Decoherence of ensembles of nitrogen-vacancy centers in diamond

We present a combined theoretical and experimental study of solid-state spin decoherence in an electronic spin bath, focusing specifically on ensembles of nitrogen-vacancy (NV) centers in diamond and the associated substitutional nitrogen spin bath. We perform measurements of NV spin free-induction decay (FID) times $T_2^{*}$ and spin-echo coherence times $T_2$ in 25 diamond samples with nitrogen concentrations [N] ranging from 0.01 to 300 ppm. We introduce a microscopic model and perform numerical simulations to quantitatively explain the degradation of both $T_2^{*}$ and $T_2$ over four orders of magnitude in [N]. Our analysis enables us to describe the NV ensemble spin coherence decay shapes as emerging consistently from the contribution of many individual NV centers.

Figure 1

(a) ${\mathrm{NV}}^{\text{−}}$ as central spin. (b) NV crystallographic defect and electronic spin-triplet ground-state level structure. (c) The free induction decay and spin-echo pulse sequence are used to measure $T_2^{*}$ and $T_2$, respectively. (d) Representative spin-echo decay fringes (blue) and envelope (red) for [N] = 80- and 6-ppm diamond samples. (e) Magnetic field dependence of $T_2$ for a select set of samples.

Figure 2

(a) NV ensemble coherence time $T_2$ as a function of [N] with orthogonal-distance regression fit to Eq. (1) (black dashed line) and range of values extracted from numerical simulation (gray band). Uncertainties are $95\%$ confidence intervals. Inset: Stretched exponential parameters $p$ with average value (black dotted line). $T_2$ measurements were taken with bias fields in the range of $2–30\mathrm{mT}$ (see Appendix pp2). (b) Similar fitting for NV ensemble dephasing time $T_2^{*}$. Measurments were taken at bias magnetic fields $2–3\mathrm{mT}$ at which any gradient-limited dephasing can be neglected.

Figure 3

Model for bath correlation time ${\tau }_c$. (a) The bath spin dynamics arise due to interaction between nearby P1 spins. (b) States $|\uparrow \downarrow \rangle$ and $|\downarrow \uparrow \rangle$ give spin bath dynamics under Eq. (2). (c) Mapping to a spin-1/2 system.

Figure 4

Simulated distribution of coupling strength $\mathrm{\Delta }$ (a) and bath correlation time ${\tau }_c$ (b) for $\left[\text{N}\right]=100\mathrm{ppm}$ extracted from ${10}^4$ spin bath configurations.