## 1 Introduction  
As requirements for high accuracy and stability frequency standards increase, optical lattice clocks based on neutral atoms with clock transition frequencies in the optical domain have been proposed [1]. After decades of hard work, these kinds of clocks have exhibited excellent performance in terms of stability and certainty [2, 3] and can be applied to not only quantum frequency standards [4, 5], but also many cutting-edge sciences [6, 7, 8, 9].  
The absolute frequency of clock transition is susceptible to the ambient environment around atoms. In addition to the Doppler effect and the vibration of experimental setups, the disturbing factors mainly come from three aspects: the electromagnetic field, such as the Stark frequency shift [10, 11, 12] and the Zeeman shift [13]; collisions, such as the frequency shift caused by collision between cold atoms [14] or collision between cold atoms and background gas [15]; and the influence of the gravitational field, such as gravitational frequency shift [16]. In terms of a one-dimensional optical lattice clock, the Zeeman shift is generally much smaller than the Stark frequency shift and the gravitational frequency shift, and is on the same order of the collision frequency shift. We need to carefully measure and correct these frequency shifts before we use a clock to generate the time standard signal.  
For the \({}^{87}\)Sr optical lattice clock, the Zeeman shift is rooted in the interaction between the hyperfine energy levels, which results in a difference of the Lande \(g\)-factors [13] between the ground state \({}^{1}S_{0}\) and the excited state \({}^{3}P_{0}\). This brings an inconsistent Zeeman shift between the ground and excited state in the same magnetic field, which finally causes the clock transition frequency to be different compared with the case of no magnetic field. The first-order Zeeman shift, which is proportional to the magnetic field intensity, could almost be cancelled by the stabilizing the clock laser to the average frequency [17] of \(m_{\rm F}=+9/2\to m_{\rm F}=+9/2\) and \(m_{\rm F}=-9/2\to m_{\rm F}=-9/2\). A residual first-order Zeeman shift could occur if the background magnetic field has a net drift while determining the center frequency of the two stretched states [18]. By contrast, the quadratic Zeeman shift is proportional to the square of the magnetic field, and there is no effective method to eliminate it. The quadratic Zeeman shift is also related to the type of atoms and the energy level of the clock transition. Relevant calculations show that under the same magnetic field, the quadratic Zeeman shift of an \({}^{87}\)Sr optical lattice clock is \(10^{8}\) times smaller than that of a cesium clock [19], and is more than one order of magnitude smaller than Hg\({}^{+}\)[20], Sr\({}^{+}\)[21], and Yb\({}^{+}\)[22] optical clocks. Three groups have experimentally measured the quadratic Zeeman shift coefficient: JILA (Joint Laboratory for Astrophysics) [23; 24; 25], LNE-SYRTE (Observatoire de Paris) [26], and PTB (Physikalisch-Technische Bundesanstalt) [27]. However, repeated measurement of the quadratic Zeeman shift coefficient of \({}^{87}\)Sr atoms is critical for precisely determining its value and depressing its statistical uncertainty.  
In this study, the quadratic Zeeman shift coefficient of the \({}^{87}\)Sr optical lattice clock was measured experimentally, and the frequency correction and uncertainty caused by the Zeeman shift of the optical lattice clock were evaluated.