## 3 Monte Carlo Simulation of Behaviors of Ensemble Atoms Associated with Spontaneous Emission  
We only consider a single atom only driven by the mean force in the modeling above. More rigorous modeling should calculate the collective behaviors of an ensemble consisting of a large number of atoms which is characterized by an initial velocity distribution and an initial position distribution. On the other hand, the random force must be considered to predict its temperature which is mainly determined by the momentum diffusion induced by spontaneous emission. The following modeling thereby takes the spontaneous emission into account. We consider the random process driven by spontaneous emissions as a Weiner process, meaning the momentums of the atoms in red MOT evolve as Brownian motions. With these considerations, the behaviors of the atom ensemble are simulated with a set of stochastic differential equations as:  
\[mdv_{z,i}(t) = F_{a,z,i}[z_{i}(t),v_{z,i}(t)]dt+\sigma[z_{i}(t),v_{z,i}(t)]dw_{i }(t), \tag{5}\] \[v_{z,i}(t)dt = dz_{i}(t), \tag{6}\]  
where the subscript \(i\) denotes the \(i\)th numbered atom, \(\sigma^{2}[z_{i}(t),v_{z,i}(t)]=D[z_{i}(t),v_{z,i}(t)]\) represents the momentum diffusion coefficient, and \(w_{i}(t)\) is associated with the Weiner process. The momentum diffusion originates from the fluctuations during the many elementary absorption-spontaneous-emission cycles, leading to a diffusion coefficient given by:  
\[D[z_{i}(t),v_{z,i}(t)]=\frac{1}{2}\hbar^{2}k^{2}\gamma[z_{i}(t),v_{z,i}(t)], \tag{7}\]  
where  
\[\gamma[z_{i}(t),v_{z,i}(t)]=\frac{F_{z,-}[z_{i}(t),v_{z,i}(t)]+F_{z,+}[z_{i}( t),v_{z,i}(t)]}{\hbar k/2} \tag{8}\]  
is equal to the number of fluorescence cycles per second. To simulate the ensemble of the atom, the initial position \(z_{i}(0)\) and initial velocity \(v_{z,i}(0)\) are randomized. Since the narrow-line red MOT is the small part centered at the broadband red MOT, we thus approximate the distribution of the \(z_{i}(0)\) as an even distribution. The initial velocity \(v_{z,i}(0)\) is  
Figure 2: Numerical simulation of the motion of a single \({}^{88}\)Sr atom in the vertical direction driven by mean force in the single-frequency, red MOT. Blue curves correspond to the case on the ground, and orange curves to the case in a \(\mu\)-gE (microgravity environments). (**a,c**) show the mean force of a rest atom depending on location. (**b,d**) show the evolution of the atom position. (**a**) and (**b**) show laser detuning of \(\Delta=-2\pi\times 1.8\) MHz, while (**c**) and (**d**) show \(\Delta=-2\pi\times 200\) kHz. supposed to be a normal distribution with a mean velocity of zero and a variance of \(k_{B}T_{0}/m\) where \(T_{0}\) is the temperature at the end of broadband red MOT. The interaction between the atoms has been neglected, so one run of an experiment with \(N_{0}\) atoms is equivalent to \(N_{0}\) runs of an experiment with a single atom. By doing \(N_{0}\) runs of independent trials of solving the above stochastic differential equations, we can obtain the collective behaviors of \(N_{0}\) atoms. Figure 3 shows a simulation of ensemble atoms. Parameters used in this simulation are \(N_{0}=1000\), \(T_{0}=12\)\(\mu\)K, while other parameters are the same as the simulation shown in Figure 2. In this simulation, the number of atoms left inside the MOT till the time of 100 ms is 771 on the ground and 1000 in \(\mu\)-gEs, indicating some of atoms escaped out of the MOT. Since we are interested more in the atoms' behavior inside the MOT, the results shown in Figure 3 only display the data of atoms left inside the MOT. Comparing the traces of the moving atoms, as shown in Figure 3(a1,b1), one can find that the atoms inside MOT in a \(\mu\)-gE are flying in sinusoid-like trajectories, while the atoms inside the MOT in the ground segment fly rapidly to the bottom. This behavior leads to a striking difference in the atom distribution, as can be visualized in Figure 3(a2,b2), which show the atom distribution at the time of 60 ms.