Analytic solutions to the central-spin problem for nitrogen-vacancy centers in diamond

Due to interest in both solid-state-based quantum computing architectures and the application of quantum mechanical systems to nanomagnetometry, there has been considerable recent attention focused on understanding the microscopic dynamics of solid-state spin baths and their effects on the coherence of a controllable, coupled central electronic spin. Using a systematic approach based on the spatial statistics of the spin-bath constituents, we develop a detailed, purely analytic theory for the central-spin decoherence problem of a nitrogen-vacancy center electron coupled to its native 1.1% bath of C-13 nuclear spins. Our theory reproduces the experimental and numerical results found in the literature, and provides a detailed theoretical description of the relevant decoherence profiles, their associated rates, corresponding electron spin-echo envelope modulations features, and an explicit analytic account of why the strength of an applied magnetic field has such a profound effect on the coherence time of the central spin.