  We survey the inclusion of interferometric elements in nonlinear spectroscopy
performed with quantum light. Controlled interference of electromagnetic fields
coupled to matter can induce constructive or destructive contributions of
microscopic coupling sequences (histories) of matter. Since quantum fields do
not commute, quantum light signals are sensitive to the order of light-matter
coupling sequence. Matter correlation functions are thus imprinted by different
field factors, which depend on that order. We identify the associated quantum
information obtained by controlling the weights of different contributing
pathways, and offer several experimental schemes for recovering it. Nonlinear
quantum response functions include out-of-time-ordering matter correlators
(OTOC) which reveal how perturbations spread throughout a quantum system
(information scrambling). Their effect becomes most notable when using
ultrafast pulse sequences with respect to the path difference induced by the
interferometer. OTOC appear in quantum-informatics studies in other fields,
including black holes, high energy, and condensed matter physics.
