# Frequency Response of Optically Pumped Magnetometer with Nonlinear Zeeman Effect
## 2 Experiment Setup
The probe laser is positively detuned by \(\sim\)400 MHz from the D2 transition (from 6 \({}^{2}\)S\({}_{1/2}\)\(F=4\) to 6 \({}^{2}\)P\({}_{3/2}\)\(F^{\prime}=5\)). The \(1/e^{2}\) diameter of the probe laser is \(\sim\)2 mm and the average power of the probe laser is \(\sim\)10 \(\mu\)W in the experiment with \(B_{z}\approx 1.15\)\(\mu\)T and is \(\sim\)50 \(\mu\)W in the experiment with \(B_{z}\approx 50\)\(\mu\)T or 70 \(\mu\)T. The precession of the atomic magnetic moment causes periodical optical rotation of the probe beam, which is analyzed with a polarimeter. The output signal from the polarimeter is demodulated with an SR865A lock-in amplifier (LIA). When the modulation frequency is tuned to the Larmor frequency, the in-phase component of the LIA is proportional to the difference between the modulation frequency and the Larmor frequency. As a result, when the modulation frequency is set to a fixed value in the nearby of the Larmor frequency, the variation of the in-phase component directly reflects the variation of the magnetic field to be measured. This is the so-called open loop mode operation of the magnetometer. Figure 1: Schematics of the Bell–Bloom optically pumped magnetometers (OPM). Cell: cylindrical paraffin-coated cesium vapor cell; AOM: acoustic-optical modulator; \(\lambda/4\): quarter-wave plate; \(\lambda/2\): half-wave plate; BPD: balanced photodetector. The cesium atoms are confined in the cylindrical paraffin-coated cesium vapor cell which is placed inside a seven-layer magnetic shielding. The biasing magnetic field is set along the axial direction of the magnetic shield, i.e., the \(z\)-axis. The atoms are optically pumped with a circularly polarized and amplitude modulated D1 laser beam, and the precessing atomic spins are probed with a linearly polarized D2 laser beam. The Faraday rotation of the probe beam is analyzed with a polarimeter which is composed of a half-wave plate, a Wollaston prism, and a balanced photodetector shown on the right side of the magnetic shielding. The detected Faraday rotation signal from the polarimeter is demodulated with a lock-in amplifier. A signal generator is used to generate the modulation signal applied to the AOM and to provide the reference signal to the lock-in amplifier for demodulation. The OPM's frequency response describes how its readout responds to magnetic-field signals with different frequencies. To extract the frequency response of the OPM, we apply a sinusoidal magnetic-field modulation with a typical amplitude of tens of pT along the \(z\)-direction and record the readout of the OPM. The OPM readout thus contains a sinusoidal component oscillating at the modulation frequency. In the ideal situation that the magnetometer could record the magnetic-field signal without time delay and distortion, this oscillating component in the OPM readout should have the same amplitude and phase as the modulated magnetic-field signal. In practice, however, there may be some amplitude scaling and phase shift between the readout of the magnetometer and the applied magnetic-field signal. The changes of the amplitude scaling factor and the phase shift with respect to the modulation frequency are the amplitude and phase frequency responses of the magnetometer, respectively, or in brief the frequency response.