# Frequency Response of Optically Pumped Magnetometer with Nonlinear Zeeman Effect
## 3 Theory
After ignoring these fast oscillating terms, the evolution of \(\tilde{\rho}_{\alpha\ \beta}^{\mathrm{F_{1}}}\) is  
\[\frac{\mathrm{d}\tilde{\rho}_{\alpha\ \beta}^{\mathrm{F_{1}}}}{\mathrm{d}t} \approx-\mathrm{i}(\alpha-\beta)[\omega_{\mathrm{L}}-(\alpha+\beta)\omega_{ \mathrm{NLZ}}/2-\omega_{\mathrm{M}}]\times\tilde{\rho}_{\alpha\ \beta}^{\mathrm{F_{1}}}+P_{\alpha\ \beta}A^{(\beta-\alpha)}-\Gamma_{\alpha\ \beta}(\tilde{\rho}_{\alpha\ \beta}^{\mathrm{F_{1}}}-\delta_{\alpha,\ \beta}/16). \tag{11}\]  
To measure the frequency response of the magnetometer, we typically add a small sinusoidal modulation component \(B_{\mathrm{S}}\mathrm{cos}(\omega_{\mathrm{S}}t)\) to the steady biasing field \(B_{0}\), and the overall magnetic field is \(B_{z}=B_{0}+B_{\mathrm{S}}\mathrm{cos}(\omega_{\mathrm{S}}t)\). As a result, the Larmor frequency is \(\omega_{\mathrm{L}}=\gamma B_{0}+\gamma B_{\mathrm{S}}\mathrm{cos}(\omega_{ \mathrm{S}}t)\). The influence of the modulation on \(\omega_{\mathrm{NLZ}}\) is small and can be neglected so that \(\omega_{\mathrm{NLZ}}=2\omega_{\mathrm{L}}^{2}/\omega_{\mathrm{H}}\approx 2( \gamma B_{0})^{2}/\omega_{\mathrm{H}}\). The response of the magnetometer to the modulation is also a sinusoidal component with the same frequency in the Bell-Bloom OPM readout signal. By extracting the amplitude scaling and phase shift of the magnetometer readout, we will derive the frequency response. We can express the frequency response as a complex function of the frequency whose amplitude indicates the amplitude scaling factor, and phase indicates the phase shift. To simplify the deduction, we could replace the modulation \(B_{\mathrm{S}}\mathrm{cos}(\omega_{\mathrm{S}}t)\) with \(B_{\mathrm{S}}\mathrm{exp}(\mathrm{i}\omega_{\mathrm{S}}t)\). After the replacement, there is an error-prone place that should be taken into consideration. Generally speaking, the density matrix \(\rho^{\mathrm{F_{1}}}\) is the Hermitian conjugate of itself and \(\rho_{\alpha\ \alpha-1}^{\mathrm{F_{1}}}-\rho_{\alpha-1\ \alpha}^{\mathrm{F_{1}}}\) in Equation (8) can be thus simplified as \(2\mathrm{i}\times\mathrm{Im}\left(\rho_{\alpha\ \alpha-1}^{\mathrm{F_{1}}}\right)\), in which \(\mathrm{Im}(Q)\) means the imaginary part of a number \(Q\). But if we replace the modulated biasing magnetic field with a complex quantity \(B_{\mathrm{S}}\mathrm{exp}(\mathrm{i}\omega_{\mathrm{S}}t)\), the density matrix is no more the Hermitian conjugate of itself. \(\rho_{\alpha\ \alpha-1}^{\mathrm{F_{1}}}-\rho_{\alpha-1\ \alpha}^{\mathrm{F_{1}}}\) in Equation (8) should not be simplified and may otherwise produce wrong results. We could perturbatively expand the \(\tilde{\rho}_{\alpha\ \beta}^{\mathrm{F_{1}}}\) with \(\gamma B_{\mathrm{S}}/\Gamma_{\alpha\ \beta}\) as  
\[\tilde{\rho}_{\alpha\ \beta}^{\mathrm{F_{1}}}=\sum_{n=0}^{+\infty}\rho_{\alpha\ \beta}^{\mathrm{F_{1}}(n)}, \tag{12}\]  
in which \(\tilde{\rho}_{\alpha\ \beta}^{\mathrm{F_{1}}(n)}\) means the \(n\)-th order expansion. Substituting Equation (12) and the expression of \(\omega_{\mathrm{L}}\) and \(\omega_{\mathrm{NLZ}}\) into Equation (11), we will get the equation for every single \(\tilde{\rho}_{\alpha\ \beta}^{\mathrm{F_{1}}(n)}\) as  
\[ \begin{split}\frac{\mathrm{d}\tilde{\rho}_{\alpha\ \beta}^{\mathrm{F_{1}}(n)}}{\mathrm{d}t}=&-\mathrm{i}(\alpha- \beta)[\gamma B_{0}-(\alpha+\beta)\omega_{\mathrm{NLZ}}/2-\omega_{\mathrm{M}} ]\times\tilde{\rho}_{\alpha\ \beta}^{\mathrm{F_{1}}(n)}-\Gamma_{\alpha\ \beta}\times\tilde{\rho}_{\alpha\ \beta}^{\mathrm{F_{1}}(n)}\\ &-\mathrm{i}(\alpha-\beta)(1-\delta_{0,\ \mu})\gamma B_{ \mathrm{S}}e^{\mathrm{i}\omega_{\mathrm{S}}t}\times\tilde{\rho}_{\alpha\ \beta}^{\mathrm{F_{1}}(n-1)}\\ &+\delta_{0,\ \mu}P_{\alpha\ \beta}A^{(\beta-\alpha)}+\Gamma_{ \alpha\ \beta}\delta_{0,\ \mu}\delta_{\alpha,\ \beta}/16.\end{split}