Stable three-axis nuclear-spin gyroscope in diamond

Gyroscopes find wide applications in everyday life from navigation and inertial sensing to rotation sensors in hand-held devices and automobiles. Current devices, based on either atomic or solid-state systems, impose a choice between long-time stability and high sensitivity in a miniaturized system. Here, we introduce a quantum sensor that overcomes these limitations by providing a sensitive and stable three-axis gyroscope in the solid state. We achieve high sensitivity by exploiting the long coherence time of the ${}^{14}$N nuclear spin associated with the nitrogen-vacancy center in diamond, combined with the efficient polarization and measurement of its electronic spin. Although the gyroscope is based on a simple Ramsey interferometry scheme, we use coherent control of the quantum sensor to improve its coherence time and robustness against long-time drifts. Such a sensor can achieve a sensitivity of $\eta \sim 0.5\left(\mathrm{mdeg}{\mathrm{s}}^{-1}\right)/\sqrt{\mathrm{Hz}{\mathrm{mm}}^3}$ while offering enhanced stability in a small footprint. In addition, we exploit the four axes of delocalization of the nitrogen-vacancy center to measure not only the rate of rotation, but also its direction, thus obtaining a compact three-axis gyroscope.

Figure 1

Conceptual design of the nNV gyro. A slab of diamond of dimensions $(2.5\times 2.5){\mathrm{mm}}^2\times 150\mu \mathrm{m}$ is anchored to the device body. Radio-frequency (rf) coils and microwave ($\mu$w) co-planar waveguides are fabricated on the diamond for fast control. NV centers are polarized by a green laser (532 nm), and state-dependent fluorescence intensity (637 nm) is collected employing a side-collection technique [11]. A second set of rf coils rotate with respect to the diamond-chip frame, for example, by being attached to one or more rings in a mechanical gimbal gyroscope (not shown). The ${}^{14}$N nuclear spins are used as probes of the relative rotation between the diamond frame and the external rf-coil frame. See Fig. 5 for a possible actual implementation.

Figure 2

nNV-gyro control sequence. rf pulses, resonant with the quadrupolar transition of the ${}^{14}$N nuclear spin, are applied in a frame rotating at a rate $\Omega$ with respect to the diamond, thus, inducing a phase $\Omega t$. The nuclear spin is first initialized by polarization transfer from the NV-electronic spin. An echo pulse is applied in the diamond frame to refocus static frequency shifts. Finally, $\Omega$ is extracted by mapping the nuclear-spin phase shift onto a population difference of the electronic spin and measuring the corresponding fluorescence intensity.

Figure 3

nNV-gyro sensitivity in $\left(\mathrm{mdeg}{\mathrm{s}}^{-1}\right)/\sqrt{\mathrm{Hz}}$ as a function of density $\eta \left(n\right)=\frac{e^{t_{\mathrm{map}}/T_{2,\mathrm{NV}}^{*}}{e}^{t/T_{2,n}}}{C\sqrt{nV/4}}\frac{\sqrt{t+t_d}}{t}$. Here, the exponential factors take into account the sensitivity degradation due to spin decoherence [12] with decay constants that depend on density, $T_{2,n},T_{2,\mathrm{NV}}^{*}\propto 1/n$ (Appendix B). We considered a diamond chip of dimensions $V=(2.5\times 2.5)\mathrm{mm}{}^2\times 150\mu \mathrm{m}$ and assumed that only 1/4 of the NV spins were along the desired direction. The interrogation times were $t=1\mathrm{ms}$ (solid lines) and $t=0.1\mathrm{ms}$ (dashed line). The sensitivity for a simple Ramsey scheme (black lines) is limited by the nuclear spin $T_{2,n}=T_2^{*}$. Using an echo scheme (red thick line) improves the sensitivity, which is now limited by the coherence time $T_{2,\mathrm{NV}}^{*}$ of the NV-electronic spin used to read out the sensor spin state since the nuclear coherence time $T_{2,n}=T_2$ is much longer.

Figure 4

Signal $S_c$ from the four classes of NV centers for a rotation $\Omega$ along an axis with angles $\theta =\pi /6$ and $\varphi =2\pi /3$ from the [111] direction (which coincides with the first class of NVs, dashed-dotted line).

Figure 5

Conceptual design of an integrated nNV-MEMS gyroscope, comprising a bulk acoustic wave (BAW) single-axis MEMS gyroscope [2] in an $\sim 800-\mu \mathrm{m}$ diamond disk implanted with NV centers, whose nuclear spins form a spin gyroscope. (a) Schematic of the nNV-gyro operation. The spins implanted in the disk are polarized by an on-chip green laser. The electrodes surrounding the disk are silvered to allow for total internal reflection, and fluorescence is side collected [11] by replacing one of them by an on-chip optical waveguide at 638 nm. Strip lines for rf or $\mu$w control are fabricated on the disk. (b) Operation of the BAW mechanical gyroscope. The BAW is electrostatically driven in the second elliptic mode by an $\sim 10-\mathrm{kHz}$ sinusoidal signal from the drive (blue) electrodes. A rotation out of the plane causes a decrease in the gap near the sense (grey) electrodes, leading to a capacitive measurement of the rotation [2]. Combinatoric filtering with the nNV measurement leads to noise rejection and improved stability.

Figure 6

Polarization of the ${}^{14}$N nuclear spin under longitudinal driving of the NV-electronic transitions with a Rabi frequency of ${\Omega }_R=500\mathrm{MHz}$, hyperfine $A\approx 2.2\mathrm{MHz}$, and a static magnetic field of 20 G. The simulation includes dephasing of the electronic spin modeled by an Ornstein-Uhlenbeck process, yielding a $T_2^{*}$ time of about 200 ns. To achieve high polarization, the process is repeated twice by repolarizing the NV-electronic spin (dashed line) via optical illumination.

Figure 7

${}^{14}$N coherence decay under a spin-echo sequence (black, dashed lines) and a Ramsey sequence (solid lines). We simulated an ensemble of 100 noninteracting nuclear spins, each subjected to a bath of $\sim 500$ electronic spins with a density (from bottom to top lines) of $n=(0.35,1.06,1.76,2.47,3.17)\times {10}^{19}{\mathrm{cm}}^{-3}$. (The lines for the echo are indistinguishable.)