  Conventional nonlinear spectroscopy uses classical light to detect matter
properties through the variation of its response with frequencies or time
delays. Quantum light opens up new avenues for spectroscopy by utilizing
parameters of the quantum state of light as novel control knobs and through the
variation of photon statistics by coupling to matter. We present an intuitive
diagrammatic approach for calculating ultrafast spectroscopy signals induced by
quantum light, focusing on applications involving entangled photons with
nonclassical bandwidth properties - known as "time-energy entanglement".
Nonlinear optical signals induced by quantized light fields are expressed using
time ordered multipoint correlation functions of superoperators. These are
different from Glauber's g- functions for photon counting which use normally
ordered products of ordinary operators. Entangled photon pairs are not
subjected to the classical Fourier limitations on the joint temporal and
spectral resolution. After a brief survey of properties of entangled photon
pairs relevant to their spectroscopic applications, different optical signals,
and photon counting setups are discussed and illustrated for simple multi-level
model systems.
