Extended Data Fig. 1: Effect of booster stage (V₀ = 2 E_rec, a = 10 a₀) and an example of the population extraction.

a, In the absence of the booster, the majority of condensed atoms remain in the central diffraction peak, with only a small fraction occupying the satellite peaks. The high atomic density of the central peak causes almost all the imaging light around this central area to be fully absorbed, leading to significant imaging saturation at k = 0. b, the booster stage promotes condensed atoms to higher diffraction orders, thus facilitating the fitting. c, Simulated diffraction pattern for the first 6 diffraction orders. The 81 peaks considered in the population count are coloured in blue, with their radius indicating the extracted population n_k. Gray dots represent the peaks that can also be observed but are not included in the population count due to their low populations. Images in (a,b) are averaged over 30 experimental shots in order to visually emphasise the signal from very weakly populated high-order diffraction peaks.