## 3 Experimental Observation of Rabi Spectroscopy and Dephasing  
### Sideband-Resolved Spectroscopy and High-Resolution Rabi Spectroscopy  
The inhomogeneity excitation, which causes the dephasing of the Rabi flopping process, originates from the thermal distribution of atoms on external states, the atomic interaction, and the misalignment angle between the clock beam and the lattice beam [13, 15, 16]. The misalignment angle of 13 mrad is constant in all measurements. The atomic temperature can be measured by the sideband-resolved spectroscopy shown in Figure 2. Herein,  
Figure 1: Experimental setup for an optical lattice trap and clock transition detection and energy level scheme of the \({}^{87}\)Sr atom. (**a**) A horizontal one-dimensional optical lattice potential is formed by overlapping the incident lattice beam and its reflection using a concave mirror (CM). The lattice light is generated by the 813 nm semiconductor laser with a tapered amplifier. The amplitude of the radio frequency signal for the acoustic optical modulator (AOM) is controlled through a voltage-controlled attenuator (VVA). The Floquet engineering (FE) is realized by applying a sinusoidal voltage to the piezoelectric (PZT) of the lattice laser. The frequency of the 698 nm clock laser is stabilized to an ultralow-expansion (ULE) glass cavity. A photomultiplier (PMT) is used to collect the fluorescence emitted by the atoms. (**b**) Energy level scheme of the \({}^{87}\)Sr atoms. the red squares correspond to atoms in the lattice loaded directly from the red MOT, and the blue triangles indicate the loading with an energy filtering process. The longitudinal trap frequency is \(\
u_{\mathrm{z}}\approx 71\) kHz inferred from the frequency gap between the sharp edge of the blue sideband and the carrier. By calculating the ratio of the integrated absorption cross sections of the red and blue sidebands, the longitudinal temperature with and without the energy filtering process is determined to be \(T_{\mathrm{H}}\approx 3.8\) uK and \(T_{\mathrm{L}}\approx 2.3\) uK, respectively. To carefully study the dephasing of the Rabi flopping process with and without FE, the high-resolution Rabi spectrum is needed, as shown in Figure 3, in which the excitation fraction depends on the duration of the clock pulse and the effective coupling strength. Figure 3a shows the high-resolution Rabi spectrum without FE, where only one peak is observed corresponding to the \({}^{1}S_{0}\) (\(m_{\mathrm{F}}=-9/2\)) \(\rightarrow\)\({}^{3}P_{0}\) (\(m_{\mathrm{F}}=-9/2\)) transition. With FE, the \({}^{1}S_{0}\) and \({}^{3}P_{0}\) states split into a series of Floquet quasi-levels, which leads to multiple transition peaks. The number of peaks and the corresponding excitation fraction of each peak with FE are determined only by the driven amplitude \(A\), and the frequency gap between adjacent peaks equals the driven frequency \(\omega_{\mathrm{s}}/2\pi\). As \(A=0.72\) and \(\omega_{\mathrm{s}}/2\pi=100\) Hz, the FE Rabi spectrum is shown in Figure 3b, where the carrier and the first-order Floquet sidebands have the same amplitude. Without considering the interaction between atoms and the tunneling effect between lattice sites, the FE spectrum can be obtained by solving the effective Hamiltonian [10], as follows:  
\[\hat{H}_{\mathrm{n}_{\mathrm{z}},\mathrm{n}_{\mathrm{r}}}(t)=\frac{\hbar}{2}( \delta+\omega_{\mathrm{z}}\cos(\omega_{\mathrm{s}}t))\hat{\sigma}^{(3)}_{ \mathrm{n}_{\mathrm{z}},\mathrm{n}_{\mathrm{r}}}+\frac{\hbar\Omega_{\mathrm{ n}_{\mathrm{z}},\mathrm{n}_{\mathrm{r}}}}{2}\hat{\sigma}^{(1)}_{\mathrm{n}_{ \mathrm{z}},\mathrm{n}_{\mathrm{r}}}, \tag{1}\]  
where \(\hat{\sigma}^{(1)}_{\mathrm{n}_{\mathrm{z}},\mathrm{n}_{\mathrm{r}}}\) and \(\hat{\sigma}^{(3)}_{\mathrm{n}_{\mathrm{z}},\mathrm{n}_{\mathrm{r}}}\) are Pauli matrices, while \(\mathrm{n}_{\mathrm{z}}\) and \(\mathrm{n}_{\mathrm{r}}\) indicate the longitudinal and transverse motional quantum numbers of the atoms, respectively.