State-dependent phonon-limited spin relaxation of nitrogen-vacancy centers

Understanding the limits to the spin coherence of the nitrogen-vacancy (NV) center in diamond is vital to realizing the full potential of this quantum system. We show that relaxation on the $|m_s=-1\rangle \leftrightarrow |m_s=+1\rangle$ transition occurs approximately twice as fast as relaxation on the $|m_s=0\rangle \leftrightarrow |m_s=\pm 1\rangle$ transitions under ambient conditions in native NVs in high-purity bulk diamond. The rates we observe are independent of NV concentration over four orders of magnitude, indicating they are limited by spin-phonon interactions. We find that the maximum theoretically achievable coherence time for an NV at 295 K is limited to 6.8(2) ms. Finally, we present a theoretical analysis of our results that suggests Orbach-like relaxation from quasilocalized phonons or contributions due to higher-order terms in the spin-phonon Hamiltonian are the dominant mechanism behind $|m_s=-1\rangle \leftrightarrow |m_s=+1\rangle$ relaxation, motivating future measurements of the temperature dependence of this relaxation rate.

Figure 1

State-dependent spin relaxation rates of nitrogen-vacancy centers. (a) Ground-state level structure of the ${\mathrm{NV}}^{-}$ electronic spin. Qubit transitions occur at rate $\mathrm{\Omega }$, while qutrit transitions occur at rate $\gamma$. An applied magnetic field lifts the $|\pm 1\rangle$ degeneracy by ${\mathrm{\Delta }}_{\pm }$. (b) Measurement of state-dependent decay curves. Relaxation out of $|0\rangle$ (cyan squares) is described by a single exponential with decay rate $3\mathrm{\Omega }$ (cyan curve). Relaxation out of $|+1\rangle$ (yellow diamonds) is described by a biexponential that depends on both $\mathrm{\Omega }$ and $\gamma$ (yellow curve). (The deviation of the yellow curve from unity at $\tau =0$ is due to $\pi$-pulse infidelity.) For data shown, $\mathrm{\Omega }=56\left(3\right){\text{s}}^{-1}, \gamma =132\left(11\right){\text{s}}^{-1}$. (c) Semilog plot of subtracted population curves, $F_{\mathrm{\Omega }}$ and $F_{\gamma }$. Displayed data is from the same measurement series as that shown in (b).

Figure 2

Rates $\gamma$ and $\mathrm{\Omega }$ for samples with differing defect concentration. (a)–(c) Confocal microscope images of samples A, B, and C, displaying NV concentrations ranging from approximately ${10}^{-5}–{10}^{-1}$ ppb. (d) Average measured rates $\gamma$ and $\mathrm{\Omega }$ at small off-axis magnetic fields in samples A, B, and C. On-axis magnetic fields are in the range $20–60\text{G}$. Measurements of both $\gamma$ and $\mathrm{\Omega }$ are equivalent to within error in all three samples, indicating no dependence on defect concentration in the measured regime. (Inset) The ratio $\gamma /\mathrm{\Omega }\approx 2$ for each pair of points shown in the main plot. Error bars are one standard error.

Figure 3

Measurements of $\mathrm{\Omega }$ and $\gamma$ as functions of $B_{\parallel }$ and $B_{\perp }$. To allow for visual separation between the points in the figure, only measurements made for magnetic fields under 65 G are shown. (See Fig. 6 for an extended version of this figure.) Green lines are weighted linear regressions intended to show correlations. The regressions account for all data, including that not shown in the plots for clarity. Error bars and shaded regions demonstrate one standard error. The data plotted in (a) and (c) and in (b) and (d) correspond to the same set of measurements of $\mathrm{\Omega }$ and $\gamma$, respectively.

Figure 4

Two examples of two-phonon processes driving the qubit transition. In both processes, a phonon of energy $\hslash {\omega }_{\text{em}}$ is emitted and a phonon of energy $\hslash {\omega }_{\text{abs}}$ is absorbed. Importantly, the emission-followed-by-absorption variant (a) is associated with a different suppressive factor than the absorption-followed-by-emission variant (b) in perturbation theory.

Figure 5

Subtraction curves $F_{\mathrm{\Omega }}$ using $P_{0,-1}$ and $P_{0,+1}$ under a magnetic field of $\sim 300\text{G}$. Both curves are well described by a single exponential, indicating ${\mathrm{\Omega }}_{0,-1}={\mathrm{\Omega }}_{0,+1}$. The qubit rates extracted from the fits agree to within error.

Figure 6

Extended version of Fig. 3, showing all data taken in this work for which the magnitude and orientation of the applied magnetic field is known. The vertical gray line at 65 G indicates the range of the data displayed in the main text version of this figure.