## 3 Experimental Observation of Rabi Spectroscopy and Dephasing
### Suppression of the Dephasing of the Rabi Flopping Process  
If the clock laser intensity keeps constant, the above analysis indicates that the coupling strength between atoms and the clock laser will be reduced with FE compared with the case of the absence of FE. This will hamper the measurement of the dephasing suppression of the Rabi flopping process by FE, as the lower coupling strength indicates the longer clock pulse duration to reach the maximum excitation fraction, and the inhomogeneous excitation will lead to additional dephasing of the Rabi flopping process. Thus, the clock laser intensity is different between FE and the absence of FE. Based on the high-resolution Rabi spectra, we measure the excitation fraction at different clock pulse durations with and without FE, as shown in Figure 4. Herein, the driving amplitude of \(A\) = 0.72 is optionally chosen, as we found that changing the driving amplitude from 0.4 to 1.5 will not lead to obvious variation to the Rabi flopping process. The coupling strengths of the carrier (corresponding to the 0 th sideband shown in Figure 3) are about 4.1 Hz in Figure 4 to avoid extra dephasing induced by inhomogeneous excitation. In this experiment, we carefully study the suppression of the dephasing by FE while considering two possible factors of interaction between atoms and the temperature of the cold ensemble. The strength of the interaction between atoms is proportional to the number of atoms, which is inferred from the atomic detection noise [27]. Figure 4a shows the excitation fraction as a function of clock pulse duration under three experimental conditions without FE. Obviously, both parameters of temperature and atomic interaction will lead to dephasing of the Rabi flopping process. The interaction between atoms causes the phenomenon of dephasing due to the following three aspects: the density-dependent frequency shift is different between lattice sites due to the Gaussian distribution of atoms [28]; the density shift in a lattice site is different for atoms populated  
Figure 3: High resolution Rabi spectra with a 0.18 s spectroscopy pulse. (**a**) The Rabi spectrum without FE. (**b**) The Rabi spectrum with FE (\(A\) = 0.72, \(\
u_{\text{s}}\) = 100 Hz). The reason why the 0 th and 1 st Floquet sidebands have approximately equal height is that the 0 th and 1 st sidebands have similar effective Rabi frequencies at this modulation amplitude. The atomic temperature and the number of atoms are 3.8 \(\mu\)K and 6300, respectively, in both figures. in different longitudinal and transverse motional quantum numbers \(\text{n}_{\text{z}}\) and \(\text{n}_{\text{r}}\)[16]; the inelastic _e_-_e_ collision causes the loss of excited atoms [15]. The temperature of the cold ensemble determines the distribution of atoms in the motional quantum states, and the higher temperature will lead to more populations in higher motional quantum states, which aggravates the dephasing of the Rabi flopping process [13]. Holding the number of atoms on \(N_{\text{L}}=2100\), the influence of the atomic temperature on the dephasing is studied by measuring the Rabi flopping processes without and with FE at low and high temperatures of \(T_{\text{L}}=2.3\)\(\mu\)K and \(T_{\text{H}}=3.8\)\(\mu\)K, respectively, shown in Figure 4b,c. To study the influence of the atomic interaction on the dephasing, keeping the temperature of the cold ensemble of \(T_{\text{H}}=3.8\)\(\mu\)K, the Rabi flopping processes with and without FE are measured under low and high atomic density (with the numbers of atoms of \(N_{\text{L}}=2100\) and \(N_{\text{H}}=6300\), respectively), as shown in Figure 4c,d. Figure 4c,d demonstrate that the excitation fractions with FE are higher than the case without FE, indicating that the dephasing of Rabi flopping process caused by high temperature and atomic interaction can be suppressed using FE. With FE, the percentages of the maximum excitation fraction in Figure 4c,d increase by 7.7% and 15.4%, respectively.