  We study the fundamental limits of the precision of estimating parameters of
a quantum matter system when it is probed by a travelling pulse of quantum
light. In particular, we focus on the estimation of the interaction strength
between the pulse and a two-level atom, equivalent to the estimation of the
dipole moment. Our analysis of single-photon pulses highlights the interplay
between the information gained from the absorption of the photon by the atom as
measured in absorption spectroscopy, and the perturbation to the temporal mode
of the photon due to spontaneous emission. Beyond the single-photon regime, we
introduce an approximate model to study more general states of light in the
limit of short pulses, where spontaneous emission can be neglected. We also
show that for a vast class of entangled biphoton states, quantum entanglement
between the signal mode interacting with the atom and the idler mode provides
no fundamental advantage and the same precision can be obtained with a
separable state. We conclude by studying the estimation of the electric dipole
moment of a sodium atom using quantum light. Our work initiates a quantum
information theoretic methodology for developing the theory and practice of
quantum light spectroscopy.
