## 2 Principle  
Specific theoretical analysis is as follows. Each mode of optical frequency comb can be expressed as  
\[F_{n}=f_{0}+nf_{rep} \tag{1}\]  
where \(f_{0}\) is the carrier-envelop offset frequency, \(f_{rep}\) is the pulse repetition rate, and n is an integer on the order of \(10^{6}\). The sum frequency of two comb modes is given by  
\[F_{sum(m+n)}=2f_{0}+(m+n)f_{rep} \tag{2}\]  
Whenever the sum frequency of two modes coincides transition frequency, the optical comb excites the two-photon transition. A hundred thousand pair modes can satisfy the resonant condition at the same time. When two comb modes are near resonance with the 5S-5P and 5P-5D transitions, the intermediate 5P state enhances stepwise two-photon transitions (S-TPT). We utilize this phenomenon to enhance the two-photon transition signal intensity. Due to the negligible Doppler broadening in cold atom experiments, the S-TPT excitation conditions are more critical. S-TPT can be achieved only when the 5S-5P and 5P-5D transition frequency difference is exactly an integral multiple of the repetition frequency. If \(f_{rep}\) deviates (\(f_{776 nm}-f_{780 mm})/n\), no S-TPT can be excited, because no other comb excites another single-photon transition when one comb frequency is aligned with a single-photon transition. To validate this feature, we consider all 14 S-TPT transition pathways for 5S\({}_{1/2}\)-5P\({}_{3/2}\)-5D\({}_{5/2}\) of \({}^{87}\)Rb. 14 S-TPT pathways and the TPT intensity of each TPT pathway are shown in Table 1.  
Each transition path has a different single-photon transition frequency, as well as a different two-step transition frequency difference, which has a different repetition frequency resonance value. The results in Figure 2 show the simulation curves of the total TPT intensities produced by these 14 S-TPT pathways with repetition rates. It can be seen that the S-TPT intensity exhibits a periodicity of about 8 kHz with \(f_{rep}\), which corresponds to the range of variation of the repetition frequency from one resonance value to the next resonance value (\(\Delta f\)/\(n\)\(\rightarrow\)\(\Delta f\)/(\(n+1\))). Take the transition path 5S\({}_{1/2}\)(F = 2)-5P\({}_{3/2}\)(F = 3)-5D\({}_{5/2}\)(F = 4) as an example. The two-step transition frequencies are \(f_{780 mm}\) = 384,228,115,203.2 kHz, \(f_{776 nm}\) = 386,341,017,529.38 kHz. The resonance frequency of the repetition  
Figure 1: The ladder-type three-level of Rb two-photon transition (TPT). frequency corresponding to the point A in the figure is \((f_{776 nm}-f_{780 nm})/16\),307 = 129,570 kHz. The next repetition resonance frequency value is \((f_{776 nm}-f_{780 nm})/16\),306 = 129,578 kHz. The difference between the two values is about 8 kHz. The calculation results of other points are also about 8 kHz, with only slight differences. So, as a whole, the TPT intensities exhibit periodicity of 8 kHz with \(f_{rep}\). Based on the above calculation and analysis, in order to further enhance the signal intensity of two-photon transition signal in cold atoms, we need to finely control \(f_{rep}\). The standard technique to change repetition rate of an Er fiber optical frequency comb is to change the laser cavity length by a piezoelectric transducer (PZT) or a translation stage. The relationship between \(f_{rep}\) and cavity length can be given as  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Rb\({}^{\bf 87}\) TPT Pathways** & **Intensity** \\ \hline \(5\)S\({}_{1/2}\)(F = 1)\(\rightarrow\)5P\({}_{3/2}\)(F = 0)\(\rightarrow\)5D\({}_{5/2}\)(F = 1) & 0.1666 \\ \(5\)S\({}_{1/2}\)(F = 1)\(\rightarrow  
\[f_{rep}=\frac{c}{k\cdot(L+l(t))} \tag{3}\]  
where \(L\) is the original length, \(l(t)\) is length of the PZT or the translation stage at time \(t\), \(c\) is the light velocity, and \(k\) is the average refractive index of the cavity.