key: cord-0497555-xo5p57iq
authors: Ara'ujo, Michelle O.; Silans, Thierry Passerat de; Kaiser, Robin
title: L'evy flights of photons with infinite mean free path
date: 2020-08-08
journal: nan
DOI: nan
sha: 52f67cd2a4a2a03b6090afe48ec2907b915ede51
doc_id: 497555
cord_uid: xo5p57iq

Multiple scattering of light by resonant vapor is characterized by L'evy-type superdiffusion with a single-step size distribution $p(x)propto 1/x^{1+alpha}$. We investigate L'evy 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 $alpha<1$ and a corresponding divergent mean step size. We extract the L'evy parameter $alphaapprox0.5$ 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.

The random walk of particles can often be described within the central limit theorem, which characterizes diffusion phenomena and implies that the mean squared displacement performed by a particle increases linearly in time. However, many physical systems exhibit superdiffusion, in which the mean squared displacement grows faster than a linear function of time [1] . A particular mechanism for superdiffusion is Lévy flights, where rare long jumps dominate the dynamics of the random walk. The occurrence of Lévy flights is not rare [2, 3, 4, 5] and they are encountered in a variety of systems, ranging from human travel [6, 7, 8] spread of diseases [9, 10, 11, 12] , trajectories of animals [13, 14, 15] , turbulence [16] and financial market [17] .

For jump size distributions p(x) decaying asymptotically with p(x) ∝ x −(1+α) , it is straightforward to show that the second moment of the jump size x 2 is finite for α > 2, and the central limit theorem will then apply. For α < 2, x 2 becomes infinite, and the random walk can no longer be described by a dif-fusion equation. This is the regime of Lévy flights. An even more extreme regime corresonds to α < 1, where even the average jump size (or scattering mean free path) diverges.

Long jumps have been recognized as an important mechanism for understanding light transport in scattering media almost 100 years ago [18] and are at the basis of many radiative transfert codes used in astrophysics [19] . Modern experimental development allowed for the investigation of Lévy flights of light in controllable and tunable systems. For instance, Lévy flights were investigated in engineered media [20, 21] and from light diffusion in atomic vapors [22, 23, 24] . The control of the optical system allows the investigation of how the light transport is impacted by, e.g., quenched [25, 26, 27] and annealed [24] disorder, correlations induced by inelastic scattering [28, 29] , fractal dimension of the random walk [30] and effects of system size [30] .

Light transport in resonant vapors is known to depend on the absorption profile and the frequency re-distribution between the scattering processes. The asymptotic decay p(x) ≈ x −(1+α) is expected to scale as α = 1 for Doppler broadened vapors and as α = 0.5 for Lorentz ones [31] . The experimental measurement of the Lévy coefficient α for a Doppler vapor was done both directly, by means of the measurement of the jump size distribution [22, 23] and indirectly, analyzing transmission signatures in the multiple scattering regime [24] . Indeed the radial profile T (r) scales with T (r) ∝ r −(3+α) for radial distances r larger than the sample thickness. Moreover, the total scattered light T diff scales with the sample opacity O as T diff ∝ O −α/2 [32, 24] .

In this work, we report on the realization of Lévy flights of light in hot atomic vapors with a Lévy coefficient α ≈ 0.5 corresponding to a regime of infinite scattering mean free path. The control of the Lévy coefficient is obtained by admixing a buffer gas of He atoms into a Rubidium vapor. Following [20, 24] we extract the Lévy exponent α by analysing the radial profile of the transmitted light and the total diffuse transmission for a multiple scattering regime.

In order to evaluate the potential of controlling Lévy flights in atomic vapors, we perform numerical simulations by describing the photon random walk in the vapor by successive absorption and emission processes. For each scattering event, the emitted frequency is redistributed and is partially correlated to the absorbed frequency. Two scenarios are usually considered for realistic vapors, namely [28] : (i) R II for combined Doppler and natural broadening, and (ii) R III for combined Doppler, natural and collisional broadening. For the R II scenario, the absorption-emission process is elastic in the atomic rest frame but the coherence is partially lost in the laboratory frame due to Doppler shifts. For the R III scenario, coherence in the atomic rest frame is lost due to collision with other atoms (e.g., with the buffer gas) and the emission is Lorentzian in the atomic rest frame. Nevertheless, partial frequency coherence is possible in the laboratory rest frame due to velocity selection in the atomic absorption. When collisions are very frequent, as will be the case in our experiment, a R III scenario will be the most appropriate one. Moreover, when collisional broadening dominates Doppler broadening, complete fre- quency redistribution occurs for a single scattering event with a Voigt emission profile with large Lorentz wings [33, 34] .

To characterize the vapor absorption properties, we first compute the dimensionless quantity f (δ) = σ(δ)/σ 0 , where σ(δ) the scattering cross section at detuning δ = ν − ν 0 of the laser frequency ν from the atomic transition frequency ν 0 and σ 0 = 3λ 2 2π Γ0 Γ is the scattering cross section of a pinned two-level atom at resonance. Here Γ = Γ 0 + Γ C is the homogeneously broadened linewidth where Γ 0 /2π = 6 MHz is the natural width of Rb and Γ C /2π ≈ 1 GHz is the collisional width at 50 torr of He [33] . We take the sum of four two-level profiles corresponding to two ground states per Rb isotope, so

where δ i is the detuning relative to transition i and β i is a weight coefficient to take into account the isotope concentration and the coupling strengths for each transition. The individual f i (δ i ) are Voigt profiles, i.e., the convolution of a Lorentz line, defined by the collisions with buffer gas, and the Maxwell-Boltzmann velocity for the atoms in the vapor,

where k = 2π λ is the wavenumber, u = 2k B T /m is the most probable atomic speed, k B the Boltzmann constant and m the atomic mass. The scattering cross section is a Voigt profile with parameter a = Γ/Γ D ≈ 3, where Γ D /2π = u/λ is the Doppler width. This allows us to compute a transmission spectrum of the light through an atomic vapor of density n and length L, yielding a resonant opacity O,

An example of a coherent transmission spectra fitted with Eqs. 2 and 1 is shown in Fig. 1b , with the opacity as the only fitting parameter. We note that a purely Doppler broadened absorption profile (shown in blue in Fig. 1b does not allow a correct description of our experiment in presence of the He buffer gas. We now turn to the multiple scattering regime. Within the R III ansatz, the step size distribution [31, 28] is given by:

with Φ(δ, δ ) the probability of having an emission at detuning δ if the incident photon is at detuning δ . For complete frequency redistribuiton (CFR) limit (valid after several scattering events), the emission profile is equal to the absorption one and p(x) = dδ φ(δ) 2 e −φ(δ)x , which decays asymptotically as p(x) → x −2 for a Doppler profile and as p(x) → x −1.5 for a Lorentz profile [31] .

To have more insight on the effect of frequency redistribution on the random walk, we have calculated p(x) from Eq. 3 for the R III scenario for a Voigt parameter a = 3. The calculated p(x) is shown in Fig. 2a together with results for CFR with Doppler and Lorentz profiles. For our Rb cell filled with 50 torr of He gas, p(x) follows the Lorentz limit for almost all opacities, since the frequency is completely redistributed for a Voigt profile with a ≈ 3.

For the range of opacities used in the experiment, most photons escapes the cell after multiple scattering. Still, information about the asymptotic behaviour of the step length distribution subsist on the transmitted light. The radial intensity distribution of the transmission T (r) for large r (r L) is dominated by single large step x L originated at r ≈ 0 [24] . The number of photons escaping the cell at radial distance between r and r + dr is proportional to the probability of the photon doing a step larger than r at an angle of cos(θ) = L/r:

which results in: for a Lorentzian vapor (α = 0.5). A line decaying as r −0.5 shows the expected rescaled behavior for a Doppler vapor with P (r) ∝ r −4 [24] . In Fig. 2c we show a local slope of the rescaled profile T (r)r 3.5 .

Information about asymptotic behavoir of the step size distribution p(x) can also be obtained from deviations of the Ohm's law. Indeed, superdiffusion favors the escape of photon relative to normal diffusion and the total transmission decays slower with system size (or opacity O) than the Ohm law T Ohm ∝ O −1 in regular diffusive samples. Details of the total diffusive transmission T diff with O depend on the input geometry and our experimental set-up corresponds to the so called non-equilibrium initial conditions [32] , for which the first scattering event occurs at z ≈ 0. Indeed, in our experiment, the incident photons are resonant with the F = 3 (or F = 2 as well) state of 85 Rb for which the absorption length inside the vapor is l a = L/O L, so the first scattering event occurs close to the input cell window. Under these conditions, the dependence of T diff with opacity O is given by [32] :

The dependency of T diff with opacity obtained in the simulations are shown in Fig. 2d together with a lines corresponding to the expected O −0.25 behavior for Lorentz vapor and O −0.5 expected for Doppler vapor [24] .

In our experiment [ Fig. 1(a) ], a disk shaped cell of radius 5.0 cm and internal thickness L = 6.3 mm is filled with a natural mixture of 85 Rb and 87 Rb isoptopes and 50 torr of He gas. As already mentioned, this pressure gives a estimated collisional broadening of Γ C /2π ≈ 1 GHz [33] , almost three times the Doppler width for the D2 line of Rb at λ = 780 nm and large enough for the scattering cross section to be approximated by a Lorentzian profile. A collimated laser beam from a Ti:Sa laser source of waist 0.88 mm and negligible linewidth (≈ 50kHz) is sent perpendicularly to the cell surface and close to its center to excite the atoms. The power of the laser beam is 2 µW, which gives a very low saturation parameter due to the important collisional broadening. Measurements at 1 µW yielded similar results, allowing to exclude nonlinear optics effects to occur [35] . In order to obtain an opaque sample around the line center, we heat the cell between T = 106 • C and T = 180 • C. Transmission spectra are obtained by scanning the laser frequency across the absorption lines allowing to extract the resonant opacity O using Eq. (2) . For this temperature range, the resonant opacities vary from O = 11 to O = 530, with a typical uncertainty of 10%.

Monitoring the linear absorption spectrum in an auxiliary Rb cell with no buffer gas, we tune the frequency to the minimum of transmission of the F = 3 hyperfine ground state of 85 Rb, taken as the reference frequency ν 0 . A CCD camera from Andor (iXon X3 885, pixel size 8.0 µm, exposure time of 1.0s) placed at an angle of 5.3 • from the transmitted laser beam records images of the scattered light from the output facet (Fig. 1a) . A lens of focal distance 50 mm is placed between the cell and the camera to provide a magnification of 9.4 for the images.

We then extract the radial profile T (r) of the scattered light at the output facet by performing an angular average of the images around the center of maximum intensity. An image without the laser beam is also recorded in order to subtract background profiles without the laser beam. Finally we adjust a powerlaw function ∝ r −s to T (r) in a region of large r in order to determine the Lévy coefficient s = 3+α [24] .

In Fig. 3a we plot the experimental radial transmission profile T (r) in log-log scale obtained from the angle averaged distribution of a single image of the CCD camera. We have cut the radial profile around r = 3 cm as for larger values of r the T (r) becomes smaller than background level. This also allows to avoid unwanted cell border effects. The observed radial profiles varies very little from image to image and have the same general behavior for the range of opacities explored. In Fig. 3b we also plot the rescaled experimental radial profile T (r)r 3.5 together with Lorentz (T (r)r 3.5 = 1) and Doppler (T (r)r 3.5 = r −0.5 ) asymptotic behavior. We clearly see that the experimental curve follows a Lorentz vapor behavior and is inconsistent with a Doppler vapor. In Fig. 3c we show a local slope of the rescaled profile T (r)r 3.5 and extract the asymptotic slope s shown in Fig. 3d . The average value is s = 3 + α = 3.45 ± 0.13 (Fig. 3d) , consistent with the expected α = 0.5 value for a Lorentz vapor. This value is constant over the full range of opacity from O = 11 to O = 530 explored here (Fig. 3d) .

For an alternative evaluation of the Lévy coefficient α via the scaling law given by Eq. 6 [32, 24] , we also compute from these radial profiles the total diffusive transmission T diff by integrating T (r) over all r up the the cell radius: Fig. 4 , in log-log scale. By fitting the experimental values with Eq. 6, we obtain for the exponent β = α/2 = 0.20 ± 0.03, corresponding to a Lévy coefficient α = 0.40 ± 0.06, in good agreement with the expected value for a Lorentz vapor.

The results for both radial and for total transmission are consistent with expected random walk for a Lorentz emission and Lorentz absorption. Using Monte-Carlo simulations we have checked that others combinations of absorption and emission profiles are not consistent with the results discussed here. For instance, for a Lorentz absorption and a Doppler emission, the result is a step length distribution that decays fast for large steps implying normal diffusion, as the Doppler emission profile decays much faster than the Lorentz absorption.

In summary, we have designed a experimental plateform allowing to control the Lévy coefficient from α ≈ 1 with an diverging diffusion coefficient [22, 23, 24] to α ≈ 0.5 where even the average step size is infinite. This plateform allows to study fundamental aspects of Lévy flights and is of interest in a range of light scattering systems, including atomic clocks based on hot vapors as used in satellite navigation systems or for refined models in radiative transfer in astrophysics [19] . So far, we have operated these experiments in a steady state regime with low power in the linear optics limit. It will be interesting to extend such experiments into time dependant regimes, where the distribution of waiting time can lead to sub-and superdiffusive spreading in so-called Lévy walks. Extending the present experiments into nonlinear optics do not pose important technical problems. If non classical light sources are used, the atomic systems might allow to study quantum correlations in Lévy flights.

We acknowledge financial support from the Brazilian Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Conselho Nacional de Desenvolvimento Cientfico e Tecnolgico (CNPq). T.P.S thanks financial support from Pronex/Fapesq-PB/CNPq. This work was conducted within the framework of the project OPTIMAL granted by the European Union by means ofthe Fond Européen de développement régional, FEDER. 

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