Light has been seen leaving an atom cloud before it entered

This part seems to be the gist of their conclusion:

Specifically, we measure mean atomic excitation times ranging from (−0.82±0.31)τ0 for the most narrowband pulse to (0.54±0.28)τ0 for the most broadband pulse, where τ0 is the non-post-selected excitation time, given by the scattering (absorption) probability multiplied by the atomic lifetime τsp. These results suggest that negative values taken by times such as the group delay have more physical significance than has generally been appreciated.

In more plain English, the "group delay" refers to how signals (light in this case) are delayed while passing through a medium. Physicists have found that some materials with certain conditions (in this example extremely cold gas) can have negative group delays which suggests that light would take a negative amount of time to travel through them.

It seems to suggest that as light is passing through a medium and exciting electrons (which normally slows it down), it's possible for atoms that the light hasn't quite reached yet to get excited BEFORE the light actually gets there, which would lead to light potentially leaving the medium before it entered.

Before this experiment, this was just a theory that hadn't been tested, but now there's experimental proof that the negative group delay values do actually mean light leaves the medium before it enters if conditions are right.