Nuclear spin relaxation in solid state defect quantum bits via electron-phonon coupling in their optical excited state

Optically accessible solid state defect spins serve as a primary platform for quantum information processing, where precise control of the electron spin and ancillary nuclear spins is essential for operation. Using the nitrogen-vacancy (NV) color center in diamond as an example, we employ a combined group theory and density functional theory study to demonstrate that spin-lattice relaxation of the $^{14}$N nuclear spin is significantly enhanced due to strong entanglement with orbital degrees of freedom in the $|^3E\rangle$ optical excited state of the defect. This mechanism is common to other solid-state defect nuclear spins with similar optical excited states. Additionally, we propose a straightforward and versatile \textit{ab initio} scheme for predicting orbital-dependent spin Hamiltonians for trigonal defects exhibiting orbital degeneracy.