## 3 The Second-Order Zeeman Coefficient Measurement and Zeeman Shift Evaluation  
### The Experimental Scheme for Measuring the Quadratic Zeeman Coefficient  
While measuring the quadratic Zeeman shift due to modulating the magnetic field, the lock-in data must be free of the first-order Zeeman shift. This was realized by using four servos, as shown in Figure 3. The red solid curve represents the spin-polarized spectra shown in Figure 2b when the bias magnetic field strength was strong (denoted by \(B_{\rm H}\)) and the blue solid curve with narrower frequency gap corresponds to the case that the bias magnetic field strength was low (denoted by \(B_{\rm L}\)). The duration of each clock feedback cycle, which included equal eight clock cycles, was 8 s. We controlled the bias magnetic field at low strength (\(B_{\rm L}\)) during clock cycles 1-4, and at stronger intensity (\(B_{\rm H}\)) during clock cycles 5-8. The FWHM of the left spin-polarized peak (\(m_{\rm F}\) = -9/2) was \(2\delta_{1}\), and the right (\(m_{\rm F}\) = +9/2) was \(2\delta_{2}\). In clock cycles 1 and 2, the clock laser frequencies were \(v_{\rm L1}\)-\(\delta_{1}\) and \(v_{\rm L1}\)+\(\delta_{1}\) respectively. After clock transition detection, the corresponding excitations were \(e_{1}\) and \(e_{2}\) respectively and the error signal \(err_{1}\) = \(e_{1}\)-\(e_{2}\) could be calculated. The feedback signal was \(\Delta f_{1}\) = \(err_{1}\) \(\times\)\(servo_{1}\) and the new frequency \(v^{\prime}\)\(v_{\rm L1}\) was obtained by directly adding the \(\Delta f_{1}\) to the input frequency \(v_{\rm L1}\). In principle,  
Figure 3: Schematic diagram of measuring the quadratic Zeeman shift. \(B_{\rm H}\) and \(B_{\rm L}\): the strength of bias magnetic field are strong and weak respectively; \(v_{\rm L1}\) and \(v_{\rm R1}\): initial center frequency of left and right peaks, when the bias magnetic field is \(B_{\rm L}\); \(v_{\rm L2}\) and \(v_{\rm R2}\): initial center frequency of left and right peaks, when the bias magnetic field is \(B_{\rm H}\); \(e_{1-8}\): excitation fraction; \(err_{1-4}\): error signal; \(Servo_{1-4}\):  
\(v^{\prime}{}_{\rm L1}\) was closer to the clock transition than \(v_{\rm L1}\) and so \(v^{\prime}{}_{\rm L1}\) replaced \(v{}_{\rm L1}\) in the next clock feedback cycle. In clock cycles 3 and 4, the frequencies of the clock laser were \(v_{\rm R1}\)\(-\)\(\delta_{2}\) and \(v_{\rm R1}\)\(+\)\(\delta_{2}\), and the new frequency \(v^{\prime}{}_{\rm R1}\)\(=\)\(\Delta f_{2}\)\(+\)\(v_{\rm R1}\) was obtained after the clock transition detection and feedback computation by the \(servo_{2}\) system. The locking processes of clock cycles 5-6 and 7-8 were similar to the 1-2 and 3-4 clock cycles, respectively, while the bias magnetic field was \(B_{\rm H}\). We were able to calculate the average frequency \(v_{\rm L0}\)\(=\)\((v^{\prime}{}_{\rm L1}\)\(+\)\(v^{\prime}{}_{\rm R1})\)/2 and \(v_{\rm H0}\)\(=\)\((v^{\prime}{}_{\rm L2}\)\(+\)\(v^{\prime}{}_{\rm R2})\)/2 after every eight clock cycles, and their difference \(\Delta v_{\rm B}\)\(=\)\(v_{\rm L0}\)\(-\)\(v_{\rm H0}\) was the quadratic Zeeman shift. In order to ensure measurement accuracy, the lock-in data were increased until their Allan deviation was below \(1\times 10^{-16}\). In this measurement, the value of \(B_{\rm L}\) was fixed at 0.1 G and we changed the \(B_{\rm H}\) in the range from 0.1 G to 1.55 G. We defined the effective magnetic field intensity \(B_{\rm eff}\)\(=\)\(\sqrt{B_{\rm H}^{2}-B_{\rm L}^{2}}\) which indicated that the value of quadratic Zeeman shift was zero when \(B_{\rm eff}\)\(=\) 0. Thus, according to the relationship between \(B_{\rm eff}\) and the quadratic Zeeman shift, the quadratic Zeeman shift coefficient could be extracted by parabola fitting. Although the clock laser frequency was stabilized to a ULE cavity by the PDH stabilization, the effective cavity length changed slowly as the cavity aged and the temperature drifted, resulting in frequency drift of the clock laser [31].