Can photons be detected without being absorbed? I am thinking about a detector that would beep if light passes through it. Is it possible?
 A: Yes, according to a paper by the Rempe group [Science 342, 1349 (2013)], photons can be detected after reflection by an optical resonator that contains a prepared atom in a superposition of two states. The reflection of the photon then results in a certain projection of the state that can be probed to detect the incident photon indirectly.
A: It is indeed possible, as demonstrated by the group of Serge Haroche in 1999 using so-called quantum non-demolition Ramsey interferometry.  The idea was to observe the presence or absence of a photon in a cavity by observing its interaction with atoms.
This beautiful experiment relies heavily on the behaviour of quantum  superposition of atomic states.  A simplified explanation is that the presence of a photon in the cavity results in an additional relative phase shift in one term of the superposition of atomic states, and this additional phase shift can be detected.  Since all the measurements are done on atomic rather than photonic states, one can infer (and thus detect) the presence of the photon without actually absorbing it.
A: A photon can (in theory) be measured as a slight impulse change on a solar sail, i.e. mirror.
