# A Synchronous Spin-Exchange Optically Pumped NMR-Gyroscope
## 3 Results
### Cross Talk  
Once we measured the stability of the PM comagnetometer, we desired to know the fidelity with which our detection separated signals from the two Xe isotopes. It is possible that the detection channel designed to measure isotope \(a\)'s phase was really measuring a linear combination of isotope \(a\) and \(b\)'s phases. If such \"cross talk\" were present then the scale factor (or how we convert the measured precession frequencies to rotation) would change [13]. Suppose there exists cross talk in both channels such that \(\omega^{a}=\gamma^{a}B_{z}+\beta\omega^{b}-\omega^{R}\) and \(\omega^{b}=\gamma^{b}B_{z}+\beta^{\prime}\omega^{a}+\omega^{R}\) where \(\beta\) and \(\beta^{\prime}\) represent the cross talk between detection channels. Solving for \(\omega^{R}\) we find  
\[\omega^{R}=\frac{\omega^{b}(\rho+\beta^{\prime})-\omega^{a}(\rho\beta+1)}{1+ \rho}, \tag{12}\]  
where if \(\beta=\beta^{\prime}=0\) we return to the expected expression for rotation without cross talk. Non-zero cross talk is undesirable because the accuracy with which it is known (or measured) limits the accuracy of conversion from measured precession frequencies to rotation (or any other non-magnetic spin-dependent interaction). A measurement of cross talk is vital since an important alleged feature of our comagnetometer is that it has a scale factor which is determined solely by \(\rho\). We characterize the cross talk present in our detection channels by looking for changes in the detected signal for one isotope when the drive of the other isotope is changed. We do this by detuning one isotope's drive frequency by \(\sim 300\) mHz and then scanning the bias pulse repetition rate. The effective resonance frequency of each Xe isotope depends on the pulsing frequency (\(\omega_{2\pi}\)) as \(\omega^{K}=\omega_{2\pi}\gamma^{K}/\gamma^{S}\). The 300 mHz detuning ensures that both isotopes are not simultaneously on resonance for a given pulsing frequency. When a Xe isotope is far from resonance, we expect the measured signals for that isotope's detection channel to be relatively flat if there is no cross talk. Cross talk manifests if the signals from the off-resonant isotope are not flat when the other isotope is scanned through resonance. Figure 8 shows a phase sensitive measurement of cross talk. When isotope \(a\) is excited and isotope \(b\) is not, isotope \(b\)'s detection channel exhibits non-zero signal, and vice versa. We estimate \(\beta=Q_{pp}^{b}/Q_{pp}^{a}=0.17\) (when isotope \(a\) is on resonance) and \(\beta^{\prime}=Q_{pp}^{a}/Q_{pp}^{b}=0.07\) (when isotope \(b\) is on resonance), where \(Q_{pp}^{K}\) is the peak-to-peak quadrature signal of isotope \(K\). Figure 8: Measurement of cross talk for detection with rectification. The Xe drive frequencies are set such that the two isotopes are not driven on resonance at the same effective bias field magnitude. The in-phase (filled symbols) and out-of-phase (un-filled symbols) components of each isotope’s detection channel (black and red correspond to isotopes \(a\) and \(b\), respectively) are shown as the bias pulsing frequency is varied. We see that when isotope \(a\) is resonant (\(f_{2\pi}\sim 15.8\) kHz) the signal on isotope \(b\)’s detection channel is not flat despite isotope \(b\) being driven many linewidths off resonance. Similarly, when isotope \(b\) is resonant (\(f_{2\pi}\sim 16\) kHz) the signal on isotope \(a\)’s detection channel is not flat despite isotope \(a\) being driven many linewidths off resonance. We observe cross talk on our detection channels even when Xe is not excited and artificial Xe signals are applied, suggesting that our observed cross talk is not due to physical interactions between Xe isotopes. We believe the measured cross talk stems from imperfect rectification of \(S_{z}\) due to optical pumping transients (i.e., gain reversals) and unaccounted-for phase shifts from high-pass filtering prior to rectifying. The optical pumping transients stem from the few ms finite response time of the magnetometer.