Cross-Sensor Feedback Stabilization of an Emulated Quantum Spin Gyroscope

Quantum sensors, such as the nitrogen-vacancy (N-V) color center in diamond, are known for their exquisite sensitivity but their performance over time is subject to degradation by environmental noise. To improve the long-term robustness of a quantum sensor, here we realize an integrated combinatorial spin sensor in the same micrometer-scale footprint, which exploits two different spin sensitivities to distinct physical quantities to stabilize one spin sensor with local information collected in real time via the second sensor. We show that we can use the electronic spins of a large ensemble of N-V centers as sensors of the local magnetic field fluctuations, affecting both spin sensors, in order to stabilize the output signal of interleaved Ramsey sequences performed on the ${}^{14}\mathrm{N}$ nuclear spin. An envisioned application of such a device is to sense rotation rates with a stability of several days, allowing navigation with limited or no requirement for geolocalization. Our results would enable stable rotation sensing for over several hours, which already reflects better performance than microelectromechanical systems (MEMS) gyroscopes of comparable sensitivity and size.

Figure 1

Control and readout of spin ensembles. (a) Schematics of the experimental device (see the main text for a description). (b) Pulsed electron spin resonance (ESR) spectrum of an ensemble of N-V centers. We use 4 points from a similar pulsed ESR spectrum (as marked on the red curve) to calibrate the N-V spin frequency prior to driving the N-V with the microwave pulses in Fig. 2. The blue curve is offset for better visualization. In blue, the N-V centers are aligned with an external magnetic field of approximately 26 G and show an equal population in each nuclear spin state. On the other hand, for a magnetic field of approximately 420 G (red), the ${}^{14}\mathrm{N}$ nuclear spin is initialized in a particular spin state ($|m_S=0,m_I=1⟩$) via a transfer of polarization that occurs close to the excited-state level anticrossing (ESLAC). (c) Coherence decay of the electronic spin. We use two subsequent Ramsey sequences, each one composed of two $\pi /2$ pulses detuned from the resonance frequency ${\nu }_e=1.704\text{GHz}$ by $\mathrm{\Delta }{\nu }_e=10\text{MHz}$. The phase of the second $\pi /2$ pulses is shifted by $\pi$ to measure both spin projections. Similarly to Eq. (1), we plot the signal difference $f$. This signal oscillates at the detuning frequency $\mathrm{\Delta }{\nu }_e$ and decays with a coherence time $T_{2e}^{\ast }=403\pm 22\text{ns}$. (d) Coherence decay of the nuclear spin. The resonance frequency ${\nu }_n$ is 4.68 MHz and the detuning $\mathrm{\Delta }{\nu }_n=5.5\text{kHz}$. The nuclear spin is prepared in a superposition of states $(|0,0⟩+|0,1⟩)/2$. We measure a coherence time $T_{2n}^{\ast }=840\pm 79\mu \text{s}$ via a Ramsey pulse sequence. A microwave $\pi$ pulse tuned at ${\nu }_e$ is used as a selective mapping ($|0,0⟩\to |-1,0⟩$, $|0,1⟩\to |0,1⟩$) between the nuclear spin state and the N-V electronic spin state to obtain higher contrast during the spin-dependent fluorescence readout. (e) Contrast response of the nuclear spin sensor to a linear change of the phase of the last $\pi /2$ pulse of the Ramsey sequence with (without) selective mapping in red (blue, Appendix 6). This is the same response as if the N-V sensor has accumulated a quantum phase $\mathrm{\Phi }$ during a time $t=600\mu \text{s}$ due to physical rotation about the N-V axis. The nuclear spin readout can be done without a mapping, releasing the constraint of a narrow-band selective pulse but with lower contrast.

Figure 2

Nuclear spin sensor with a stabilized readout. (a) The sequence consisting of alternating measurements of the quantum phase via a Ramsey sequence with the ${}^{14}\mathrm{N}$ nuclear spins ($t=600\mu \text{s}$) and measurements of the energy shift via a set of ESR measurements with the N-V spins. The full sequence is symmetrized to reject common noise. (b) Nuclear Ramsey contrast $F$ as defined by Eq. (1) with an uncorrected (blue) and a corrected (red) selective mapping. A slowly varying magnetic field shifts the energy levels and perturbs the mapping. (c) Allan deviations of the Ramsey signals of (b). The maximum averaging time $\tau$ used to calculate the Allan deviation is about one third of the total acquisition time of 1.15 d (see Appendix 7). The right-hand axis is rescaled using a factor $s=4.4\times {10}^6(\mathrm{deg}{\text{s}}^{-1}{)}^{-1}$, calculated at the steepest point of curve $F$ in Fig. 1.

Figure 3

Stabilized nuclear spin sensor. (a) Dual-sensor scheme. The ESR shift ${\nu }_{\mathrm{ESR}}$ collected from the N-V spins also serves to apply feedback on the nuclear spin control parameters $\delta {\nu }_{\mathrm{N}}$ to stabilize the ${}^{14}\mathrm{N}$ nuclear Ramsey contrast $G$ with respect to a noise common to both spins. (b) Allan deviation of the nuclear spin signal for different correction strengths. We directly collect the nuclear-spin-state-dependent fluorescence without mapping pulse to isolate the perturbation effect on the nuclear spin signal. The right-hand axis is rescaled using a factor $s=1.4\times {10}^6(\mathrm{deg}{\text{s}}^{-1}{)}^{-1}$, calculated at the steepest point of curve $G$ in Fig. 1.

Figure 4

Simulation of history-based feedback protocols. (a) We simulate a sinusoidal perturbation with stochastic noise of varying amplitude (red). A polynomial (here linear, blue dashed line) fit is used on a certain number $N$ of points $[i,i+N]$ to guess the value of the point $i+N+1$, generating the guessed signal (orange). In blue, we plot the error between red and orange. (b) In the case of the absence of noise, we plot the error as previously defined for different degrees $d$ of the fitting polynomial. The number of points used is set at $N=d+1$. (c) We plot the average error, defined as the mean value of the error [blue curve in (a)] averaged over 20 realizations, as a function of the noise amplitude and the degree of the polynomial. We note that the trend is different depending on the noise amplitude. (d) From experimental ESR data, we estimate a ratio between the magnetic perturbation and our experimental noise of 0.1. This is a regime in which no improvement can be achieved by using the history of the measurements, as one can see in the red region of (c).

Figure 5

Nuclear Ramsey signal collected directly from the N-V fluorescence, without mapping from the nuclear to the electronic spin states.