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Physical review. E

Sample size effects for Lévy flight of photons in atomic vapors

A. S. M. Macedo, J. P. Lopez, Thierry Passerat de Silans

Published Nov 30, 2021• 5 citations
Publisher

Abstract

Lévy flight superdiffusion consists of random walks characterized by very long jumps that dominate the transport. However, the finite size of real samples introduces truncation of long jumps and modifies the transport properties. We measure typical Levy flight parameters for photon diffusion in atomic vapor characterized by a Voigt absorption profile. We observe the change of Lévy parameter as a function of truncation length. We associate this variation with size-dependent contributions from different spectral regions of the emission profile with the Doppler core dominating the transport for thin samples and Lorentz wings for thick samples. Monte Carlo simulations are implemented to support the interpretation of results.

DOI
10.1103/physreve.104.054143

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Lévy flight superdiffusion consists of random walks characterized by very long jumps that dominate the transport. However, the finite size of real samples introduces truncation of long jumps and modifies the transport properties. We measure typical Levy flight parameters for photon diffusion in atomic vapor characterized by a Voigt absorption profile. We observe the change of Lévy parameter as a function of truncation length. We associate this variation with size-dependent contributions from different spectral regions of the emission profile with the Doppler core dominating the transport for thin samples and Lorentz wings for thick samples. Monte Carlo simulations are implemented to support the interpretation of results.
The propagation of a photon in an atomic medium can be seen as a L\'evy-flight random walk where the probability distribution P(z)P(z)P(z) for the photon step length zzz scales with ${z}^{\ensuremath{-}(\ensuremath{\alpha}+1)}$, with $0<\ensuremath{\alpha}<2$. In atomic vapors, previous work reported the value of $\ensuremath{\alpha}$ obtained from direct measurement of P(z)P(z)P(z) and also from the dependence of transmission on sample opacity, both for Doppler and Lorentz profiles. In this work, we report the measurement of $\ensuremath{\alpha}$ in a Cs cell from the scaling of transmission with starting point, i.e., the average penetration depth of the photon in the medium before the first scattering event. We show that the parameter $\ensuremath{\alpha}$ depends on the size of the system and on the probability that an atom suffers a collision before spontaneous emission. The measured $\ensuremath{\alpha}$ parameter corresponds to the expectation value for P(z=L)P(z=L)P(z=L), with LLL the size of the system, which is consistent with the so-called single-big-jump principle that states that a single jump rules the transport.
Multiple scattering of light by resonant vapor is characterized by Lévy-type superdiffusion with a single-step size distribution p(x)∝1/x^{1+α}. We investigate Lévy flight of light in a hot rubidium vapor collisional-broadened by 50 torr of He gas. The frequent collisions produce Lorentzian absorptive and emissive profiles with α<1 and a corresponding divergent mean step size. We extract the Lévy parameter α≈0.5 in a multiple-scattering regime from radial profile of the transmission and from violation of the Ohm's law. The measured radial transmission profile and the total diffusive transmission curves are well reproduced by numerical simulations for Lorentzian line shapes.
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(ℓ)∼ℓ−1−α(ℓ)P(\ell)\sim \ell^{-1-α(\ell)}P(ℓ)∼ℓ−1−α(ℓ), with the Lévy index α(ℓ)α(\ell)α(ℓ) 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(ℓ=L)α=α(\ell=L)α=α(ℓ=L). Simulations are made for walkers alternating between two Lévy like step-length
Photon trajectories in incoherent radiation trapping for Doppler, Lorentz, and Voigt line shapes under complete frequency redistribution are shown to be Lévy flights. The jump length (r) distributions display characteristic long tails. For the Lorentz line shape, the asymptotic form is a strict power law r(-3/2), while for Doppler the asymptotic is r(-2)(ln(r)(-1/2). For the Voigt profile, the asymptotic form always has a Lorentz character, but the trajectory is a self-affine fractal with two characteristic Hausdorff scaling exponents.