  When a single photon traverses a cloud of 2-level atoms, the average time it
spends as an atomic excitation -- as measured by weakly probing the atoms --
can be shown to be the spontaneous lifetime of the atoms multiplied by the
probability of the photon being scattered into a side mode. A tempting
inference from this is that an average scattered photon spends one spontaneous
lifetime as an atomic excitation, while photons that are transmitted spend zero
time as atomic excitations. However, recent experimental work by some of us
[PRX Quantum 3, 010314 (2022)] refutes this intuition. We examine this problem
using the weak-value formalism and show that the time a transmitted photon
spends as an atomic excitation is equal to the group delay, which can take on
positive or negative values. We also determine the corresponding time for
scattered photons and find that it is equal to the time delay of the scattered
photon pulse, which consists of a group delay and a time delay associated with
elastic scattering, known as the Wigner time delay. This work provides new
insight into the complex and surprising histories of photons travelling through
absorptive media.
