## 2 Experimental Apparatus  
The details of the \({}^{87}\)Sr OLC setup in our experiment have been described in previous work [22; 23; 24]. Here, we briefly recapitulate some basic parts and key improvements of the apparatus. The setup for the optical lattice trap and clock transition detection is shown in Figure 1a, and the partial level scheme of \({}^{87}\)Sr is shown in Figure 1b. The strontium atoms in the oven are heated to 770 K to produce atomic flux, and then collimated with a 461 nm collimating beam (with a red frequency detuning of 15 MHz from the 5s\({}^{2}\)\({}^{1}\)S\({}_{0}\) (\(F=9/2\)) \(\rightarrow\) 5s5p \({}^{1}\)\(P_{1}\) (\(F=11/2\)) transition shown in Figure 1b) to reduce the transverse divergence angle. Before the magneto-optical trap (MOT), a Zeeman slower is used to increase the number of atoms with a velocity below 50 m/s. The first stage of cooling (blue MOT) is based on the 5s\({}^{2}\)\({}^{1}\)S\({}_{0}\) (\(F=9/2\)) \(\rightarrow\) 5s5p \({}^{1}\)\(P_{1}\) (\(F=11/2\)) transition with a red frequency detuning of 35 MHz. The large scatter rate of 2\(\pi\times\) 32 MHz leads to atomic temperature of 2 mK after the blue MOT. For further cooling of the atoms to \(\sim\)uK, the second stage of cooling (red MOT) based on 5s\({}^{2}\)\({}^{1}\)S\({}_{0}\) (\(F=9/2\)) \(\rightarrow\) 5s5p \({}^{3}\)\(P_{1}\) (\(F=11/2\)) transition at 689 nm is applied. After two stages of cooling, roughly 10\({}^{6}\) atoms are trapped in the red MOT with a temperature of 3 uK. The number of atoms trapped in the red MOT can be controlled by changing the power or frequency of the collimating beam without changing the atomic temperature. At the end of the red MOT, the magnetic field for the MOT and all the cooling lasers is switched off, and the cooled atoms are automatically loaded into a one-dimensional horizontal optical lattice trap. The lattice beam with the \"magic wavelength\" of 813.42 nm has a maximum one-way power of 450 mW and is focused on the center of the red MOT with a waist radius of 50 \(\upmu\)m, which corresponds to a trap depth of \(U=107\)\(E_{\text{R}}\) (\(E_{\text{R}}\) is the lattice photon recoil energy) [13]. Then, atoms are polarized to the Zeeman level of \(m_{\text{F}}=+9/2\) (\(m_{\text{F}}=-9/2\)) by the \(\sigma+\) (\(\sigma-\)) polarized light corresponding to 5s\({}^{21}\)S\({}_{0}\) (\(F=9/2\)) \(\rightarrow\) 5s5p \({}^{3}\)\(P_{1}\) (\(F=9/2\)) transition at 689 nm. At the spin-polarized stage, using three pairs of Helmholtz coils, the residual magnetic fields in the horizontal direction are compensated to near zero and the magnitude of the vertical magnetic field is about 50 mG to define the quantum axis along the vertical (gravitational) direction. Then, the energy filtering process is used to remove hot atoms by ramping down the lattice depth from 107 \(E_{\text{R}}\) to 58 \(E_{\text{R}}\) within 15 ms, holding for 20 ms, and ramping back to 107 \(E_{\text{R}}\) within 15 ms [25]. The energy filtering stage allows us to control the temperature of the atoms trapped in the lattice by changing the magnitude of the lowest trap depth. After completing the preparation of the cold ensemble, the clock laser with 698 nm is used to detect the transition probability of the 5s\({}^{2}\)\({}^{1}\)S\({}_{0}\) (\(m_{\text{F}}=\pm 9/2\)) \(\rightarrow\) 5s5p \({}^{3}\)\(P_{0}\) (\(m_{\text{F}}=\pm 9/2\)) clock transition. The clock laser is frequency stabilized on a 20 cm long ultralow-expansion (ULE) cavity via the Pound-Drever-Hall method. The clock laser beam used for frequency stabilization passes through an acoustic optical modulator (AOM). The power of the locking beam is stabilized by changing the amplitude of the AOM. The linear frequency drift of the cavity is compensated by sweeping the frequency applied on the AOM, which reduces the frequency drift of the clock laser from 80 mHz/s to below 1 mHz/s.