The study of collective modes emerging from spontaneous symmetry breaking in interacting Fermi systems has recently gained a great interest in the condensed matter physics community. Experimental advance in time-resolved spectroscopic techniques (THz pump-probe spectroscopy) has allowed to access their typical excitation and relaxation time-scale. Even though much emphasis has been put so far on the superconducting amplitude (Higgs) mode, it has been recently shown that its detection is made difficult in BSC superconductors because of the weak coupling of the Higgs with the external field . Here we show that the situation is different in charge-density-wave systems (CDW), where the amplitude fluctuations of the CDW order parameter can become spectroscopically accessible via their coupling to the CDW phonon ,. By means of an effective-action approach we show that the non-linear optical response of a CDW system displays two distinguishable peaks: one at the threshold 2ΔCDW connected with the electron-hole pairs excitations and the other one at the renormalized phonon energy ωph ≤ 2ΔCDW. These features are responsible for the oscillations seen in pump-probe experiments, as recently shown experimentally .
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