  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 absorbed or transmitted? In particular, if it is
$\textit{not}$ absorbed, does it cause atoms to spend any time in the excited
state, and if so, how much? In an experiment with ultra-cold Rubidium atoms, we
simultaneously record whether atoms are excited by incident ("signal") photons
and whether those photons are transmitted. We measure the time spent by atoms
in the excited state by using a separate, off-resonant "probe" laser to monitor
the index of refraction of the sample - that is, we measure the nonlinear phase
shift written by a signal pulse on this probe beam - and use direct detection
to isolate the effect of single transmitted 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 average time atoms spend in the excited state due to one transmitted
photon is not zero, but rather (77 $\pm$ 16)% of the time the average incident
photon causes them to spend in the excited state. We attribute this observation
of "excitation without loss" to coherent forward emission, which can arise when
the instantaneous Rabi frequency (pulse envelope) picks up a phase flip - this
happens naturally when a broadband pulse propagates through an optically thick
medium with frequency-dependent absorption [1]. These results unambiguously
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.
