Coherence of single spins coupled to a nuclear spin bath of varying density

The dynamics of single electron and nuclear spins in a diamond lattice with different $^{13}\text{C}$ nuclear spin concentration is investigated. It is shown that coherent control of up to three individual nuclei in a dense nuclear spin cluster is feasible. The free-induction decays of nuclear spin Bell states and single nuclear coherences among $^{13}\text{C}$ nuclear spins are compared and analyzed. Reduction in a free-induction-decay time ${T}_{2}^{\ensuremath{\ast}}$ and a coherence time ${T}_{2}$ upon increase in nuclear spin concentration has been found. For pure diamond, ${T}_{2}^{\ensuremath{\ast}}$ as long as $30\text{ }\ensuremath{\mu}\text{s}$ and ${T}_{2}$ of up to 0.65 ms for the electron spin has been observed. The $^{13}\text{C}$ concentration dependence of ${T}_{2}^{\ensuremath{\ast}}$ is explained by Fermi contact and dipolar interactions with nuclei in the lattice. It has been found that ${T}_{2}$ decreases approximately as $1/n$, where $n$ is $^{13}\text{C}$ concentration, which corresponds to the reported theoretical line of ${T}_{2}$ for an electron spin interacting with a nuclear spin bath.