Amplitude collective modes in superconductors and their coupling to charge-density waves

At low temperature the long-wavelength perturbations of the amplitude of the superconducting gap propagate as an undamped collective mode with a finite frequency. The dispersion and damping of this mode are calculated. The phase or the Bogoliubov modes of a superconductor are strongly affected by Coulomb interactions and rendered indistinguishable from plasmons. By contrast, the amplitude modes are shown not to perturb the charge density thus remaining unaffected by the Coulomb interactions. Under certain conditions long-wavelength phonons couple to this mode. This coupling is derived and the observation of the amplitude mode through Raman scattering experiments in the charge-density-wave compound Nb${\mathrm{Se}}_2$ are quantitatively explained.