Timekeeping with electron spin states in diamond

Frequency standards based on atomic states, such as Rb or Cs vapors, or single-trapped ions, are the most precise measures of time. Here we propose and analyze a precision oscillator approach based upon spins in a solid-state system, in particular, the nitrogen-vacancy defect in single-crystal diamond. We show that this system can have stability approaching portable atomic standards and is readily incorporable as a chip-scale device. Using a pulsed spin-echo technique, we anticipate an Allan deviation of ${\sigma }_y={10}^{-7}{\tau }^{-1/2}$ limited by thermally-induced strain variations; in the absence of such thermal fluctuations, the system is limited by spin dephasing and harbors an Allan deviation nearing $\sim {10}^{-12}{\tau }^{-1/2}$. Potential improvements based upon advanced diamond material processing, temperature stabilization, and nanophotonic engineering are discussed.

Figure 1

Nitrogen-vacancy-center energy levels and resonant response. (a) The lowest lying triplet (${}^{3}E$, ${}^{3}A$) and singlet (${}^{1}E$, ${}^{1}A$) orbital states of the NV${}^{-}$ center. $I$, $\gamma$, $\kappa$, and $\lambda$ represent the absorption, PL, intersystem crossing, and deshelving rates, respectively. Sublevels 0 and 2 are $S_z$ eigenstates $|0\rangle$, whereas 1 and 3 are $|\pm 1\rangle$, with the degeneracy lifted by small crystal strain or applied magnetic field. The model is simplified by approximating both singlet states as a single metastable level. (b) The steady-state fluorescence emission of the NV center under continuous optical and microwave irradiation, detuned from resonance by $\Delta$. The PL response function, as derived from a master equation approach, is approximated by a Lorentzian: $F\left(\Delta \right)=I_0(1-\frac{C\delta {\nu }^2}{{(\Delta /\pi )}^2+\delta {\nu }^2})$, where $C$ is the modulation depth and $\delta \nu$ is the FWHM [18].

Figure 2

The spin-1 ground state $|0\rangle$ state is prepared by optical pumping. 45${}^{\circ }$ rotations ensure that the $S_z$ term evolution averages within the echo sequence, while the $S_z^2$ does not, as illustrated in the Euler projections of the state (inset). These projections are to show the equipotential surface plots, $\mathcal{E}(\kappa ,\phi )$ the density matrix element $|0\rangle \langle 0|$ for each point of the sequence after undergoing rotations $\kappa$ and $\phi$ about the $S_y$ and $S_z$ axes, respectively: $\mathcal{E}=\langle 0\left|R^{†}\rho R\right|0\rangle$, $R=e^{i\phi S_y}{e}^{i\kappa S_z}$ and provide a visual representation of the coherences between states, akin to the Bloch sphere for two-level systems [19]. Note that the $S_z$ term is completely refocused by the echo sequence, while the $S_z^2$ operator evolves, resulting in a different magnitude between the first and sixth states. A transient fluorescence (TF) measurement records the photocurrent for $\sim 300$ ns timed with a pulse of green light.

Figure 3

Allan deviation for atomic and solid-state standards. Note that in moving from a CW scheme with a single NV center to an ensemble of centers with pulsed excitation and detection, we gain almost six decades of improvement. Cited deviations are as follows: Al-ion clock [2, 5]; Sr lattice clock [4]; thorium clock (theoretical) [35]; Cs chip clock [7]; TXCO and commercial Rb figures available from Stanford Research Systems (www.thinksrs.com); surface acoustic wave (SAW) oscillator is quote from Epson Toyocom Corporation EG-4101$/$4121CA datasheet. Pulsed echo NV samples are assumed to have 0.01 ppb NV center concentrations for a 1 mm${}^3$-sized sample. Note the difference of roughly three orders of magnitude achieved by synchronizing the temperature compared to the uncompensated thermal drift. The ideal echo scheme assumes no temperature dependence, with sensitivity limited by the spin lifetime.

Figure 4

Schematic of the diamond frequency standard. A thin (100 $\mu$m) diamond chip is surrounded by dielectric stacks (Bragg reflectors) on both sides to create a resonant cavity for 532 nm excitation in order to reduce the power requirements. On-chip 532 nm excitation comes from a doubled 1064 nm surface emitting laser (not shown). Silicon photodetectors underneath the diamond serves to collect emission. It may also be advantageous to collect emission from the near side of the device [22]. Microwaves, which address the NV magnetic sublevels, are applied to the entire sample by a planar stripline.

Figure 5

Temperature stabilization using two clocks. (a) The two diamond clocks have differing $d{D}_{\text{gs}}/d\theta$ since they are clamped to substrates with different thermal expansion coefficients, ${\eta }_{1,2}$ and different Young's moduli, $E_{1,2}$. (b) Schematic illustration of a synchronized composite clock setup. (c) Alternatively, using strain engineering, a single clamp may be designed to fully cancel the temperature dependence of the NV ZFS.

Figure 6

Simulated $D_{\text{gs}}$ variation with temperature for tungsten (green) and brass (blue) clamps. (b) Spatial frequency variation as a function of position within the diamond disk. (c) Clamp and diamond geometry.