Nuclear-spin relaxation in solid-state-defect quantum bits via electron-phonon coupling in the optically 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. With use of the nitrogen-vacancy color center in diamond as an example, by a combined group theory and density functional theory study we demonstrate that spin-lattice relaxation of the ${}^{14}\mathrm{N}$ nuclear spin is significantly enhanced due to strong entanglement with orbital degrees of freedom in the ${|}^{3}E⟩$ optically excited state of the defect. This mechanism is common to other solid-state-defect nuclear spins with similar optically excited states. Additionally, we propose a straightforward and versatile ab initio scheme for predicting orbital-dependent spin Hamiltonians for trigonal defects exhibiting orbital degeneracy.