# Thermalization of a Trapped Single Atom with an Atomic Thermal Bath
## 3 Results
For this figure panel, there are no datapoints for radial direction below \(T_{A}=1\)\(\upmu\)k as the probability to be in \(n_{x}=n_{y}=0\) state is practically 100% and, hence, further cooling is not possible. Figure 2c shows the variation of the thermalization time with respect to the bath temperature, where \(T_{A}=10\)\(\upmu\)k for all points. Here we find that, as the bath temperature is reduced, the thermalization time required for the radial direction increases. This is because, according to Equation (7), the time between collisions increases when the bath temperature decreases. The axial direction does not show such a simple behavior, the thermalization time remains initially fairly constant until \(T_{B}\simeq 1\)\(\upmu\)k, when it has a minimum, and then increases below such temperature. This is because although the time between collisions increases, the number of collisions required to cool the single atom reduces as \(p_{m\to n}/p_{n\to m}\) increases, where \(\mathbf{n}<\mathbf{m}\). The above results tell us that (as expected) the thermalization of the axial direction is by far the limiting factor in all the conditions we have explored. Additionally, we have found that in order to effectively cool the trapped atom, there exists an optimal bath temperature that minimizes the axial thermalization time and therefore could be used to speed up the entire thermalization process. In Figure 3 we show the results of a Monte Carlo simulation where the trapped atom is immersed in a bath whose temperature is decreasing due to evaporative cooling. In evaporative cooling, the temperature, the number of atoms, and the trapping frequency of the bath atoms reduce continuously. As we need to calculate \(p_{n\to m}\) for each \(T_{B}\), we consider the change in these parameters in step-wise manner. The figure shows how the average vibrational state of the trapped atom changes as we move along the evaporation ramp. The length of the ramp has been chosen to ensure \(T_{A}\simeq T_{B}\) at every time. An alternative, faster approach is shown in Figure 4a,b. In this case, the radial vibration levels are able to follow the bath temperature because of the fast thermalization with the bath. The axial direction is always lagging behind as the time taken for the thermalization of the axial direction is higher compared to timescales of evaporation. In order for the axial direction to catch up, we then add a waiting time. This allows enough time for all the degrees of freedom of the trapped atom to thermalize with the bath atoms and reach \(T_{A}=T_{B}\). Notably, with this second method, we are able to sensibly reduce the evaporation time compared to the first ramp. To further speed up the thermalization, i.e., to increase the axial direction's thermalization rate, one could either increase the density of the bath or increase the scattering length. Both of these strategies also lead to an increase in the harmful three-body collision rate, as shown in Figure 5a,b. We convert the three-body loss rate to total probability of loss during the evaporation of the bath for our case, which is shown in Figure 5a. The three-body loss rate here is calculated as \(\tau_{loss}=4\pi\hbar a^{4}n_{B}^{2}/m_{B}\)[15]. In Figure 5b, we show the effect of change in scattering length to \(\tau\) and the loss probability, \(1-e^{-\tau/\tau_{loss}}\), due to three-body collisions during thermalization time. Figure 4: Trapped atom interacting with a bath that is undergoing fast evaporative cooling for three seconds. (**a**) Radial and axial mean vibrational occupation numbers are shown by red solid and blue dashed lines, respectively. Here, the final occupation numbers are \(\langle n_{r}^{\prime}\rangle=0.00\) and \(\langle n_{z}^{\prime}\rangle=0.23\), meaning that the atom always ends up in the ground state of the radial direction. (**b**) The temperature and the atomic density of the bath atoms are shown by red solid and blue dashed lines, respectively.