Collective Modes of the Superconducting Condensate

When a continuous symmetry is spontaneously broken, collective modes emerge. Usually, their spectrum is dominated by the low-energy physics of massless Goldstone modes. Superconductors, that break U(1) symmetry, are different. Here, the Goldstone boson is gapped out due to the Anderson-Higgs mechanism. The superconducting condensate can therefore host a zoo of massive collective excitations that are stable for lack of a gapless decay channel. The most prominent of them is the Higgs mode. Spectroscopy of collective modes can be a probe to reveal the nature of the superconducting state. In this thesis, we study the signatures of collective modes in nonlinear optical experiments. We ex- plore the theoretical description of a new excitation scheme. We show how impurity scattering significantly enhances the optical Higgs mode response. We apply group theoretical methods to multi-order-parameter theories and investigate the microscopic signature of coupled modes in third harmonic generation experi- ments. Finally, we study the phenomenology and collective mode spectrum of an exotic system of twisted cuprate bilayers that supports topological superconductivity.