# Frequency Response of Optically Pumped Magnetometer with Nonlinear Zeeman Effect
## 3 Theory
The pump beam is amplitude modulated at frequency \(\omega_{\rm M}\) in an on-off manner in our experiment,so we can assume that \(A(t)\) is also modulated by a square wave. The Fourier expansion of \(A(t)\) is as follows  
\[A(t)=\sum_{m=-\infty}^{\infty}A^{(m)}e^{im\omega_{\text{M}}t}, \tag{5}\]  
in which \(A^{(m)}\) is the amplitude of the \(m\)-th order expansion. We can choose a proper initial phase so that each \(A^{(m)}\) is a real number and \(A^{(m)}=A^{(-m)}\). The relaxation of each density matrix element can be assumed to be an exponential decay as  
\[\frac{\text{d}^{(2)}\rho_{\alpha\;\beta}^{F_{1}}}{\text{d}t}=-\Gamma_{\alpha\; \beta}(\rho_{\alpha\;\beta}^{F_{1}}-\delta_{\alpha,\;\beta}/16), \tag{6}\]  
in which \(\Gamma_{\alpha\;\beta}\) is a real number describing the relaxation rate of the density matrix element \(\rho_{\alpha\;\beta}^{F_{1}}\). The final value to which the matrix element decays is \(\delta_{\alpha,\;\beta}/16\). The diagonal elements of \(\rho^{F_{1}}\) eventually decay to \(1/16\), which is the population of each Zeeman sublevels when the atoms are in the unpolarized state. The off-diagonal elements decay to \(0\), which corresponds to no coherence in the unpolarized state. As the density matrix \(\rho^{F_{1}}\) is Hermitian, \(\Gamma_{\alpha\;\beta}\) can be assumed to be equal to \(\Gamma_{\beta\;\alpha}\). After substituting Equations (3)-(6) into Equation (1), we will get the equation depicting the overall evolution of density matrix \(\rho^{F_{1}}\) as  
\[\frac{\text{d}\rho_{\alpha\;\beta}^{F_{1}}}{\text{d}t}=-\text{i}(\alpha-\beta) [\omega_{\text{L}}-(\alpha+\beta)\omega_{\text{NLZ}}/2]\times\rho_{\alpha\; \beta}^{F_{1}}+P_{\alpha\;\beta}\sum_{m=-\infty}^{\infty}A^{(m)}e^{im\omega_{ \text{M}}t}-\Gamma_{\alpha\;\beta}(\rho_{\alpha\;\beta}^{F_{1}}-\delta_{ \alpha,\;\beta}/16). \tag{7}\]  
In the detection of atomic polarization, the Faraday rotation angle for the near-resonant linearly polarized probe light is proportional to the atomic polarization along the propagating direction of the probe beam [36], which is  
\[\langle F_{y}\rangle^{F1}=\sum_{\alpha=-3}^{4}\frac{1}{2\text{i}}\left(\rho_{ \alpha\;\alpha-1}^{F_{1}}-\rho_{\alpha-1\;\alpha}^{F_{1}}\right)\times C_{ \alpha}. \tag{8}\]  
Therein, \(C_{\alpha}\) is a real number. In the case of an optically thin medium, the output signal from the polarimeter is proportional to the Faraday rotation angle, which is also proportional to \(\langle F_{y}\rangle^{F1}\) in Equation (8). To make it easier to solve the evolution of the density matrix \(\rho^{F_{1}}\), we transfer \(\rho^{F_{1}}\) into a rotating frame whose rotation frequency is the modulation frequency \(\omega_{\text{M}}\). The rotation direction is the same as the direction of the Larmor precession. The relation between the density matrix in the Lab frame, i.e., \(\rho^{F_{1}}\), and that in the rotating frame, i.e., \(\tilde{\rho}^{F_{1}}\), is  
\[\tilde{\rho}_{\alpha\;\beta}^{F_{1}}=\rho_{\alpha\;\beta}^{F_{1}}e^{\text{i}( \alpha-\beta)\omega_{\text{M}}t}. \tag{9}\]  
The evolution of \(\tilde{\rho}^{F_{1}}\) is  
\[ \begin{split}\frac{\text{d}\tilde{\rho}_{\alpha\;\beta}^{F_{1}}} {\text{d}t}=&-\text{i}(\alpha-\beta)[\omega_{\text{L}}-(\alpha+ \beta)\omega_{\text{NLZ}}/2-\omega_{\text{M}}]\times\tilde{\rho}_{\alpha\; \beta}^{F_{1}}\\ &+P_{\alpha\;\beta}\sum_{m=-\infty}^{\infty}A^{(m)}e^{\text{i}im \omega_{\text{M}}t}e^{\text{i}(\alpha-\beta)\omega_{\text{M}}t}\\ &-\Gamma_{\alpha\;\beta}(\rho_{\alpha\;\beta}^{F_{1}}-\delta_{ \alpha,\;\beta}/16).\end{split}  \tag{10}\]In the contributions from the optical-pumping process, or \(P_{\alpha\ \beta}\sum_{m=-\infty}^{\infty}A^{(m)}e^{im\omega_{\mathrm{M}}t}e^{ \mathrm{i}(\alpha-\beta)\omega_{\mathrm{M}}t}\), the terms with \(m+\alpha-\beta\
eq 0\) have fast oscillations. Their effects cancel out and the net effect is negligible.