Calculating how much time resonant photons spend as atomic excitations before being transmitted

When a single photon traverses a cloud of 2-level atoms on resonance, how much time does it spend as an atomic excitation, as measured by weakly probing the atoms? It turns out that the answer, on average, is simply the spontaneous lifetime, multiplied by the probability of the photon being scattered into a side-mode. An obvious inference from this is that photons that are transmitted spend no time as excited atoms, and photons that are scattered spend, on average, one spontaneous lifetime. Our recent experimental work (PRX Quantum 3, 010314) shows that this inference is incorrect. However, a complete theoretical treatment of the open-system dynamics for such a system has never, to our knowledge, been carried out. We examine this problem using the weak-value formalism, and find that transmitted photons in general spend a non-zero amount of time as atomic excitations, and that this time can even be negative. We also determine the corresponding time for scattered photons, which turns out to be related to the "Wigner time" associated with elastic scattering. This work provides insight into the complex histories of photons travelling through absorptive media.