  Time-resolved spectroscopies using intense THz pulses appear as a promising
tool to address collective electronic excitations in condensed matter. In
particular recent experiments showed the possibility to selectively excite
collective modes emerging across a phase transition, as it is the case for
superconducting and charge-density-wave (CDW) systems. One possible signature
of these excitations is the emergence of coherent oscillations of the
differential probe field in pump-probe protocols. While the analogy with the
case of phonon modes suggests that the basic underlying mechanism should be a
sum-frequency stimulated Raman process, a general theoretical scheme able to
describe the experiments and to define the relevant optical quantity is still
lacking. Here we provide this scheme by showing that coherent oscillations as a
function of the pump-probe time delay can be linked to the convolution in the
frequency domain between the squared pump field and a Raman-like non-linear
optical kernel. This approach is applied and discussed in few paradigmatic
examples: ordinary phonons in an insulator, and collective charge and Higgs
fluctuations across a superconducting and a CDW transition. Our results not
only account very well for the existing experimental data in a wide variety of
systems, but they also offer an useful perspective to design future experiments
in emerging materials.
