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

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