# Frequency Response of Optically Pumped Magnetometer with Nonlinear Zeeman Effect
## 3 Theory  
In this work, we use the density matrix to model the cesium atomic states in the magnetometer. A complete description of the atomic states should involve both, the ground and excited states. Considering the fact that both, the pump and probe laser, are weak in our experiment, i.e., the light power densities of the pump and probe laser beams are so low that the rate at which atoms in the ground states are optically pumped to the excited states is much slower than the rate of the spontaneous emission which brings atoms in the excited states to the ground states, most of the cesium atoms lay in the ground states. As a consequence, we only model the evolution of the ground states. The ground states of cesium atoms are composed of two hyperfine states, i.e., \(F=4\) and \(F=3\). There are altogether 16 different Zeeman sublevels in the ground states. As a result, the density matrix \(\rho\) describing both these two hyperfine states is a \(16\times 16\) matrix. When the density matrix is expressed in the coupled basis, the terms corresponding to the coherence between the two different hyperfine states are fast oscillating and averaged to zero. So the only non-zero terms are the matrix blocks corresponding to the \(F=4\) hyperfine state, or \(\rho^{F_{1}}\), and that corresponding to the \(F=3\) hyperfine state, or \(\rho^{F_{2}}\). In our experiment, we are probing the magnetic resonance in the \(F=4\) hyperfine state. Only \(\rho^{F_{1}}\) has thus a direct influence on the probe signal. As a result, we can use the matrix block corresponding to the \(F=4\) hyperfine state, i.e., \(\rho^{F_{1}}\), to describe the evolution of the atomic state. The evolution of \(\rho^{F_{1}}\) follows the following Liouville equation  
\[\frac{\mathrm{d}\rho^{F_{1}}}{\mathrm{d}t}=-\frac{\mathrm{i}}{\hbar}[H,\,\rho ^{F_{1}}]+\frac{\mathrm{d}^{(1)}\rho^{F_{1}}}{\mathrm{d}t}+\frac{\mathrm{d}^{ (2)}\rho^{F_{1}}}{\mathrm{d}t}, \tag{1}\]  
in which the square brackets denote the commutator, \(H\) is the Hamiltonian of the magnetic-field interaction that includes the linear Zeeman effect and the NLZ effect. The term \((\mathrm{d}^{(1)}/\mathrm{d}t)\rho^{F_{1}}\) describes the optical pumping and \((\mathrm{d}^{(2)}/\mathrm{d}t)\rho^{F_{1}}\) is the relaxation term which involves the contributions from the wall-collision relaxation, power broadening due to the pump and probe beams, broadening due to the magnetic field gradient, and et al. It should be noticed that, for the more general case where the spatial distribution and diffusion of the density matrix are taken into consideration, the partial derivations should be used in this Liouville equation. In our experiment, we use an anti-relaxation-coated evacuated vapor cell in which cesium atoms traverse the cell within their ground-state Zeeman coherence lifetime. As a result, the atomic polarization is uniformly distributed in the vapor cell. Consequently, we can neglect the non-uniform distribution of the density matrix and simplify the partial derivations to the absolute derivations. We choose the quantization axis along the direction of the magnetic field \(B_{z}\) and extract the second-order Taylor expansion of the Breit-Rabi formula [33], so that \(H\) is a diagonal matrix in the coupled basis, with  
\[H_{\alpha\,\beta}=\delta_{\alpha,\beta}\hbar(\alpha\omega_{\mathrm{L}}-\alpha ^{2}\omega_{\mathrm{NLZ}}/2+8\omega_{\mathrm{NLZ}}), \tag{2}\]wherein \(\alpha\) and \(\beta\) are the magnetic quantum numbers ranging from \(-4\) to \(4\), \(\omega_{\rm L}=\gamma B_{z}\) is the Larmor frequency, where \(\gamma\) is the gyromagnetic ratio, \(B_{z}\) is the strength of the magnetic field, and \(\omega_{\rm NLZ}\approx 2\omega_{\rm L}^{2}/\omega_{\rm HF}\) is the splitting of the magnetic resonance induced by the NLZ effect in the Earth's field range, with \(\omega_{\rm HF}=2\pi\times 9.19\) GHz being the ground state hyperfine splitting of cesium atoms.