  Vacancy centers in diamond have proven to be a viable solid-state platform
for quantum coherent opto-electronic applications. Among the variety of vacancy
centers, silicon-vacancy (SiV) centers have recently attracted much attention
as an inversion-symmetric system that is less susceptible to electron-phonon
interactions. Nevertheless, phonon-mediated processes still degrade the
coherent properties of SiV centers, however characterizing their
electron-phonon coupling is extremely challenging due to their weak
spectroscopic signatures and remains an open experimental problem. In this
paper we theoretically investigate signatures of electron-phonon coupling in
simulated linear and nonlinear spectra of SiV centers. We demonstrate how even
extremely weak electron-phonon interactions, such as in SiV centers, may be
completely characterized via nonlinear spectroscopic techniques and even
resolved between different fine-structure transitions.
