## 3 Experiment & Results  
The experiment setup for our system is shown in Figure 3. An Erbium-fiber-based frequency comb with the 1556 nm center wavelength is used as the source to excite atoms. The Er frequency comb emits 100 fs pulses with 100 mW average power and a repletion rate of 129 MHz. The cavity length can be adjusted via a PZT and a translation stage in the cavity. A home-built two-stage erbium-doped fiber amplification module is used to amplify the output of 1556 nm mode-locked laser to 200 mw. In order to improve pulse peak power, the comb is compensated the pulse broadening in optical fiber with a pair of prisms made of silica. The properties of the optical pulse are adjusted to focus the power of the optical frequency comb spectrum to the vicinity of 1556 nm. Then we focus the beam into a periodically poled lithium niobate crystal (PPLN) by a lens to double the frequency of the comb. A 20 mW frequency comb with center wavelength of 778 nm is obtained in the end. Secondly, the 778 nm frequency comb is directed into magneto optical trap (MOT) cell to probe the cold \({}^{87}\)Rb atom cloud which is cooled and trapped in MOT. To eliminate the influence of magnetic fields in MOT, we design a timing control cycle. We switched off the magnetic fields in MOT when the 778 nm frequency comb is directed into MOT cell to excite the Rb atom for 2 ms. Then the magnetic fields are switched on and the atom cloud is captured and cooled in MOT for 8 ms, and the 778 nm frequency comb is switched off by Acousto-optical Modulators (AOM) at the same time. Repeating this 10 ms cycle, we demonstrate the excitation of the two-photon transition. To verify the relationship between \(f_{rep}\) and TPT intensity, we scan \(f_{rep}\) and detect 420 nm fluorescence signal from cascade decay via the 6P-5S state with a photomultiplier (PMT), used in photon counting mode. First, in order to verify the simulation results in Figure 2, we fix the pump current at 92.2 mA in experiments and adjust the repetition frequency to observe the fluorescence signal at the same time. When we scan \(f_{rep}\) 50 Hz, the comb teeth at 780 nm will scan 129 MHz. So, we can get the direct frequency comb spectroscopy which illustrates all 5S\(\rightarrow\)5D transition lines by scanning \(f_{rep}\) 50 Hz with controlling voltage on PZT. The intensity of each line depends on the center scanning frequency of \(f_{rep}\). The peak intensity of the DFCS is recorded. Then, we change the center \(f_{rep}\) scanning frequency by adjusting the translation stage in cavity and repeat the above steps. The obtained result is shown in Figure 4. Blue dots show the peak intensity of the DFCS with different center \(f_{rep}\) scanning frequency. Red lines show simulation results. It can be seen that the experiment data and the simulation results are almost perfectly matched, which fully illustrates the influence of the repetition frequency on the two-photon intensity and verifies our theoretical analysis. Then, we verify the small change of repetition frequency caused by the pump power change. In ensuring the laser cavity length and average power fixed, we change the pump power and measure TPT intensity, and the results are shown in Figure 5. We can observe the small shift of the TPT peaks in the spectrum because the frequency of the two photon transitions is fixed. It can be seen that different pump powers affect the repetition frequency of the frequency comb. Figure 4: Relationship between repetition rate of comb and two-photon transition intensity. Red lines show simulation results. Blue dots show experiment results. The dashed line shows the average peak intensity of direct frequency comb spectroscopy (DFCS) without optimization. Figure 3: Simplified block diagram for stabilization system. MLL: Mode-locked laser; EDF: Er-doped fiber; ISO: isolator; HNLF: High non-linear fiber; PPLN: Periodically Poled Lithium Niobate; PMT: photon-multiplier; MOT: magneto optical trap; QWP: quarter-wave plate; HWP: half-wave plate.