Dynamical Quantum Phase Transitions (DQPTs) are nonanalyticities in the Loschmidt echo, a quantity that characterizes the time evolution of quantum systems. These are analogous to thermal phase transitions, which are nonanalyticities in the thermal partition function. Over the past decade, DQPTs have been shown to have a close relationship with the equilibrium phases of quantum systems. DQPTs can serve as a probe of the underlying quantum critical points of a system.

In this thesis, we explore how DQPTs can occur in superconductors. We analytically show that DQPTs can occur in topological superconductors but not in the topologically trivial s-wave superconductor. We extend this analysis to study the spectral form factor, which is a related quantity that tells us about the time evolution of systems. We see how the spectral form factor of unconventional superconductors can have nonanalyticities, whereas that of the s-wave superconductor is featureless. These nonanalyticities arise due to the nontrivial structure of the superconducting gap function in momentum space, which in turn result from the symmetries of the Cooper pair wavefunction and the underlying lattice. We propose a method by which the spectral form factor can be measured if the superconducting Hamiltonian is simulated using qubits realized as Anderson pseudospins.

We find that the Schwinger-Keldysh mean field formalism is insufficient to evaluate these nonanalyticities in superconductors, and therefore develop a generalized mean field theory for this purpose. We present a simple model of a flat-band superconductor for which we can explicitly verify the validity of our generalized mean field theory.