Weakly measuring the time atoms spend in the excited state due to a photon they don't absorb

When a resonant photon traverses a sample of absorbing atoms, how much time do atoms spend in the excited state? Does the answer depend on whether the photon is ultimately scattered or transmitted? In particular, if it is not scattered, meaning that it emerges in the direction it entered, does it cause atoms to spend any time in the excited state?, and if so, how much? We report an experimental measurement of the time a transmitted photon causes atoms to spend in the excited state. The experiment is carried out in an ultra-cold gas of Rubidium 85 atoms and involves simultaneously recording changes to the excited state occupation number induced by incident photons and whether those photons are transmitted. For short pulses and an optically thick medium, we find that the average time atoms spend in the excited state due to one transmitted photon is not zero, but rather (77 +/- 16)% of the time the average incident photon causes them to spend in the excited state. This observation of "excitation without loss" is attributed to coherent forward emission, which arises, for instance, due to the sign-flip the instantaneous Rabi frequency (pulse envelope) picks up when a broadband pulse propagates through an optically thick medium with highly frequency-dependent absorption. These results reveal the complex history of photons as they propagate through an absorbing medium and illustrate the power of utilizing post-selection to experimentally investigate the past behaviour of observed quantum systems.