Measuring the topology of quantum degeneracies using an NV center platform

In a recent experiment Brown et al. have used a cold atom platform to demonstrate a new protocol to characterize quantum degeneracies by driving the quantum state non-adiabatically through the degeneracy and making use of projective measurements. Using this protocol they were able to topologically distinguish conical degeneracies from quadratic ones in the band structure of their cold atom lattice. We show that their protocol provides the same topological characterization of the degeneracy as Berry's phase which is measured by adiabatic transport of the quantum state for the cases of conical and quadratic degeneracies considered by Brown et al, as well as more generally. We consider prospects for experimentally implementing the same protocol on an NV center platform.