So what's happening is negative group delay as a pulse of light travels through a medium. (In this case a very cold 85Rb atom fog -- a few microkelvins above absolute zero.)

+ Post-selection, in that they only look at cases where group photons get transmitted rather than absorbed.

+ Weak-value formalism, where a measurement can yield values outside the usual spectrum if you measure "weakly" and then condition on a specific outcome.

In concrete terms, you'll never see a photon exit the fog before the pulse strikes the fog. All that's happened is that the group's peak shifted. "Negative time" is in the measured "delay" or "excitation time" of the entire pulse, which has a minus sign in front of it. It's interesting but not very physically meaningful.