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When a single photon traverses a cloud of 2-level atoms, the average time itspends as an atomic excitation -- as measured by weakly probing the atoms --can be shown to be the spontaneous lifetime of the atoms multiplied by theprobability of the photon being scattered into a side mode. A temptinginference from this is that an average scattered photon spends one spontaneouslifetime as an atomic excitation, while photons that are transmitted spend zerotime as atomic excitations. However, recent experimental work by some of us[PRX Quantum 3, 010314 (2022)] refutes this intuition. We examine this problemusing the weak-value formalism and show that the time a transmitted photonspends as an atomic excitation is equal to the group delay, which can take onpositive or negative values. We also determine the corresponding time forscattered photons and find that it is equal to the time delay of the scatteredphoton pulse, which consists of a group delay and a time delay associated withelastic scattering, known as the Wigner time delay. This work provides newinsight into the complex and surprising histories of photons travelling throughabsorptive media.

How much time does a photon spend as an atomic excitation before being transmitted?