# A Synchronous Spin-Exchange Optically Pumped NMR-Gyroscope
## 3 Results  
### NMR Excitation and Detection  
Figure 3 shows the time series of the measured \(S_{z}\) with and without rectification when we drive both isotopes near resonance simultaneously using \(S_{+}\). We see that rectification reveals the sinusoidal precession of each isotope. The outlying data on the rectified signal, which occur when the polarization is reversed, are due to the finite response time of the Rb magnetometer. Although rectification collects the many Xe signal sidebands into the two Xe carrier frequencies (see power spectrum on right), it also maps \(1/f\) detection noise on \(S_{z}\) to the carrier frequencies. The mapping of low-frequency \(S_{z}\) noise to the carrier frequencies can be prevented by high-pass filtering \(S_{z}\) with a 1 Hz corner prior to rectification.  
Figure 4 shows the detected NMR signals for each isotope. These data were acquired by driving one isotope on resonance while varying the other isotope's drive frequency and recording its \(K_{x}\) and \(K_{y}\) derived using demodulation. We see that the lineshapes are nearly Lorentzian with linewidths (half-width at half-max) of \(\sim\)15 mHz and amplitudes of approximately 60 uG. The on-resonance amplitudes are in agreement with estimates similar to those outlined in [3]. The implied \(T_{2}^{a}\), \(T_{2}^{b}\) from the fits are in good agreement with independent measurements of each isotope's \(T_{1}\). The \(\sim\)15 mHz linewidths are only possible because of two features of our experiment; (i) the use of a Rb hydride cell coating (without which \(T_{2}^{b}\) would be substantially shorter and \(T_{2}^{a}\) much longer) [15], and (ii) the application of a magnetic compensation field \(B_{x}\) that cancels the Rb spin-exchange field experienced by the Xe.  
Figure 5 demonstrates the amplitude spectral density of the phase noise measured for each isotope under simultaneous resonant excitation conditions. We see that for frequencies less than 1 Hz the spectra are dominated by \(1/f\) noise which is about \(\rho=\gamma^{a}/\gamma^{b}\) greater for isotope \(a\) (black traces) than for isotope \(b\) (red traces) suggesting the dominant source of \(1/f\) noise is magnetic in nature. Also shown is the phase noise measured when the Xe isotopes are driven off resonance (not excited) and a so-called \"artificial\" Xe signal is applied as an AC field, \(B_{y}\). The amplitudes of this signal \(A^{a}\sin(\omega_{d}^{a}t)+A^{b}\sin(\omega_{d}^{b}t)\) are chosen to produce the same size magnetometer signal as real Xe. We note that this artificial Xe signal is planar, unlike the real Xe signal which rotates. The artificial signal allows us to measure the signal-to-noise ratio (SNR) of the Rb magnetometer. These signals do not show \(1/f\) dependence because, unlike the real Xe phase, the SNR of the magnetometer does not depend on the bias magnetic field to first order. The detection phase noise from each artificial Xe measurement is uncorrelated and limits the possible field suppression when performing comagnetometry. The SNR for each isotope is \(\sim\)5000\(\sqrt{\rm Hz}\). While the detection of artificial Xe is insensitive to \(1/f\) bias magnetic field noise, \(1/f\)\(S_{z}\) noise can still be mapped to the artificial Xe frequencies via rectification, and so the addition of a 1 Hz high-pass filter prior to rectification was essential for realizing such an SNR.  
Figure 4: Measured NMR lineshapes of each species. Filled circles are \(K_{x}\), open circles are \(K_{y}\), and lines are Lorentzian fits.  
Figure 3: Measured square wave PM signals. Top: \(S_{+}(t)\). Middle: \(S_{z}(t)\). Bottom: Rectified \(S_{z}(t)\). The plot on the right shows the amplitude spectral density with (red) and without (blue) rectification.