# Frequency Response of Optically Pumped Magnetometer with Nonlinear Zeeman Effect
## 3 Theory
According to Equations (1) and (2), the evolution of the density matrix element \(\rho_{\alpha\;\beta}^{F_{1}}\) due to the magnetic field is  
\[-\frac{\rm i}{\hbar}[H,\rho^{F_{1}}]_{\alpha\;\beta}=-\frac{\rm i}{\hbar}(H_{ \alpha\;\alpha}-H_{\beta\;\beta})\times\rho_{\alpha\;\beta}^{F_{1}}=-\rm i( \alpha-\beta)[\omega_{\rm L}-(\alpha+\beta)\omega_{\rm NLZ}/2]\times\rho_{ \alpha\;\beta}^{F_{1}}, \tag{3}\]  
which means that \(\rho_{\alpha\;\beta}^{F_{1}}\) tends to oscillate at frequency \(-(\alpha-\beta)[\omega_{\rm L}-(\alpha+\beta)\omega_{\rm NLZ}/2]\). During the optical pumping process, as the pump laser is resonant with the \(6\;^{2}S_{1/2}\;F=3\) to \(6\;^{2}P_{1/2}\;F^{\prime}=4\) transition, the atoms on the \(F=3\) ground state are pumped into the \(F^{\prime}=4\) excited state, and then decay into both the \(F=4\) and \(F=3\) ground states through the spontaneous emission. Atomic polarizations are generated in both, the \(F=4\) and \(F=3\) ground states, during this process. The atomic polarization in the ground states are generated in two ways, i.e., the depopulation pumping and the repopulation pumping [34]. The depopulation pumping originates from the different coupling strength between different ground Zeeman sublevels and makes the sublevels be pumped out at different rates and causes population imbalance [34]. The repopulation pumping is caused by the decay of the excited states to the ground states during which the atomic polarization in the excited state is transferred into the ground states [34]. In our experiment, the atomic polarization in the \(F=3\) ground hyperfine state is generated from both, the depopulation pumping and the repopulation pumping, while the polarization in the \(F=4\) ground hyperfine state is generated only from the repopulation pump, as the \(F=4\) state is not directly coupled with the pump laser. The magnetic resonance in the \(F=4\) state is thus less affected by the power broadening induced by the depopulation pump [35], which helps achieve higher sensitivity of the OPM. As a result, we use a linearly polarized probe laser beam, which is positively detuned by \(\sim\)400 MHz from \(6\;^{2}S_{1/2}\;F=4\) to \(6\;^{2}P_{3/2}\;F^{\prime}=5\) transition to detect the magnetic resonance in the \(F=4\) state. The frequency of the probe laser beam is optimized to get the largest magnetic resonance signal. The evolution of \(\rho^{F_{1}}\) due to the repopulation pump is described by Equation (4) as  
\[\frac{\rm d^{(1)}\rho_{\alpha\;\beta}^{F_{1}}}{\rm d}t=A(t)P_{\alpha\;\beta}, \tag{4}\]  
in which \(A(t)\) is a real number which is related to the pump rate and \(P\) is a matrix determined by the density matrix of the \(F=3\) ground hyperfine state, i.e., \(\rho^{F_{2}}\), and the polarization of the pump beam. Since the difference between the gyromagnetic ratio of \(F=3\) and \(F=4\) states is about \(2\pi\times 11\) Hz/\(\rm\upmu T\), the difference between the Larmor frequency of \(F=3\) and \(F=4\) states in the Earth's field is several hundreds of Hertz, which is much larger than the magnetic resonance linewidth. As we are detecting the magnetic resonance of the \(F=4\) state, the modulation frequency of the pump beam is resonant with the Larmor frequency of the \(F=4\) state in our experiment. The modulation frequency is thus far off-resonant with the Larmor frequency of the \(F=3\) ground state, indicating that the transverse atomic polarization of the \(F=3\) ground state is not synchronized by the modulated pump beam. As a result, when it comes to the steady state, the \(F=3\) state is almost unaffected by the modulation frequency of the pump beam and the transverse atomic polarization of \(F=3\) state decays to \(0\), which means that \(\rho^{F_{2}}\) is a constant diagonal matrix. In this case, if we ignore the phase shift due to the Larmor precession of the excited states, one can show that the matrix \(P\) induced by the circularly polarized pump beam that transmits along the \(x\)-direction is a constant matrix and each of its matrix elements is a real number.