# Frequency Response of Optically Pumped Magnetometer with Nonlinear Zeeman Effect
## 2 Experiment Setup  
The overall scheme of our Bell-Bloom cesium magnetometer is shown in Figure 1. The cesium atoms are contained in a cylindrical paraffin-coated vapor cell with a diameter and length of 25 mm. The paraffin coating preserves the atomic polarization when the alkali atoms collide with the cell wall [31], which helps to prolong the lifetime of the ground-state Zeeman coherence and leads to a narrow magnetic resonance with a half width at half maximum of about 2 Hz in a small magnetic field (1.15 \(\upmu\)T). The vapor cell is kept at room temperature during the experiment and is placed in a cylindrical magnetic shielding with seven layers of mu-metal. The magnetic shielding prevents the external magnetic field from entering its inner space and provides a residual magnetic field \(\sim\)nT (1 nT = 10\({}^{-9}\) T) inside the shielding. A magnetic coil, with a field-current constant of \(\sim\)0.115 \(\upmu\)T/mA, is placed inside the magnetic shielding to control the inner magnetic field along the axial direction (the \(z\)-direction). The strength of the magnetic field can be set to any value below 115 \(\upmu\)T by adjusting the current applied to the magnetic coil. The biasing magnetic field \(B_{z}\) in our setup is set along the axial direction, or the \(z\)-axis. The magnetic coil used in our experiment is a cylindrical commercial magnetic coil which is composed of eight circular coils connected in series. Figure 1 is only a schematic and is not drawn to scale. The diameter of the coil is about 30 cm and the number of turns and the position of each circular coil are optimized to minimize the magnetic field gradient. There are four holes on the lateral side of the coil to let the beams pass and no circular coil passes through these holes. To estimate the broadening of the magnetic resonance due to the magnetic inhomogeneity, we measure the distribution of the biasing magnetic field \(B_{z}\) along the \(z\)-axis and along the \(x\)-axis with a fluxgate. The result shows that the relative magnetic field inhomogeneity in the central \(\pm\)1.25 cm is about 0.067%, which is limited by the inhomogeneity along the \(z\) axis. The magnetic resonance broadening induced by this magnetic field inhomogeneity in the evacuated cesium vapor cell is estimated to be 4.3 (8.5) Hz for 50 (70) \(\upmu\)T biasing field (the broadening will be larger if the vapor cell is not exactly placed in the center of the coil) [32], which is about 5 times smaller than the experimental measured overall magnetic resonance linewidth for the corresponding biasing field. Consequently, the broadening due to the magnetic field inhomogeneity is not the main reason for linewidth broadening in our setup. During the experiment, we use a circularly-polarized pump laser beam with a \(1/e^{2}\) diameter of \(\sim\)5 mm (\(e\) is the base of the natural logarithms. The light intensity falls to \(1/e^{2}\approx 0.135\) times the maximum value at the edge of the \(1/e^{2}\) diameter) to polarize the cesium atoms inside the vapor cell. Although the pump beam enters the vapor cell through the lateral cylindrical wall, which is curved, the distortions of the beam shape and the polarization are negligible because the beam diameter is much smaller than the cell diameter and the wall where the pump beam passes through is relatively flat. The pump laser is resonant with the cesium D1 transition (from 6 \({}^{2}\)S\({}_{1/2}\)\(F=3\) to 6 \({}^{2}\)P\({}_{1/2}\)\(F^{\prime}=4\), where \(F\) and \(F^{\prime}\) are the total angular momentum numbers). Its amplitude is modulated with an acoustic-optical modulator (AOM) about the Larmor frequency of cesium atoms and a modulation duty cycle is 20%. The average power of the pump beam is \(\sim\) 50 \(\mu\)W. The macroscopic magnetic moment produced by the pump beam precesses at the modulation frequency of the pump beam and is monitored with a linearly polarized probe laser which propagates along the axis of the cylindrical vapor cell.