Microscopic theory of electron spin relaxation in N@C$_{60}$

Endohedral N@C$_{60}$ exhibits an extremely long electron spin relaxation time and offers a great potential in storing and processing quantum information. Here we present a microscopic theory of electron spin relaxation in N@C$_{60}$. The theory combines (1) the spin-orbit interaction of N $2p$ electrons, which mixes the ground state $^4S$ with excited $^2P$ and $^2D$ states, and (2) the coupling between the N $2p$ electrons and C$_{60}$ $H_g$ vibrations, which facilitates transitions between $^2P$ and $^2D$ states. The spin relaxation occurs via a two-phonon (Raman) process by absorbing a $H_g$ phonon and emitting another at the (approximately) same frequency. The theory consistently explains measured spin relaxation time $T_1$ and its temperature dependence, and predicts two distinct spin decoherence $T_2$ constants. In addition, the excellent agreement between theory and experiment suggests a universal importance of the two-phonon Raman process in determining spin relaxation in nanostructures such as quantum dots, where a one-phonon process is ineffectual in flipping electron spins because of the lack of low-energy phonons in nanostructures.