# A Synchronous Spin-Exchange Optically Pumped NMR-Gyroscope
## 3 Results
### Comagnetometry  
We perform comagnetometry by subtracting magnetic field correlations between the two isotope's precession frequencies. Since our device measures phase, we need to know or measure the transfer function from phase to frequency. In Section 1.1 we derived the transfer function (see Equation (7)). We measured the transfer function of each isotope by recording the response of the measured phase \(\delta\) to sinusoidal modulation of \(B_{z}\). Figure 6 shows the measured transfer function for isotope \(a\). We use a chirp waveform to modulate the bias field \(B_{z}^{mod}(t)=B_{0}\sin(2\pi[e^{t/T_{2}}-1-t/T_{2}])\) (where \(T_{2}=1/2\pi\Gamma_{2}\)), the time series of which is shown in the inset of Figure 6. This modulation waveform allows us to measure the transfer function from 0.002 to 0.1 Hz with good SNR in a single data acquisition. The transfer function is the ratio \(\gamma^{K}\tilde{B}_{z}^{mod}/\delta\). We fit the data according to Equation (7) and find excellent agreement with the linewidth derived from the fits in Figure 4.  
Although conversion from phase to frequency for the measured Xe phases is possible using a measured transfer function, feedback is desirable because (in the high gain limit) the performance of the comagnetometry becomes insensitive to changes in the transfer function. We used the measured precession phase of isotope \(a\) to stabilize the bias field and the measured transfer function of isotope \(b\) to convert its measured phase noise to frequency noise. Under such conditions the frequency noise of isotope \(b\) is proportional to rotation. We apply a feedback field, \(B_{f}\), to hold the measured phase of isotope \(a\), \((\delta^{a}-\epsilon_{z})\), equal to zero. The field stabilization can then be written \(\tilde{B}_{f}=\tilde{G}(\delta^{a}-\tilde{\epsilon}_{z})\). In the high gain limit, this becomes  
\[\lim_{\tilde{G}\rightarrow\infty}\tilde{B}_{tot}=\frac{1}{\gamma^{a}}(\tilde {\omega}^{R}+i\omega\tilde{\epsilon}_{z}), \tag{10}\]  
where \(B_{tot}=B_{z0}+B_{p}+B_{f}\), and where we made substitutions using Equation (7) and \(\tilde{\Delta}^{a}=-\gamma^{a}\tilde{B}_{z}-\tilde{\omega}^{R}\) (since \(\omega_{d}^{a}\) is held constant). The measured phase of isotope \(b\), \((\delta^{b}+\epsilon_{z})\), converted to frequency is  
\[\tilde{\omega}^{b}\equiv(\tilde{\omega}^{b}+\tilde{\epsilon}_{z})(\Gamma_{2} ^{b}+i\omega)=(\rho^{-1}+1)(\tilde{\omega}^{R}+i\omega\tilde{\epsilon}_{z}). \tag{11}\]  
We see that in the high gain limit, when correcting the bias field to keep the measured phase of isotope \(a\) equal to zero, the rotation is simply \(\tilde{\omega}^{R}=\rho\ \tilde{\omega}^{b}/(1+\rho)\) assuming \(\tilde{\epsilon}_{z}\) is negligible. The best performance we observed with bias field feedback activated is shown in Figure 7. The feedback consisted of two analog inverted zero gain stages. The influence of bias field feedback  
Figure 5: Measured phase noise of each species. Traces labeled “Real” are recorded when the Xe isotopes are excited on resonance. Traces labeled “Artificial” are recorded when the Xe isotopes are both driven off resonance (not excited) and an AC \(B_{y}\) is applied to the magnetometer at the off-resonance drive frequencies. is dramatic from 0.1 to 200 mHz. The servo suppresses \(\tilde{\omega}^{a}\) to below 1 \(\upmu\)Hz/\(\sqrt{\text{Hz}}\) at low frequency, which is nearly \(10^{4}\)x less than the open loop noise. Because magnetic noise dominates each isotope's precession, servoing the measured phase of isotope \(a\) also greatly suppresses \(\tilde{\omega}^{b}\). We observe at least 100\(\times\) improvement in \(\tilde{\omega}^{b}\) due to feedback. The modified Allan deviation [18] suggests a rotation ARW sensitivity of \(\sqrt{2}\) 15 \(\upmu\)Hz/\(\sqrt{\text{Hz}}\)\(\frac{\rho}{1+\rho}\)\(\sim\) 16 \(\upmu\)Hz/\(\sqrt{\text{Hz}}\) and a rotation bias instability of 1 \(\upmu\)Hz \(\frac{\rho}{1+\rho}\)\(\sim\) 800 mHz.