New frontiers in the optical detection of superconducting collective modes

Spontaneous symmetry breaking across the superconducting critical temperature is characterized by the emergence of a finite order parameter and by collective electronic excitations connected to its fluctuations. The amplitude fluctuations are usually named Higgs mode for their analogy with the massive boson of the standard model, while phase fluctuations identify the Goldstone massless excitation expected when the broken symmetry is a continuous one. Their description needs to go beyond BCS theory, which nonetheless explains with great accuracy the conventional spectroscopies in standard superconductors. The reason is that these collective excitations are spectroscopically inert in ordinary single-band BCS superconductors. In the last few years, a number of experiments have been proposed to detect and identify the superconducting Higgs and Goldstone mode. Here I will review our theoretical understanding of the optical properties of superconducting collective excitations in a variety of systems, considering the peculiar effects due to space inhomogeneity, coexisting states and non-linear optical response.