Coherent forward broadening in cold atom clouds

It is shown that homogeneous line-broadening in a diffuse cold atom cloud is proportional to the resonant optical depth of the cloud. Furthermore, it is demonstrated how the strong directionality of the coherent interactions causes the cloud's spectra to depend strongly on its shape, even when the cloud is held at constant densities. These two numerical observations can be predicted analytically by extending the single-photon wave-function model. Lastly, elongating a cloud along the line of laser propagation causes the excitation probability distribution to deviate from the exponential decay predicted by the Beer–Lambert law to the extent where the atoms at the back of the cloud are more excited than the atoms at the front. These calculations are conducted at the low densities relevant to recent experiments.

Figure 1

The scattered radiation is studied for (i) clouds increasing in the number of atoms while being held at constant average density and shape, and (ii) clouds with varying shapes with respect to $\widehat{\mathit{k}}$ held at constant average density and atom number. The figure also shows the level diagram of the transitions focused on in this paper: the ${}^{88}\mathrm{Sr} J=0$ to $J=1$ intercombination line where the ${}^{3}P_1$ level can be either (a) degenerate in $M_J$ or (b) Zeeman split.

Figure 2

(a) Photon-scattering rate $\gamma$ in arbitrary units versus detuning divided by $\mathrm{\Gamma } \left(\delta \equiv \mathrm{\Delta }/\mathrm{\Gamma }\right)$, for a symmetric Gaussian cloud with $\overline{\rho }=5\times {10}^{17}{\text{m}}^{-3}$. (b) Broadening normalized by ${\overline{\rho }}^{2/3},\eta \equiv ({\mathrm{\Gamma }}^{\prime }-\mathrm{\Gamma })/\left(\mathrm{\Gamma }{\rho }^{2/3}\right)$, in units of ${10}^{-12}{\text{m}}^2$ for densities ${10}^{15}{\text{m}}^{-3}\text{to}5.0\times {10}^{17}{\text{m}}^{-3}$ compared to the broadening predicted by using the single-photon wave-function model. These densities span a range $\overline{\rho }/k^3=1.3\times {10}^{-6}\text{to}6.6\times {10}^{-4}$.

Figure 3

Photon-emission rate $\gamma$ in arbitrary units versus $\delta$ (see Fig. 2) for an axially symmetric Gaussian cloud of ${10}^4$ atoms at $\overline{\rho }=5\times {10}^{17}{\text{m}}^{-3}$ for different values of $\xi$. All calculations are for the same intensity laser.

Figure 4

The average probability of an individual atom being excited in a ${10}^4$ atom cloud for various values of $\delta$ (see Fig. 2) and shapes $\xi$ versus $\widehat{\mathit{k}}·\mathit{r}$ in units of ${10}^5{a}_0$ (setting the center of the cloud as the origin). (a) The average probability of excitation $\left(P\right)$ for a cloud driven by a red-detuned laser $\left(\delta =-3.1\right)$ for increasing values of $\xi$. It should be noted that, in the figure, plots with larger values of $\xi$ average over fewer atoms since those clouds are stretched beyond the region plotted. (b) The “normalized” average probability of excitation $\left(P^{\prime }\right)$ for a cloud at $\xi =5$ for positive and negative laser detunings, where “normalized” means that each curve is divided by its initial point.