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arXiv:1311.0253 · Contemporary Physics

Radiation trapping and Lévy flights in atomic vapours: an introductory review

Martine Chevrollier

Published May 1, 2012• 22 citations

Abstract

Multiple scattering is a process in which a particle is repeatedly deflected by other particles. In an overwhelming majority of cases, the ensuing random walk can successfully be described through Gaussian, or normal, statistics. However, like a (growing) number of other apparently inoffensive systems, diffusion of light in dilute atomic vapours eludes this familiar interpretation, exhibiting a superdiffusive behaviour. As opposed to normal diffusion, whereby the particle executes steps in random directions but with lengths slightly varying around an average value (like a drunkard whose next move is unpredictable but certain to within a few tens of centimetres), superdiffusion is characterised by sudden abnormally long steps (Lévy flights) interrupting sequences of apparently regular jumps which, although very rare, determine the whole dynamics of the system. The formal statistics tools to describe superdiffusion already exist and rely on stable, well understood distributions. As scientists become aware of, and more familiar with, this non-orthodox possibility of interpretation of random phenomena, new systems are discovered or re-interpreted as following Lévy statistics. Propagation of light in resonant atomic vapours is one of these systems which have been studied for decades and have only recently been shown to be the scene of Lévy flights.

DOI: 10.1080/00107514.2012.684481

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Foreword

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Long ago, Paul Lévy invented a strange family of random walks – where each segment has a very broad probability distribution. These flights, when they are observed on a macroscopic scale, do not follow the standard Gaussian statistics. When I was a student, Lévy's idea appeared to me as (a) amusing, (b) simple – all the statistics can be handled via Fourier transforms – and (c) somewhat baroque: where would it apply? As often happens with new mathematical ideas, the fruits came later. For example, é. Bouchaud proved that adsorbed polymer chains often behave like Lévy flights. In a very different sector, J.P. Bouchaud showed the role of Lévy distributions in risk evaluation. Now we meet a third major example, which is described in this book: cold atoms. The starting point is a jewel of quantum physics: we think of an atom in a state of 0 translational momentum p = 0 (zero Doppler effect), inside a suitably prescribed laser field. For instance, with an angular momentum J = 1 we can have two ground states │+〉 and │−〉, and one excited state │0〉. The particular state │+〉+│−〉 has an admirable property: it is entirely decoupled from the radiation and can live for an indefinitely long time. It is thus possible to create a trap (around p = 0 in momentum space) in which the atoms will live for very long times: this so-called ‘ subrecoil laser cooling’ has been a major advance of recent years.

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Multiple Scattering of Light in Atomic Gases: From Levy Flights to Random Lasers

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Cold atoms have emerged as interesting quantum system to study coherent transport properties of light. First experiments have established that dilute samples with large optical thickness allow studying weak localization of light. The present goal of this research is to study coherent transport of photons in dense samples. The quest for Anderson localization of light is among the most interesting questions currently studied. One important aspect in the quest of Anderson localization of light with cold atoms is the relation of Anderson localization and Dicke super- or subradiance. With large and dense samples of cold atoms the interplay between these different phenomena can be studied. During this research one also has to study anomalous diffusion (Lévy flights of photons) and new situations such as a random lasing can arise if one combines gain with multiple scattering. Cold atoms thus continue to allow for new exciting physics to be investigated.

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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.

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Multiple scattering is a process in which a particle is repeatedly deflected by other particles. In an overwhelming majority of cases, the ensuing random walk can successfully be described through Gaussian, or normal, statistics. However, like a (growing) number of other apparently inoffensive systems, diffusion of light in dilute atomic vapours eludes this familiar interpretation, exhibiting a superdiffusive behaviour. As opposed to normal diffusion, whereby the particle executes steps in random directions but with lengths slightly varying around an average value (like a drunkard whose next move is unpredictable but certain to within a few tens of centimetres), superdiffusion is characterised by sudden abnormally long steps (Lévy flights) interrupting sequences of apparently regular jumps which, although very rare, determine the whole dynamics of the system. The formal statistics tools to describe superdiffusion already exist and rely on stable, well understood distributions. As scientists become aware of, and more familiar with, this non-orthodox possibility of interpretation of random phenomena, new systems are discovered or re-interpreted as following Lévy statistics. Propagation of light in resonant atomic vapours is one of these systems which have been studied for decades and have only recently been shown to be the scene of Lévy flights.

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.

Long ago, Paul Lévy invented a strange family of random walks – where each segment has a very broad probability distribution. These flights, when they are observed on a macroscopic scale, do not follow the standard Gaussian statistics. When I was a student, Lévy's idea appeared to me as (a) amusing, (b) simple – all the statistics can be handled via Fourier transforms – and (c) somewhat baroque: where would it apply? As often happens with new mathematical ideas, the fruits came later. For example, é. Bouchaud proved that adsorbed polymer chains often behave like Lévy flights. In a very different sector, J.P. Bouchaud showed the role of Lévy distributions in risk evaluation. Now we meet a third major example, which is described in this book: cold atoms. The starting point is a jewel of quantum physics: we think of an atom in a state of 0 translational momentum p = 0 (zero Doppler effect), inside a suitably prescribed laser field. For instance, with an angular momentum J = 1 we can have two ground states │+〉 and │−〉, and one excited state │0〉. The particular state │+〉+│−〉 has an admirable property: it is entirely decoupled from the radiation and can live for an indefinitely long time. It is thus possible to create a trap (around p = 0 in momentum space) in which the atoms will live for very long times: this so-called ‘ subrecoil laser cooling’ has been a major advance of recent years.

Cold atoms have emerged as interesting quantum system to study coherent transport properties of light. First experiments have established that dilute samples with large optical thickness allow studying weak localization of light. The present goal of this research is to study coherent transport of photons in dense samples. The quest for Anderson localization of light is among the most interesting questions currently studied. One important aspect in the quest of Anderson localization of light with cold atoms is the relation of Anderson localization and Dicke super- or subradiance. With large and dense samples of cold atoms the interplay between these different phenomena can be studied. During this research one also has to study anomalous diffusion (Lévy flights of photons) and new situations such as a random lasing can arise if one combines gain with multiple scattering. Cold atoms thus continue to allow for new exciting physics to be investigated.

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.