## 2 Optical Lattice Clock Based on \({}^{87}\)Sr
### Closed-Loop Operation of \({}^{87}\)Sr Optical Lattice Clock  
In the experimental setup for our \({}^{87}\)Sr optical lattice clock (Figure 6), a clock laser operating at 698-nm, locked to an ULE cavity using the PDH technique, delivers a beam to a cold atom sample. The clock transition is detected using the electron-shelving technique for the transition \((55^{2})^{1}S_{0}\)-\((555p)^{1}P_{1}\). A normalization scheme is used to eliminate the shot-to-shot noise [24]. To stabilize the clock laser to resonate at the atomic transition of \((55^{2})^{1}S_{0}\)-\((555p)^{3}P_{0}\), we applied the modulation method. We stabilized the clock laser to the average frequency of the two \(m_{F}=+9/2\)-\(m_{F}=+9/2\) and \(m_{F}=-9/2\)-\(m_{F}=-9/2\) Zeeman transitions. The error signal is calculated digitally from four clock cycle. After filtering using a PID (proportional, integral, derivative) controller, feedback is provided by a frequency correction signal to modulate the AOM2 [25]. Figure 5: (**a**) Spectra of the \(\pi\) transitions in the presence of a 0.3-G magnetic field; (**b**) Spin-polarized spectra for the transitions \(m_{F}=+9/2\)-\(m_{F}=+9/2\) and \(m_{F}=-9/2\)-\(m_{F}=-9/2\); (**c**) Narrowest linewidth of the magnetic sublevel transitions. Figure 6: Schematic diagram of the experimental setup for \({}^{87}\)Sr optical lattice clock. (DDS: direct digital synthesizer; PMT: photomultiplier tube; PDH: Pound-Drever-Hall). The error signal was obtained after the clock laser had been locked to the atomic transition (Figure 7a) along with the in-loop error signal stability of the \({}^{87}\)Sr optical lattice clock (Figure 7b). Short-term stability is limited by the clock laser and is affected by the length fluctuation of the ULE cavity; long-term stability is also limited by the clock transition of \({}^{87}\)Sr. The fractional frequency instability obtained was \(2.8\times 10^{-17}\) for an averaging time of 2000 s. The Allan deviation is fitted with \(1.6\times 10^{-15}/\tau^{1/2}\), which is limited mainly by white-frequency-noise. Experimental Determination of the Lande \(g\)-Factors for (5s\({}^{2}\))\({}^{1}\)\({}_{5}\)\({}_{0}\) and (5s5p)\({}^{3}\)\({}_{0}\) States of \({}^{87}\)Sr  
Because the clock transition frequency is sensitive to both first- and second-order Zeeman shifts, the value of the Lande \(g\)-factors of the clock states need to be determined accurately [26]. We measured experimentally the \(g\)-factors of the ground state (5s\({}^{2}\))\({}^{1}\)\({}_{5}\)\({}_{0}\) and the differential \(g\)-factor \(\delta g\) between the (5s5p)\({}^{3}\)\({}_{0}\) and (5s\({}^{2}\))\({}^{1}\)\({}_{5}\)\({}_{0}\) states of \({}^{87}\)Sr with \(\pi\)-polarized and \(\sigma^{\pm}\)-polarized interrogation beams, respectively, at different magnetic field strengths [27]. A typical spectrum of the \(\pi\)-transition is shown in Figure 5a. When the polarization direction of the clock laser is perpendicular to the direction of the magnetic field, the \(\sigma^{\pm}\)-transition spectrum is obtained (Figure 8a). These peaks correspond to the transitions between the ground and excited sublevels. From the \(\sigma^{\pm}\)-transition spectrum, \(\delta g\) can be expressed as  
\[\delta g=\frac{g\left(5s^{2}\,{}^{1}\!S_{0}\right)}{2f_{\sigma^{+},m_{F}}}-1 =\frac{\left(1-\sigma_{d}\right)g_{I}}{2f_{\sigma^{\pm},m_{F}}}-1 \tag{1}\]  
where the diamagnetic correction \(\sigma_{d}=0.00345\)[26],\(f_{\sigma^{\pm},mF}\) is the frequency gap between neighboring lines for the \(\sigma^{\pm}\) transitions, \(f_{d,mF}\) is the frequency difference between \(\sigma^{+}\) and \(\sigma^{-}\) transition from the same sublevel, \(g_{I}=\mu_{1}/I\mu_{0}\) with \(\mu_{0}\) the vacuum permeability, and the nuclear magnetic moment \(\mu_{1}=-1.0924(7)\,\mu_{\rm N}\)[28] with \(\mu_{\rm N}\) the nuclear magneton.