How much time do transmitted photons spend as atomic excitations? Theory and experimental progress

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. A tempting inference from this is that photons that are scattered spend, on average, one atomic lifetime as atomic excitations, and photons that are transmitted through the medium spend no time at all. Our previous experimental work (PRX Quantum 3, 010314) shows that this inference is incorrect, and that transmitted photons can indeed spend time as atomic excitations. 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. Progress towards an experimental test of this theory using cold 85Rb atoms will also be presented. This work provides new insight into the complex and surprising histories of photons travelling through absorptive media.