We present a single-scattering formalism for incoherent resonant light scattering by dilute quantum gas systems such as the atomic-trap Bose-Einstein condensates. We show that resonant scattering gives access to more information than the dynamical structure factor, familiar from non-resonant scattering. In particular, we show that the detuning dependence of the incoherent scattering cross-section allows the direct determination of the BEC pairing density
<ψψ>, which is a broken symmetry and provides evidence that the condensate is not in a good number state.
We report a combined experimental and theoretical investigation of near-resonance light scattering from a high-density and ultracold atomic
87Rb gas. The atomic sample, having a peak density $\ensuremath{\sim}$$5\ifmmode\times\else\texttimes\fi{}{10}^{13}$ atoms/cm
3, temperature $\ensuremath{\sim}$65 $\ensuremath{\mu}$K, and initially prepared in the
F=1 lower-energy
87Rb hyperfine component, is optically pumped to the higher-energy
F=2 hyperfine level. Measurements are made of the transient hyperfine pumping process and of the time evolution of scattering of near-resonance probe radiation on the $F=2\ensuremath{\rightarrow}{F}^{\ensuremath{'}}=3$ transition. Features of the density, detuning, and temporal dependence of the signals are attributed to the high density and consequent large optical depth of the samples.