  We consider the estimation of a Hamiltonian parameter of a set of highly
photosensitive samples, which are damaged after a few photons $N_{\rm abs}$ are
absorbed, for a total time $T$. The samples are modelled as a two mode photonic
system, where photons simultaneously acquire information on the unknown
parameter and are absorbed at a fixed rate. We show that arbitrarily intense
coherent states can obtain information at a rate that scales at most linearly
with $N_{\rm abs}$ and $T$, whereas quantum states with finite intensity can
overcome this bound. We characterise the quantum advantage as a function of
$N_{\rm abs}$ and $T$, as well as its robustness to imperfections (non-ideal
detectors, finite preparation and measurement rates for quantum photonic
states). We discuss an implementation in cavity QED, where Fock states are both
prepared and measured by coupling atomic ensembles to the cavities. We show
that superradiance, arising due to a collective coupling between the cavities
and the atoms, can be exploited for improving the speed and efficiency of the
measurement.
