# Frequency Response of Optically Pumped Magnetometer with Nonlinear Zeeman Effect
## 3 Theory
 \tag{13}\]  
The steady state solutions of the first two orders of expansion are  
\[\tilde{\rho}_{\alpha\ \beta}^{\mathrm{F_{1}}(0)}=\frac{P_{\alpha\ \beta}A^{(\beta-\alpha)}+ \Gamma_{\alpha\ \beta}\delta_{\alpha,\ \beta}/16}{\mathrm{i}(\alpha-\beta)\delta\omega_{\alpha\ \beta}+\Gamma_{\alpha\ \beta}}, \tag{14}\]  
in which \(\delta\omega_{\alpha\ \beta}=\gamma B_{0}-(\alpha+\beta)\omega_{\mathrm{NLZ}}/2- \omega_{\mathrm{M}}\), and  
\[ \begin{split}\tilde{\rho}_{\alpha\ \beta}^{\mathrm{F_{1}}(1)}&=\frac{-\mathrm{i}(\alpha- \beta)\gamma B_{\mathrm{S}}\mathrm{exp}(\mathrm{i}\omega_{\mathrm{S}}t)}{ \mathrm{i}\omega_{\mathrm{S}}+\mathrm{i}(\alpha-\beta)\delta\omega_{\alpha\ \beta}+\Gamma_{\alpha\ \beta}}\times\tilde{\rho}_{\alpha\ \beta}^{\mathrm{F_{1}}(0)}\\ &=\frac{-\mathrm{i}(\alpha-\beta)\gamma B_{\mathrm{S}}\mathrm{exp}( \mathrm{i}\omega_{\mathrm{S}}t)}{\mathrm{i}\omega_{\mathrm{S}}+\mathrm{i}( \alpha-\beta)\delta\omega_{\alpha\ \beta}+\Gamma_{\alpha\ \beta}}\times\frac{P_{\alpha\ \beta}A^{(\beta-\alpha)}+\Gamma_{\alpha\ \beta}\delta_{\alpha,\ \beta}/16}{\mathrm{i}(\alpha-\beta)\delta\omega_{\alpha\ \beta}+\Gamma_{\alpha\ \beta}}.\end{split}  \tag{15}\]According to Equation (8), the output signal from the polarimeter is  
\[ \begin{split}\langle F_{y}\rangle^{F1}&=\sum_{\alpha=-3}^{ 4}\frac{1}{2\mathrm{i}}\left(\bar{\rho}_{\alpha\;\alpha-1}^{F_{1}(0)}e^{- \mathrm{i}\omega_{\mathrm{M}}t}-\bar{\rho}_{\alpha-1\;\alpha}^{F_{1}(0)}e^{ \mathrm{i}\omega_{\mathrm{M}}t}+\bar{\rho}_{\alpha\;\alpha-1}^{F_{1}(1)}e^{- \mathrm{i}\omega_{\mathrm{M}}t}-\bar{\rho}_{\alpha-1\;\alpha}^{F_{1}(1)}e^{ \mathrm{i}\omega_{\mathrm{M}}t}\right)\times C_{\alpha}\\ &=\sum_{\alpha=-3}^{4}\frac{1}{2\mathrm{i}}\left(\bar{\rho}_{ \alpha\;\alpha-1}^{F_{1}(0)}-\bar{\rho}_{\alpha-1\;\alpha}^{F_{1}(0)}+\bar{ \rho}_{\alpha\;\alpha-1}^{F_{1}(1)}-\bar{\rho}_{\alpha-1\;\alpha}^{F_{1}(1)} \right)\times C_{\alpha}\mathrm{cos}\omega_{\mathrm{M}}t\\ &\quad-\sum_{\alpha=-3}^{4}\frac{1}{2}\left(\bar{\rho}_{\alpha\; \alpha-1}^{F_{1}(0)}+\bar{\rho}_{\alpha-1\;\alpha}^{F_{1}(0)}+\bar{\rho}_{ \alpha\;\alpha-1}^{F_{1}(1)}+\bar{\rho}_{\alpha-1\;\alpha}^{F_{1}(1)}\right) \times C_{\alpha}\mathrm{sin}\omega_{\mathrm{M}}t.\end{split}  \tag{16}\]  
After demodulating the output signal at \(\omega_{\mathrm{M}}\) with the LIA, the in-phase component (the term proportional to \(\mathrm{cos}\omega_{\mathrm{M}}t\)) is composed of the steady part  
\[Out_{\mathrm{In}}^{(0)}=\sum_{\alpha=-3}^{4}\frac{1}{2\mathrm{i}}\left(\bar{ \rho}_{\alpha\;\alpha-1}^{F_{1}(0)}-\bar{\rho}_{\alpha-1\;\alpha}^{F_{1}(0)} \right)\times C_{\alpha}=-\sum_{\alpha=-3}^{4}\frac{\delta\omega_{\alpha\; \alpha-1}}{\delta\omega_{\alpha\;\alpha-1}^{2}+\Gamma_{\alpha\;\alpha-1}^{2}} \times C_{\alpha}P_{\alpha\;\alpha-1}A^{(-1)}, \tag{17}\]  
and the oscillating part  
\[ \begin{split} Out_{\mathrm{In}}^{(1)}(t)=&\sum_{ \alpha=-3}^{4}\frac{1}{3\mathrm{i}}\left(\bar{\rho}_{\alpha\;\alpha-1}^{F_{1}( 1)}-\bar{\rho}_{\alpha-1\;\alpha}^{F_{1}(1)}\right)\times C_{\alpha}\\ =&-\sum_{\alpha=-3}^{4}\frac{C_{\alpha}P_{\alpha\; \alpha-1}A^{(-1)}\gamma B_{\mathrm{S}}e^{\mathrm{i}\omega_{\mathrm{S}}t}}{2 \left(\mathrm{i}\omega_{\mathrm{S}}+\mathrm{i}\delta\omega_{\alpha\;\alpha-1} +\Gamma_{\alpha\;\alpha-1}\right)\times\left(\mathrm{i}\delta\omega_{\alpha\; \alpha-1}+\Gamma_{\alpha\;\alpha-1}\right)}\\ &-\sum_{\alpha=-3}^{4}\frac{C_{\alpha}P_{\alpha\;\alpha-1}A^{(-1)} \gamma B_{\mathrm{S}}e^{\mathrm{i}\omega_{\mathrm{S}}t}}{2\left(\mathrm{i} \omega_{\mathrm{S}}-\mathrm{i}\delta\omega_{\alpha\;\alpha-1}+\Gamma_{\alpha\; \alpha-1}\right)\times\left(-\mathrm{i}\delta\omega_{\alpha\;\alpha-1}+\Gamma_ {\alpha\;\alpha-1}\right)}.\end{split}  \tag{18}\]  
The steady part Equation (17) corresponds to the conventional magnetic resonance signal, which is a sum of eight Lorentz dispersive curves with different central frequencies.