Quantitative absorption imaging of optically dense effective two-level systems

Absorption imaging is a commonly adopted method to acquire, with high temporal resolution, spatial information on a partially transparent object. It relies on the interference between a probe beam and the coherent response of the object. In the low saturation regime, it is well described by a Beer Lambert attenuation. In this paper we theoretically derive the absorption of a $\sigma$ polarized laser probe by an ensemble of two-level systems in any saturation regime. We experimentally demonstrate that the absorption cross section in dense $^{87}$Rb cold atom ensembles is reduced, with respect to the single particle response, by a factor proportional to the optical density b of the medium. To explain this reduction, we developed a model that incorporates, in the single particle response, the incoherent electromagnetic background emitted by the surrounding ensemble. We show that it qualitatively reproduces the experimental results. Our calibration factor that has a universal dependence on optical density $b$ for $\sigma$ polarized light : $\alpha$ = 1.17(9) + 0.255(2)b allows to obtain quantitative and absolute, in situ, images of dense quantum systems.