# Thermalization of a Trapped Single Atom with an Atomic Thermal Bath
## 3 Results  
We first explore how the rate of thermalization for the trapped atom changes with respect to various parameters. Second, we simulate what happens when the trapped atom is interacting with a bath undergoing evaporative cooling. To illustrate a specific case, in our simulations we use \({}^{87}\) Rb and \({}^{41}\) K as the bath atoms and the single atom, respectively, similar to what was done in [9]. We are interested in the rate at which the trapped atom thermalizes with the bath atoms. We ran the Monte Carlo simulations to know how many _elastic_ collisions it takes for the radial and axial vibrational states of the trapped atom to thermalize with the bath. To calculate the thermalization time constant, \(\tau\), we multiplied the number of collisions with the time between collisions, which is given by  
\[\tau_{\rm col}=1/(4\pi a^{2}n_{B}\varrho), \tag{7}\]  
where \(n_{B}\) is the density of bath atoms, \(a\) is the heteronuclear s-wave scattering length between the bath and the trapped atom, \(\tilde{v}=\sqrt{8k_{B}T_{B}/(\tau\mu)}\) is the mean velocity at bath temperature (\(T_{B}\)), and \(\mu\) is the reduced mass between the bath atom and the single atom. For the simulations, we consider bath temperatures in the range 0.1 \(\mu\)K-10 \(\mu\)K, and the atomic density in the bath to be \(\sim\)10\({}^{12}\) cm\({}^{-3}\). These parameters can be produced in current experimental setups [13]. For the interspecies scattering length that sets the strength of the interaction between the trapped atom and the bath, we have chosen \(a=160a_{0}\), which can be achieved exploiting one of the KRb heteronuclear Feshbach resonances [14]. Unless specified otherwise, we have set \(\omega=100\) kHz, \(\eta_{y}=0.98\) and \(\eta_{z}=0.105\), which are  
Figure 1: Pictorial representation (not to scale) of the system under study: a single atom (red sphere) is trapped in an optical tweezer (yellow beam) and is immersed in a bath of atoms of a different species (blue cloud). realistic values for tweezers experiments [10]. As \(\omega_{x}\simeq\omega_{y}\), from here, we refer to either of them as the radial frequency \(\omega_{r}\) and we identify the radial direction with both \(x\) and \(y\). The results of our simulations for various parameters are shown in Figure 2, where we report mean values and standard deviations of the mean of \(\tau\) obtained with 6 repetitions of 5000 trajectories each. As expected, from Figure 2a we can note that, because the trapping frequency is low in the axial direction (denoted by \(z\)), the time required for thermalization is higher than in the radial direction (denoted by \(r\)). The thermalization time does not show a strong dependence on the trapping frequency within the parameters of interest because the conditions \(\hbar\omega\sim k_{B}T_{B}\) and \(\hbar\eta_{z}\omega\ll k_{B}T_{B}\) do not change substantially for the radial and  
Figure 2: The variation of thermalization time constants of radial and axial vibrational levels of the trapped atom with respect to (**a**) the trapping frequency, (**b**) the starting temperature of the trapped atom, and (**c**) temperature of the bath. The red circles are for radial direction and the blue triangles are for the axial direction. Here \(a=160a_{0}\), \(n_{B}=7.7\times 10^{12}\) cm\({}^{-3}\), \(\omega/(2\pi)=100\) kHz for (**b**,**c**), \(T_{B}=4\)\(\mu\)k, and \(T_{B}=0.1\)\(\mu\)k for (**a**,**b**) respectively. \(T_{A}=8\)\(\mu\)k and \(T_{A}=10\)\(\mu\)k for (**a**,**c**) respectively. For all the data points \(\eta_{y}=0.98\) and \(\eta_{z}=0.105\). the axial directions, respectively. In Figure 2b, we keep \(T_{B}\) constant at 0.1 \(\upmu\)k and vary the starting temperature of the trapped atom, \(T_{A}\). Here, \(\tau\) increases for both directions as the initial \(T_{A}\) increases, but the increase is rather contained.