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 eventually transmitted? If the photon is not absorbed, do atoms spend any time in the excited state, and if so, how much? In an experiment with ultra-cold Rubidium atoms, we simultaneously measure whether atoms are excited by incident photons and whether those photons are transmitted. We measure the time spent by atoms in the excited state by using a weak off-resonant probe laser to monitor the index of refraction of the sample (in other words, we measure the nonlinear phase shift written by a "signal" pulse on a second "probe" beam) and separately use direct detection to distinguish transmitted and absorbed photons. For short pulses (10 ns, to be compared to the 26 ns atomic lifetime) and an optically thick medium (peak OD = 4, leading to 60% absorption given our broad bandwidth), we find that the time atoms spend in the excited state due to transmitted photons is not zero, but rather 0.79 +/- 0.16 of the time spent in the excited state due to the average incident photon. We attribute this observation of "excitation without loss" to coherent forward emission related to the frequency-dependent absorption experienced by a broadband pulse propagating through an optically thick medium. These results raise intriguing new questions about the history of post-selected photons, which motivate further experiment, and demand a fully quantized theoretical treatment.

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