## 2 Single-Atom Magnetic Optical Trap  
Ideally, the lifetime of single-atom in the optical tweezer is identical to the lifetime of the atom state, which means the lifetime for a ground-state atom can be extremely long. The trapping life of the ground state atoms in the optical tweezers is limited due to the influence of background atoms collision, the parametric heating of the optical tweezers and the residual thermal motion of the atoms. The individual atom trapped in the optical tweezer will collide with the free atoms in the Cs gas chamber. In this process the single-atom gains kinetic energy, heats up and eventually escapes from the optical tweezer, resulting in the reduction of the atomic trapping lifetime. In the experiment, we can reduce the collision probability of induced heating by increasing the background vacuum of the cesium atom gas chamber and reducing the number of background atoms. This can extend the trapping lifetime to some extent. Since the probe of the thermal ionization gauge in the vacuum system is far away from the cesium atom gas chamber, the measured vacuum may be different from the actual vacuum degree in the Cs atomic vapor cell. By measuring the typical time of the MOT, we can know the pressure level in the Cs atomic gas chamber more accurately. In our scheme, due to MOT capturing the atoms in the chamber, it is impossible to isolate trapped atoms from background atoms. Thus, we measure and fit the loading curve instead of using release and recapture to get the typical time of MOT [17]. The background pressure is related to the atomic density \(N\) in the vapor cell [18]  
\[N=\frac{1}{\tau\sigma}\sqrt{\frac{M}{3k_{B}T}} \tag{1}\]  
where \(\tau\) is the atomic trapping lifetime in the MOT, \(\sigma\) is the atomic collision cross-section of Cs, \(M\) is the mass of Cs atom, \(k_{B}\) is the Boltzmann constant, according to the pressure formula \(p=Nk_{B}T\) the pressure (\(p\)) in the vapor cell is [18]  
\[p=\frac{\sqrt{3 Mk_{B}T}}{3\tau\sigma} \tag{2}\]Figure 1 shows the loading curve of the magneto-optical trap when the cooling laser power of the MOT is 150 \(\upmu\)W, and the magnetic field gradient is 151.41 Gauss/cm. For atom number \(N(t=0)=0\) as initial condition and fit the curve with \(N(t)=N_{\text{S}}\big{(}1-e^{-t/\tau}\big{)}\)[18], the typical time of the MOT is \(5.81\pm 0.14\) s, and the pressure in the chamber is about \(6\times 10^{-7}\) Pa. The temperature of atoms in the MOT is \(\sim\)105 \(\upmu\)K under similar conditions [19]. Adjusting the spatial overlap and temporal overlap of the MOT and the optical tweezer can transfer the single-atom between the traps efficiently. The loading rate of the MOT \(R_{L}\) is sensitive to the axial gradient of the quadrupole magnetic field \(\left(\frac{dB}{dz}\right)\), and \(R_{L}\propto\left(\frac{dB}{dz}\right)^{-\frac{14}{3}}\)[19; 20]. The smaller the axial gradient of the quadrupole magnetic field, the higher the MOT loading rate and the more captured atoms. Conversely, the higher the magnetic field gradient, the lower the MOT loading rate and the fewer atoms trapped. Since the loading rate of the MOT is pretty sensitive to the axial gradient of quadrupole magnetic field, we can use a trigger loop to control the loading rate of the MOT automatically. As shown in Figure 2a, the fluorescent photons are collected into the single-photon counting module (SPCM), and the output pulses of SPCM enter the pulse counter. Then we can set the output voltage of the pulse counter for different circumstances that represents the atom numbers in the MOT. The output voltage of the pulse counter enters the quadrupole magnetic field power supply (Model SM 70-22, DELTA, Holland) as control voltage. The output current of the power supply will change related to the control voltage, and the quadrupole magnetic field gradient and the loading rate of the MOT change, too. Figure 2b and Table 1 shows the working principle of the trigger loop.