytical model based on microscopic first principles including Doppler broadening we find anomalous Lévy-type superdiffusion corresponding to a single-step size distribution $P\left(x\right)\propto x^{-(1+\alpha )}$, with $\alpha \approx 1$. We show that this step size distribution leads to a violation of Ohm's law [$T_{\mathrm{diff}}\propto L^{-\alpha /2}\ne L^{-1}$], as expected for a Lévy walk of independent steps. Furthermore, the spatial profile of the transmitted light develops power-law tails [$T_{\mathrm{diff}}\left(r\right)\propto r^{-3-\alpha }$]. In an experiment using a slab geometry with hot Rb vapor, we measured the total diffuse transmission $T_{\mathrm{diff}}$ and the spatial profile of the transmitted light $T_{\mathrm{diff}}\left(r\right)$. We obtained the microscopic Lévy parameter $\alpha$ under macroscopic multiple scattering conditions paving the way to investigation of Lévy flights in different atomic physics and astrophysics systems.

Figure 1

Notations used in a slab geometry to derive the radial dependance of the transmitted flux in the case of Lévy flights.

Figure 2

(a) Experimental radial profile (in linear scale) of the scattered light transmitted through the cell (blue solid line). Gaussian profile with the same width at half maximum as the experimental profile (black dashed line). (b) Experimental radial profile (in log scale) of the scattered light transmitted through the cell (blue solid line). Numerical prediction from a Monte Carlo simulation of the transport equation (dotted red line). Fit of the experimental data in the range $[1.1,2.1]\mathrm{c}\mathrm{m}$ (black dashed line). The power law obtained is $T_{\mathrm{diff}}\left(r\right)\propto r^{-4.03\pm 0.15}$. The temperature is $T={114}^{\circ }$ C corresponding to an opacity of $O=1.15\times {10}^4$ and an optical thickness of $b=540$. (c) Power-law exponent of the radial profile tail of the transmitted intensity $T_{\mathrm{diff}}\left(r\right)$ as a function of the opacity $O$ tuned by varying the temperature. This exponent is close to $-4$ for a wide range of opacity showing the robustness of the Lévy character of the system. Vertical error bars correspond to standard deviation of the fit to extract the power law of $T_{\mathrm{diff}}\left(r\right)$. Error bars increase for large opacities due to the decrease of the relevant signal.

Figure 3

Transmission profiles (experimental, with milk, and for a Monte Carlo simulation) and corresponding fits with an analytical model for diffusion, Eq. (28). (a) Experimental transmission profile (${\ell }^{*}=0.05\pm 1.5\mathrm{m}\mathrm{m}$, $L=14.82\pm 0.05\mathrm{m}\mathrm{m}$). Also shown is the contribution of a single term, the first and slowest decaying of the series, the one responsible for the quasi-exponential asymptotic. (b) Monte Carlo (input parameters from the fit of the milk) for an infinite slab, fitted with the analytical model Eq. (28) (three fitting parameters: width, transport mean-free path, and scale; ${\ell }^{*}=0.09\pm 2.5\mathrm{m}\mathrm{m}$, $L=14.7\pm 3.5\mathrm{m}\mathrm{m}$), with a Gaussian (two parameters: width and scale) and with a power-law asymptotic law (two parameters: power-law parameter and scale, plus a cutoff distance; here $20\mathrm{m}\mathrm{m}$). The diffusion model is indistinguishable from the Monte Carlo data (even in a log scale spanning four orders of magnitude). The Gaussian and the power law are inadequate.

Figure 4

(a) Experimental setup: A laser beam is incident on a flat cylindrical cell filled with a Rb vapor. The transmitted ballistic light spectrum is measured via a photodiode aligned with the laser beam. The scattered light coming out of the slab is imaged on the CCD camera misaligned by an angle of $3^{\circ }$ with the laser. (b) Signal of the photodiode: The transmission of the laser scanned in frequency throughout the rubidium cell shows the D2 hyperfine absorption of ${}^{85}\mathrm{Rb}$ and ${}^{87}\mathrm{Rb}$. An ab initio fit of this signal is used to deduce the opacity $O$. Even if the system is strongly scattering, a coherent transmission measurement can still be done especially out of resonance where the optical thickness is weaker. (c) Image of the radial profile observed on the CCD camera.

Figure 5

Total diffuse transmission in log scale as a function of the opacity. Experimental data (blue crosses). Each point corresponds to the sum of all pixels of a CCD image for a given opacity. Fit of the experimental data (black solid line). The power law obtained is $T_{\mathrm{diff}}\propto O^{-0.516\pm 0.024}$.