Enhancement of radiation trapping for quasiresonant scatterers at low temperature

We present a transport equation for the incoherent propagation of radiation inside a quasiresonant atomic gas at low temperature. The derivation is based on a generalized Bethe-Salpeter equation taking into account the motion of the atoms. The obtained equation is similar to the radiative transfer equation. It is solved numerically by an original Monte Carlo approach in the case of a slab geometry. The partial frequency redistribution caused by the small velocity of the scatterers makes the emitted flux outside the system and the energy density inside the medium to behave differently than in the case of complete frequency redistribution. In particular, the long time dependence of the specific intensity (escape factor) is slightly different from the Holstein prediction.

Figure 1

(Color online) Geometry of the system studied numerically. The transmitted and reflected fluxes are integrated over all directions.

Figure 2

(Color online) Spectral distribution of the transmitted photons for five different times in the case of a thin slab of optical thickness $b=10$. The spectral broadening is due to the accumulation along a multiple scattering path of small Doppler shifts (here typically $0.02\Gamma$ at each scattering event). At short time, the broadening is small compared to the natural linewidth $\Gamma$, resulting in a Gaussian line shape. At long times, the spectral distribution broadening saturates, with an approximate Gaussian line shape. Note that the total transmitted flux is larger at $\Gamma t=20$ than at $\Gamma t=10$ because the photons has to be multiply scattered before escaping in the forward direction. The decay at longer time is due to the finite trapping time in the medium.

Figure 3

(Color online) Density of photons $U\left(L/2,t,\delta \right)$ at the center of the slab versus time for three different frequencies (left plot) and energy density versus frequency for four different times (right plot). We also plot the temporal behavior of the transmission for a pinned system (i.e., $\overline{v}=0$). The optical thickness of the system is $b=40$.

Figure 4

(Color online) Transmission $T\left(t,\delta \right)$ versus time for three different frequencies (left plot) and transmission versus frequency for four different times (right plot). We have also plotted the temporal behavior of the transmission for a pinned system (i.e., $\overline{v}=0$). The incoming pulse is at resonance ${\delta }_L=0$, the optical thickness of the medium is $b=40$, and $k\overline{v}/\Gamma =0.02$. Note that the time scale which governs the temporal evolution is large compared to ${\Gamma }^{-1}$ while the spectral scale is of the order of ${\Gamma }^{-1}$. This validates the independence of the temporal and spectral variables in the specific intensity.

Figure 5

(Color online) Relative transmission $T\left(t,\delta \right)$ versus frequency for four different times in the case of a detuned excitation ${\delta }_L=\Gamma$. The spectral behavior at long time tends to be symmetric, as in the resonant case. The optical thickness of the system at resonance is 40.

Figure 6

(Color online) Temporal decay rate $\tau$ of the outgoing flux and the energy density versus the system size. Crosses correspond to the full Monte Carlo simulations [i.e., resolution of Eq. (31)]; the solid line is related to the numerical resolution of the dispersion relation [Eq. (41)]. For small $L$ (but large enough to be in the diffusive regime), the behavior is quadratic as for pinned scatterers (dotted line). This is the quadratic regime, which extends roughly up to $L/{\ell }_0\approx 30$, as predicted by Eq. (44). For very large systems, the behavior is completely different from the Holstein prediction (dashed line). This is the Doppler regime. The agreement between the two numerical methods is very good, the small difference coming from approximating the cosine of the scattering angle by its average (anisotropy factor).