# A Synchronous Spin-Exchange Optically Pumped NMR-Gyroscope
## 1 Introduction
### Bloch Equation  
The spin dynamics of the polarized noble gas transverse (\(K_{+}=K_{x}+iK_{y}\)) and longitudinal (\(K_{z}\)) spins are described by the Bloch equations  
\[\frac{dK_{+}}{dt}=-(\mp i\Omega_{z}^{K}+\Gamma_{2})K_{+}+\Gamma_{ S}^{K}S_{++}+\mp i\Omega_{+}^{K}K_{z}, \tag{1a}\] \[\frac{dK_{z}}{dt}=\mp(\Omega_{y}^{K}K_{x}-\Omega_{x}^{K}K_{y})- \Gamma_{1}K_{z}, \tag{1b}\]  
where \(\mathbf{\Omega}^{K}\) is the Xe resonance frequency arising from both the Larmor precession \(\gamma^{K}\mathbf{B}\) and from rotation \(\omega^{K}\mathbf{\hat{z}}\), \(\Gamma_{1}\) (\(\Gamma_{2}\)) is the longitudinal (transverse) relaxation rate, \(\mathbf{S}\) is the Rb polarization, and \(\Gamma_{S}^{K}\) is the spin-exchange rate constant. Let superscript \(a\) represent \({}^{129}\)Xe and superscript \(b\) represent \({}^{131}\)Xe. Since \(\gamma^{a}<0\) and \(\gamma^{b}>0\) we find it useful to write the gyromagnetic ratio \(\gamma\) (and hence \(\Omega\)) as a positive value. The sign is written explicitly out front (top sign is for \(a\), bottom sign is for \(b\)). We null the transverse fields experienced by the Xe, including the spin-exchange field \(b_{S}^{K}S_{+}\) (where \(b_{S}^{K}\) is the spin-exchange coefficient characterizing the influence of the Rb polarization on the Xe) [12]. This suppresses \(K_{z}\) such that \(K_{+}\) is much more sensitive to \(\Omega_{z}^{K}\) than \(\Omega_{+}^{K}\). Assuming our transverse optical pumping produces negligible \(S_{z}\), the Xe resonance frequencies are then \(\Omega_{z}^{K}=\gamma^{K}(B_{z0}+B_{p})\mp\omega^{R}\), where \(B_{z0}\) is the stray field inside our magnetic shields and \(B_{p}\) is the field from the bias pulses. Since the Xe precess on the order of only 2\(\pi/10^{3}\) radians per pulse, the pulsed field can be approximated as a continuous field with \(B_{p}=\omega_{p}/\gamma^{S}\) for pulsing frequency \(\omega_{p}\). The transverse Rb polarization (\(S_{+}\)) is used to polarize the Xe via spin-exchange optical pumping, and the longitudinal Rb polarization (\(S_{z}\)) is used to detect the Xe precession. For small Rb precession (\(\Omega^{S}<<\Gamma^{\prime}\)), the time-average solution to the Bloch equation for the Rb polarization can be expanded as  
\[\mathbf{S}=\frac{\mathbf{R}}{\Gamma^{\prime}}+\frac{\mathbf{\Omega}^{S} \times\mathbf{R}}{\Gamma^{\prime 2}}+\frac{\mathbf{\Omega}^{S}\times\mathbf{\Omega}^{S} \times\mathbf{R}}{\Gamma^{\prime 3}}+..., \tag{2}\]where \(\mathbf{R}=\int dv\Phi(\
u)\sigma(\
u)\mathbf{p}/A\) is the pumping rate, \(\sigma(\
u)\) is the cross section of the atomic transition, \(\int dv\Phi(\
u)=P/h\
u\) is the total photon flux for a beam of power P incident on area A, \(\mathbf{p}\) is the photon spin, \(\Gamma^{\prime}\) is the total relaxation rate (including pumping), and \(\mathbf{\Omega}^{S}=\gamma^{S}(\mathbf{B}_{0}+b_{a}^{S}\mathbf{K}^{a}+b_{b}^ {S}\mathbf{K}^{b})\). The magnetic field experienced by the Rb includes the stray fields \(\mathbf{B}_{0}\) and the spin-exchange fields (\(b_{K}^{S}\) is the spin-exchange coefficient characterizing the influence of the Xe polarization on the Rb), but it does not include the bias field. This is because the bias field is applied as a sequence of low duty cycle pulses, the area of which correspond to \(2\pi\) precession of the Rb atom. These equations assume negligible back polarization from the Xe to the Rb [14], and that \(\mathbf{K}\) precesses slowly such that \(S_{z}\) responds adiabatically.