Redistribution of light frequency by multiple scattering in a resonant atomic vapor

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 Lévy 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.

Figure 1

Probability density function of the velocity component of absorbing atoms parallel to the direction of incident photons, for different excitation detunings ${\delta }_I$. The values of $V_{\left|\right|}$ are normalized to the half width of the Maxwell-Boltzmann distribution at temperature of 294 K. (a) In log scale, (b)–(e) in linear scale (zoom). (b) For ${\delta }_I=-\frac{1}{3}{\mathrm{\Gamma }}_D$ and (c) for ${\delta }_I=-{\mathrm{\Gamma }}_D$. Photons with a small detuning are preferentially absorbed by atoms whose parallel velocity component Doppler compensates for the detuning. (d) For ${\delta }_I=-1.7{\mathrm{\Gamma }}_D$ and (e) for ${\delta }_I=-2{\mathrm{\Gamma }}_D$. Very few atoms absorb the far-detuned photons at the line center in the atomic rest frame (see small narrow peaks). Most atoms absorb instead in the far-reaching wings of the Lorentz lineshape, where favorable MB probability compensates for the very weak absorption probability (this one decaying as ${\delta }^{-2}$ in the wings).

Figure 2

Probability density function for the frequency emitted after the first scattering event (${\delta }_1$). The arrows in the figure indicate the excitation frequency (${\delta }_I$) for each curve. Calculated $R({\delta }_1,{\delta }_I)$ using Eq. (3.12.1) given by Hummer [15] are shown as solid lines.

Figure 3

Probability density function of the speed of the absorbing atoms for different excitation detunings ${\delta }_I$. The magenta solid line superposed to the ${\delta }_I=0$ curve is a 2D MB distribution at 294 K. The green solid line superposed to the ${\delta }_I=-2{\mathrm{\Gamma }}_D$ curve is a 3D MB distribution at 294 K.

Figure 4

Probability density function of the emitted frequency for different scattering event number $i$ (denoted by the numbers in the figure). The initial excitation is at the line center (${\delta }_I=0$).

Figure 5

Probability density function of the emitted frequency for different scattering event number $i$ (denoted by the numbers in the figure). The initial excitation is ${\delta }_I=-{\mathrm{\Gamma }}_D$.

Figure 6

Probability density function of the emitted frequency for different scattering event number $i$ (denoted by the numbers in the figure). The initial excitation is at ${\delta }_I=-2{\mathrm{\Gamma }}_D$.

Figure 7

(a) Frequency diffusion during the random walk of a photon in a vapor as a function of the number of scattering events. Two photon realizations (red diamond and black triangle) are shown for ${\delta }_I=-2{\mathrm{\Gamma }}_D$. Dotted lines in (a) correspond to $\pm {\delta }_L$. (b) PDF of the number of scattering events before $|{\delta }_i|<{\delta }_L$ for ${\delta }_I=-2{\mathrm{\Gamma }}_D$ (left axis) and its cumulative probability (right axis).

Figure 8

Full width at $1/e$ of the maximum of the emitted frequency PDF as a function of the scattering event number for various excitation frequencies ${\delta }_I$. For ${\delta }_I=-2{\mathrm{\Gamma }}_D$ only the peak around the line center is reported. For the first three scattering events such peak is not resolved and is not reported here.

Figure 9

Probability density function of the step length [$P\left(l\right)$] of the photon before being reabsorbed by the vapor for different scattering event number $i$ (denoted as numbers in the figure). Excitation is at the line center ${\delta }_I=0$. The step length is normalized by the absorption coefficient at the line center ${\alpha }_0$.

Figure 10

Probability density function of the step length [$P\left(l\right)$] of the photon before being reabsorbed by the vapor for different scattering event number $i$ (denoted as numbers in the figure). Excitation is at ${\delta }_I=-{\mathrm{\Gamma }}_D$. The step length is normalized by the absorption coefficient at the line center ${\alpha }_0$.

Figure 11

Probability density function of the step length [$P\left(l\right)$] of the photon before being reabsorbed by the vapor for different scattering event number $i$. The arrow indicates the sequence of curves with $i=1,2,4,6,8,9,10$. Excitation is at ${\delta }_I=-2{\mathrm{\Gamma }}_D$. The step length is normalized by the absorption coefficient at the line center ${\alpha }_0$.