  Silicon-vacancy (SiV) center in diamond is a photoluminescence (PL) center
with a characteristic zero-phonon line energy at 1.681 eV that acts as a
solid-state single photon source and, potentially, as a quantum bit. The
majority of the luminescence intensity appears in the zero-phonon line;
nevertheless, about 30\% of the intensity manifests in the phonon sideband.
Since phonons play an essential role in the operation of this system, it is of
importance to understand the vibrational properties of the SiV center in
detail. To this end, we carry out density functional theory calculations of
dilute SiV centers by embedding the defect in supercells of a size of a few
thousand atoms. We find that there exist two well-pronounced quasi-local
vibrational modes (resonances) with $A_{2u}$ and $E_u$ symmetries,
corresponding to the vibration of the Si atom along and perpendicular to the
defect symmetry axis, respectively. Isotopic shifts of these modes explain the
isotopic shifts of prominent vibronic features in the experimental SiV PL
spectrum. Moreover, calculations show that the vibrational frequency of the
$A_{2u}$ mode increases by about 30\% in the excited state with respect to the
ground state, while the frequency of the $E_u$ mode increases by about 5\%.
These changes explain experimentally observed isotopic shifts of the
zero-phonon line energy. We also emphasize possible dangers of extracting
isotopic shifts of vibrational resonances from finite-size supercell
calculations, and instead propose a method to do this correctly.

Quantum Physics;Materials Science;Computational Physics