Optical $\Lambda$ transitions and quantum computing in the $^{15}$N-V$^{-}$ Center in Diamond

We present a thorough derivation of the excited state energy levels of the negatively charged $^{15}$N-V$^{-}$ center in diamond for the strong applied electric field case. We show that in the $^{15}$N-V$^{-}$ center a spin non-conserving two-photon $\Lambda$ transition exists that is mediated by the hyperfine interaction, which provides the possibility to write quantum information. Using second order perturbation theory we obtain a $\Lambda$ transition rate of the order of 10 MHz at room temperature, which allows for approximately $10^4$ quantum logic operations within the spin coherence time $\tau_d(T=300 K)\approx 1m$s of the $^{15}$N-V$^{-}$ center.