Identity Test of Single ${\mathrm{NV}}^{-}$ Centers in Diamond at Hz-Precision Level

Atomiclike defects in solids are not considered to be identical owing to the imperfections of host lattice. Here, we found that even under ambient conditions, negatively charged nitrogen-vacancy (${\mathrm{NV}}^{-}$) centers in diamond could still manifest identical at Hz-precision level, corresponding to a ${10}^{-7}$-level relative precision, while the lattice strain can destroy the identity by tens of Hz. All parameters involved in the ${\mathrm{NV}}^{-}-{}^{14}\mathrm{N}$ Hamiltonian are determined by formulating six nuclear frequencies at 10-mHz-level precision and measuring them at Hz-level precision. The most precisely measured parameter, the ${}^{14}\mathrm{N}$ quadrupole coupling $P$, is given by $-4945754.9(8)\mathrm{Hz}$, whose precision is improved by nearly 4 orders of magnitude compared with previous measurements. We offer an approach for performing precision measurements in solids and deepening our understandings of NV centers as well as other solid-state defects. Besides, these high-precision results imply a potential application of a robust and integrated atomiclike clock based on ensemble NV centers.

Figure 1

Characterization of the NV center coupled with the attendant ${}^{14}\mathrm{N}$ nuclear spin and its spin states in the ground state ${}^3{A}_2$. (a) Atomic structure of the NV center in diamond lattice. The orange sphere denotes the nitrogen atom while the white one the vacancy and the black one carbon atoms. (b) Level diagrams of the ${\mathrm{NV}}^{-}$ electron and the coupled electron and nuclear spins in the ground state. The state marked by the gray circle is the initialized state. The gray, black, and orange arrows indicate the six nuclear spin transitions driven by rf pulses. The blue arrow shows the electron spin transition driven by MW pulses. The MW, RF1, and RF2 pulses are used in the pulse sequence in Fig. 2. (c) Level structure of the ${\mathrm{NV}}^{-}-{}^{14}\mathrm{N}$ system described by the principal term $H_{\parallel }$ of the entire Hamiltonian. The lines with two arrowheads denote the transition matrix elements from the perturbative term $H_{\perp }$. The dashed-dotted (purple), dashed (blue), and dotted (red) lines stand for the transverse hyperfine interaction, the transverse Zeeman terms for the electron spin and the nuclear spin in $H_{\perp }$, respectively. (d) Reduced three-level system in analogy to CSRT. Energy level repulsions induced by the transverse hyperfine interaction and the transverse Zeeman term of the electron spin are displayed by dashed-dotted and dashed lines. The values are the transition matrix elements of three transverse items.

Figure 2

Ramsey interferometry and interference pattern. (a) The pulse sequence of laser, MW, and rf for Ramsey interference between the state $|m_S=-1,m_I=0⟩$ and the state $|m_S=-1,m_I=-1⟩$. The sequence encircled by the dashed box is aimed at population transfer to the state for performing Ramsey interference. Find the corresponding manipulation of spin states in Fig. 1 according to the names or the colors. (b) Resulting interference pattern after applying the sequence above to the last NV center in Fig. 3. The black line is data fitting using the function $\{a\sin [2\pi (\delta f)t+{\varphi }_0]+b\}{e}^{-(t/T_2^{*}{)}^p}+c$ with $\delta f$ meaning the detuning.

Figure 3

Identity test. From top to bottom: the ${}^{14}\mathrm{N}$ quadrupole coupling $P$, the longitudinal component $A_{\parallel }$, the transverse component $A_{\perp }$ of the hyperfine interaction, and the ratio ${\gamma }_e/{\gamma }_n$ of the gyromagnetic ratio of the ${\mathrm{NV}}^{-}$ electron to that of ${}^{14}\mathrm{N}$ for seven NV centers. Among them, the first five NVs (red) are far away from SILs and the last two NVs (black) are inside a SIL. The values represented by the dashed lines are the weighted averages of the NVs far away from SILs for $P$ and $A_{\parallel }$, and all seven NVs for $A_{\perp }$ and ${\gamma }_e/{\gamma }_n$.

Figure 4

Allan deviation of the proposed NV clock. The solid black line shows the fractional frequency instability at 1 s of integration as a function of the number of ensemble NV centers (bottom) and the density of NV centers (top) for a $1{\mathrm{mm}}^3$ diamond. For comparison, the fractional frequency instability is $2.5\times {10}^{-10}$ for a Cs chip clock [46], $2\times {10}^{-11}$ for a commercial Rb clock [47], and $1.2\times {10}^{-11}$ for a commercial Cs clock [48].