## 2 Principle and Overall Design of Frequency Stabilization System  
### Basic Principle  
The Rb element contains two isotopes, \({}^{85}\)Rb and \({}^{87}\)Rb, with natural abundances of 72.15% and 27.85%, respectively. Figure 1 shows the hyperfine energy level structure of the Rb atom [21; 22; 23], and its transitions from \(5^{2}S_{1/2}\) to \(5^{2}P_{3/2}\) and \(5^{2}P_{3/2}\) are respectively called D1 and D2 lines. The numbers between energy levels represent the energy level spacing (unit MHz). Figure 1: Hyperfine energy level structure of the Rb atom. The laser from the same laser is split into a stronger pump light and a weaker probe light. When the laser frequency scans the transition line of the atoms, the group of atoms with zero velocity along the laser beam axis absorbs photons from both beams. The atoms are saturated with the pump light absorption of high power, so the absorption of the probe light is weak, resulting in strong transmission of the probe light. The raised peak is observed on the absorption spectrum, i.e., the intrinsic saturation absorption peak, and the corresponding spectral line is called the SAS. In addition, cross-saturation absorption peaks occur on the SAS when the frequency is scanned to the crossing line of the atoms [12]. Figure 2 shows the corresponding saturated absorption peaks are the \(F_{\text{g}}\) = 2\(\rightarrow\)\(F_{\text{e2}}\) = (1,3) cross line (\(C_{\text{13}}\)), \(F_{\text{g}}\) = 2\(\rightarrow\)\(F_{\text{e2}}\) = (2,3) cross line (\(C_{\text{23}}\)), and \(F_{\text{g}}\) = 2\(\rightarrow\)\(F_{\text{e2}}\) = 3 transition line (\(T_{\text{3}}\)) of \({}^{87}\)Rb and \(F_{\text{g}}\) = 3\(\rightarrow\)\(F_{\text{e2}}\) = (2,4) cross line (\(C_{\text{24}}\)), \(F_{\text{g}}\) = 3\(\rightarrow\)\(F_{\text{e2}}\) = (3,4) cross line (\(C_{\text{34}}\)), and \(F_{\text{g}}\) = 3\(\rightarrow\)\(F_{\text{e2}}\) = 4 transition line (\(T_{\text{4}}\)) of \({}^{85}\)Rb, and the corresponding wavelengths and frequencies are shown in Table 1.  
Laser frequency stabilization is a dynamic balance process, and the control system constantly compares the laser frequency with the reference frequency to generate an error signal. The error signal is not zero when the reference frequency offsets the laser frequency. The signal is fed back to the current or external cavity of the ECDL through a feedback loop, which dynamically compensates for the change in the laser frequency and makes it return to the reference frequency to achieve frequency stabilization [24]. The differential curve of SAS is used as the error signal to reduce Doppler background noise and improve the signal-to-noise ratio (SNR). Figure 3a-c, respectively, show the primary, secondary, and tertiary differential curves of Rb atomic SAS (In Figure 2). We can see that the amplitude changes significantly at the saturation absorption peak, while it is almost zero at other positions. The secondary differential curve is not zero at the center of the spectral line. It cannot distinguish between positive and negative laser frequency detunings. Compared to the tertiary differential curve, the linear region of the zero crossing of the primary differential \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Transition**} & \multicolumn{1}{c}{**Wavelength/nm**} & \multicolumn{1}{c}{**Frequency/GHz**} \\ \hline \multirow{3}{*}{\({}^{87}\)Rb (\(F_{\text{g}}\) = 2\(\rightarrow\)\(F_{\text{e2}}\))} & \(C_{\text{13}}\) & 780.246451 & 384,227.9034 \\  & \(C_{\text{23}}\) & 780.246292 & 384,227.9819 \\  & \(T_{\text{3}}\) & 780.246021 & 384,228.1152 \\ \hline \multirow{3}{*}{\({}^{85}\)Rb (\(F_{\text{g}}\) = 3\(\rightarrow\)\(F_{\text{e2}}\))} & \(C_{\text{24}}\) & 780.243920 & 384,229.1497 \\  & \(C_{\text{34}}\) & 780.243856 & 384,229.1814 \\ \cline{1-1}  & \(T_{\text{4}}\) & 780.243733 & 384,229.2417 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Frequency and wavelength of the saturated absorption peak of Rb atoms in vacuum [22; 23]. Figure 2: SAS of the Rb D2 line transition.