## 2 Analysis of the Working Principle of MOTs
### Action Mechanism  
Taking atoms moving along the z-axis as an example, the operational mechanism of the MOT is illustrated in Figure 2. It is assumed that the ground state has an orbital angular momentum of \(S=0\) and that the excited state has an orbital angular momentum of \(S=1\). When subjected to a gradient magnetic field, the excited state experiences Zeeman splitting, dividing into three magnetic sublevels, \(m_{S}=-1\), \(m_{S}=0\), and \(m_{S}=1\), as depicted in Figure 2. The extent of sublevel splitting is influenced by the magnetic field strength and is linearly correlated with the position. The Zeeman shift of the magnetic sublevel decreases linearly along the z-axis for \(m_{S}=-1\), remains constant for \(m_{S}=0\), and increases linearly along the z-axis for \(m_{S}=1\). The two beams traveling in opposite directions are polarized in opposite directions. The laser frequency \(\omega\) is lower than the transition frequency of the atom, indicating negative laser detuning \(\delta\).  
In the region where z > 0, the Doppler effect causes the \(\sigma^{-}\) light received by the atom to be closer to the atomic transition frequency. As a result, the atom absorbs more \(\sigma^{-}\) photons and experiences a scattering force along the z-axis. Similarly, in the region where z < 0, the atom also receives \(\sigma^{+}\) light closer to the atomic transition frequency, leading to increased \(\sigma^{+}\) photon absorption and a scattering force along the z-axis. The scattering force on an atom is influenced not only by its velocity but also by its position in space. As the atom moves closer to the center, the detuning value increases gradually, leading to a gradual decrease in the scattering force. This allows atoms to be trapped in the central region.  
From an energy conversion standpoint, when an atom in motion is exposed to a counterpropagating laser, the perceived frequency of the laser by the atom will be slightly higher than its actual frequency \(\omega\). Upon reaching the transition frequency, the atom will undergo excited absorption, leading to a transition to the excited state upon absorbing the counterpropagating photon. However, as the atom returns to its ground state through spontaneous radiation from the excited state, the momentum of the emitted photon will not exhibit a specific direction. Through numerous iterations, the average momentum of the spontaneous radiation on the atom converges to zero, resulting in a scenario in which the atom consistently absorbs the momentum of the counterpropagating photon, leading to a continuous decrease in its velocity. By expanding this principle into three-dimensional space, a stable MOT can be established.  
Figure 2: Schematic diagram of a one-dimensional MOT.  
The preceding analysis provides a qualitative depiction of the operational mechanism of the MOT. However, the distribution of energy levels in atoms and the spatial arrangement of magnetic and light fields are intricate, necessitating the consideration of additional factors [41; 42; 43; 44]. For example, in the context of the spontaneous radiation process of atoms, if the atom's ground state comprises multiple sublevels, subsequent to spontaneous radiation, the atoms may revert to different sublevels, impeding their involvement in the subsequent stimulated absorption. In such cases, it is necessary to add a pump laser to sustain the cooling process.