Double-Quantum Spin-Relaxation Limits to Coherence of Near-Surface Nitrogen-Vacancy Centers

We probe the relaxation dynamics of the full three-level spin system of near-surface nitrogen-vacancy (NV) centers in diamond to define a $T_1$ relaxation time that sets the $T_2\le 2T_1$ coherence limit of the NV’s subset qubit superpositions. We find that double-quantum spin relaxation via electric field noise dominates $T_1$ of near-surface NVs at low applied magnetic fields. Furthermore, we differentiate $1/f^{\alpha }$ spectra of electric and magnetic field noise using a novel noise-spectroscopy technique, with broad applications in probing surface-induced decoherence at material interfaces.

Figure 1

(a) Surface-noise spectroscopy with the triplet ground state of a shallow NV center. The double-quantum relaxation channel (orange, $\gamma$) is sensitive to electric field noise, and the single-quantum channel (blue, $\mathrm{\Omega }$) is sensitive to magnetic field noise. An applied dc magnetic field tunes the DQ transition frequency ${\omega }_{\pm 1}/2\pi$. (b),(c) Measurement sequences to extract the relaxation rates $\mathrm{\Omega }$ and $\gamma$. The spin is initialized into population (b) ${\rho }_0=1$ or (c) ${\rho }_{-1}=1$ by a green laser pulse and, for ${\rho }_{-1}=1$, a microwave ${\pi }_{0,-1}$ pulse. After a dark time $\tau$, any of the three spin state populations ${\rho }_j$ can be read out by a choice of ${\pi }_{0,\pm 1}$ pulse before photoluminescence detection, giving signal $S_{i,j}$. (d) Population decay data with three-level relaxation model fits as solid lines.

Figure 2

Enhancement of SQ coherence time using $\mathrm{CPMG}\text{−}N$ for shallow NVs under conditions of (a) large $\gamma$ at ${\omega }_{\pm 1}/2\pi =37.1\mathrm{MHz}$ and (b) small $\gamma$ at ${\omega }_{\pm 1}/2\pi =1376\mathrm{MHz}$. Data shown are Hahn echo (green diamonds) and $\mathrm{CPMG}\text{−}N$ (gray squares), where $N$ is the number of $\pi$ pulses, and solid lines are fits to $\exp [-(T/T_2{)}^n]$. Dashed red lines are reference plots of $\exp (-T/T_1)$ using the measured $T_1={(3\mathrm{\Omega }+\gamma )}^{-1}$.

Figure 3

(a) Measured DQ relaxation rates $\gamma$ for three shallow NV centers versus DQ frequency splitting $f={\omega }_{\pm 1}/2\pi$ tuned by applied field $B_z$. Each symbol type refers to one NV. The $1/f^{\alpha }$-type dependence is attributed to surface-related electric field noise and saturation at large ${\omega }_{\pm 1}$ is attributed to bulk effects. (b) Measured SQ relaxation rate $\mathrm{\Omega }$ for the same NVs.

Figure 4

Measured noise spectra in terms of coupling power (${\mathrm{Hz}}^2/\mathrm{Hz}$) and transverse electric field power (${\mathrm{V}}^2{\mathrm{m}}^{-2}/\mathrm{Hz}$) for shallow NVs A1 (a),(b) and A8 (c),(d) using dephasing spectroscopy (left plots) and DQ relaxation spectroscopy (right plots). Each NV data set is jointly fit to a noise model (green solid line) of three parts: $1/f^{\alpha }$-like electric fields (blue dashed-dotted line), magnetic fields (red dashed line), and a minimum relaxation rate ${\gamma }_{\infty }$ due to bulk effects (horizontal dashed line).