# A Synchronous Spin-Exchange Optically Pumped NMR-Gyroscope
## 2 Materials and Methods
The maximum and minimum voltages of the EOM drive waveform are chosen to produce \(\pm\lambda/2\) retardance. The quarter wave plate at the output of the EOM converts the EOM output at \(V_{max}(V_{min})\) to be \(\sigma^{+}(\sigma^{-})\). The collimated output of each EOM is coupled into the vapor cell from opposing directions. The overall beam resizing is set to somewhat overfill the aperture of the ceramic heater. Fine tuning of each pump beam's pointing is controlled using long focal length lenses mounted on two axis translation stages just outside the magnetic shield. The position of each steering lens is chosen to optimize the magnetometer gain. The power and detuning of each pump laser is chosen to approximately cancel the AC Stark effect. While we could also suppress the AC Stark effect by pumping with a single laser tuned on resonance, we would not get good spatial uniformity of the Rb polarization without significantly increasing our laser power due to optical thickness effects. To detect \(S_{z}\), approximately one mW of linearly polarized light from the output of a third distributed feedback laser diode, tuned near the Rb D2 line, is directed through the center of the cell and parallel to \(\hat{z}\) onto a balanced Faraday detector. Figure 2: Experimental setup for PM excitation (not to scale). DM: dichroic mirror, Pol: polarizer, HWP: half wave plate, QWP: quarter wave plate, PBS: polarizing beam splitter, WP: Wollaston prism, PD: photodiode, EOM: electro-optic modulator, TS: two-axis translation stage with lens. The three-axis magnetic shim and pulsing coils are not shown. The setup fits on a four foot square optical table. We choose to modulate the Rb polarization by switching between just two polarization states, \(\sigma^{\pm}\). We modulate the x-component of **R** according to  
\[R(t)=R_{0}\ \text{sign}[\cos\left(\frac{\omega_{d}^{a}+\omega_{d}^{b}}{2}t+2\cos( \frac{\omega_{d}^{a}-\omega_{d}^{b}}{2}t)\right)]. \tag{8}\]  
This waveform is advantageous in that the smallest separation between reversals is larger than the finite response times of the EOMs and of the optical pumping of the Rb atoms. Modulating the polarization as a sum of two sine waves would also keep the modulations within the bandwidth of the EOMs and the Rb magnetometer, but we find that producing such a PM waveform in our optically thick vapor cell is very challenging. We use the same modulated cosine waveform to apply a compensation field, the amplitude of which is set to cancel the spin-exchange field experienced by the Xe from the Rb. Doing so suppresses the production of \(K_{z}\) and narrows the NMR linewidths, \(\Gamma_{2}\)[12]. From Equation (3) we find that the detected longitudinal Rb polarization \(S_{z}\) is  
\[S_{z}=-\frac{R(t)}{\Gamma^{\prime 2}}[b_{a}^{S}K_{\perp}^{a}\sin(\delta^{a}+ \alpha^{a}-\epsilon_{z})+b_{b}^{S}K_{\perp}^{b}\sin(\delta^{b}+\alpha^{b}+ \epsilon_{z})], \tag{9}\]  
where \(\delta=\phi-\alpha\) is the difference between the drive phase \(\alpha=\int dt\omega_{d}\) and the Xe precession phase \(\phi\) for each isotope, \(\epsilon_{z}\) is the magnetometer phase shift, and \(R\sim S_{+}\) is the optical pumping rate of the Rb. Although lock-in detection can be accomplished on \(S_{z}\) as it appears in Equation (9), the phase sensitivity is diminished due to the presence of \(R(t)\) which effectively mixes some Xe signal to DC. A more effective approach is to \"rectify\" the \(S_{z}\) signal such that \(R(t)\) is removed. This is accomplished by multiplying \(S_{z}\) by \(R^{-1}(t)\). The resulting signal is then sent into two separate lock-in amplifiers, each referenced to a different isotope's drive frequency. From these demodulations, we arrive at the measured phases \(\delta^{a}-\epsilon_{z}\) and \(\delta^{b}+\epsilon_{z}\).