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arXiv:2603.18967

Temporal dynamics of Levy flights of photons in a three-dimensional geometry for hot vapor

Ricardo V. M. de Almeida Filho, J. C. de Aquino Carvalho, Thierry Passerat de Silans, Marcio H. G. de Miranda, M. Araujo

Published Mar 19, 2026• 0 citations
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Abstract

Multiple scattering of light by resonant vapor is characterized by Lévy-type superdiffusion with a step size distribution $P(x) \propto 1/x^{1+α}$, with $0 < α< 2$. The Lévy parameter $α$ was measured from $P(x)$, steady fluorescence, frequency-dependent fluorescence and time-resolved transmission, where the latter was observed for quenched disordered media, not yet for annealed disordered media. Here we report first measurements of this quantity from time-resolved fluorescence for photons that are forward diffused from light pulses exciting a hot rubidium vapor, a medium characterized by annealed disorder, where correlations between the photon steps are absent. Also, the vapor is contained in a three-dimensional geometry, a turned cylinder with respect of the incident photon direction. We show experimentally that $α$ can be extracted from this setup, and the results are consistent with those given by the steady frequency-dependent forward fluorescence. Theoretical simulations are consistent with these results. Also, our simulations show the validity of the one-dimensional theories for our actual geometry and other three-dimensional ones, as thin and thick disks, and a measurement of the number of steps that the photon performed before leaving the medium.

DOI
10.48550/arxiv.2603.18967

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Multiple scattering of light by resonant vapor is characterized by Lévy-type superdiffusion with a step size distribution P(x)∝1/x1+αP(x) \propto 1/x^{1+α}P(x)∝1/x1+α, with 0<α<20 < α< 20<α<2. The Lévy parameter ααα was measured from P(x)P(x)P(x), steady fluorescence, frequency-dependent fluorescence and time-resolved transmission, where the latter was observed for quenched disordered media, not yet for annealed disordered media. Here we report first measurements of this quantity from time-resolved fluorescence for photons that are forward diffused from light pulses exciting a hot rubidium vapor, a medium characterized by annealed disorder, where correlations between the photon steps are absent. Also, the vapor is contained in a three-dimensional geometry, a turned cylinder with respect of the incident photon direction. We show experimentally that ααα can be extracted from this setup, and the results are consistent with those given by the steady frequency-dependent forward fluorescence. Theoretical simulations are consistent with these results. Also, our simulations show the validity of the one-dimensional theories for our actual geometry and other three-dimensional ones, as thin and thick disks, and a measurement of the number of steps that the photon performed before leaving the medium.
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 present theoretical and experimental results of Lflights 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 micro- scopic first principles including Doppler broadening we find anomalous Lsuperdiffusion corresponding to a single-step size distribution P (x) ∝ x −(1+α) , with α ≈ 1. We show that this step size distribution leads to a violation of Ohm's law (Tdiff ∝ L −α/2 � L −1 ), as expected for a L´ evy walk of independent steps. Furthermore, the spatial profile of the transmitted light develops power-law tails (Tdiff(r) ∝ r −3−α ). In an experiment using a slab geometry with hot Rb vapor, we measured the total diffuse transmission Tdiff and the spatial profile of the transmitted light Tdiff(r). We obtained the microscopic L´ evy parameter α under macroscopic multiple scattering conditions paving the way to investigation of L´ evy flights in different atomic physics and astrophysics systems.
The propagation of light in a resonant atomic vapor can a priori be thought of as a multiple scattering process, in which each scattering event redistributes both the direction and the frequency of the photons. Particularly, the frequency redistribution may result in L\'evy flights of photons, directly affecting the transport properties of light in a resonant atomic vapor and turning this propagation into a superdiffusion process. Here, we report on a Monte Carlo simulation developed to study the evolution of the spectrum of the light in a resonant thermal vapor. We observe the gradual change of the spectrum and its convergence towards a regime of complete frequency redistribution as the number of scattering events increases. We also analyze the probability density function of the step length of photons between emissions and reabsorptions in the vapor, which governs the statistics of the light diffusion. We observe two different regimes in the light transport: superdiffusion when the vapor is excited near the line center and normal diffusion for excitation far from the line center. The regime of complete frequency redistribution is not reached for excitation far from resonance even after many absorption and reemission cycles due to correlations between emitted and absorbed frequencies.
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.
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