# Thermalization of a Trapped Single Atom with an Atomic Thermal Bath
## 2 The System  
The system under study is pictorially represented in Figure 1. A single atom is trapped in an optical tweezer and interacts with a cloud of thermal atoms. Our aim was characterizing how the interaction with the bath leads to the thermalization of the single atom, meaning that it reaches the Gibbs state \(G(T)=1/Z\sum_{n}e^{-E_{n}/k_{B}T}\), with \(Z\) the partition function, \(E_{n}\) the energy of the nth level, \(k_{B}\) the Boltzmann's constant and \(T\) the temperature of the bath. Therefore, in the following, we refer to the the _temperature of the single atom_\(T_{A}\) as the temperature associated with \(G(T_{A})\).  
For a single atom in an optical tweezer, when the trap depth is significantly greater than the temperature of the atom, we can approximate the tweezer as a three-dimensional harmonic trap. Since optical tweezers are produced with a single beam, they are characterized by a strong anisotropy, with the axial direction of propagation of the beam providing a weaker confinement than the radial directions. In addition, we consider the bath as made of thermal bosonic atoms. After a collision with the bath atom, the trapped atom can undergo a change in the vibrational level it occupies. Which level it goes to depends on the overlap of the initial and the final wavefunctions. If the atom changes its vibrational state from \(\mathbf{n}=(n_{x},n_{y},n_{z})\) to \(\mathbf{m}=(m_{x},m_{y},m_{z})\), the relative rate is given by [11; 12],  
\[\gamma_{\mathbf{n},\mathbf{m}}=C\int dt\int d\vec{k}a\vec{k}^{\prime}n(\vec{k})(n(\vec {k}^{\prime})+1)|T(\mathbf{n},\mathbf{m},k,k^{\prime})|^{2}e^{\left(\frac{\mathbf{k}^{2}}{ 2\mathbf{m}_{B}}-\frac{\mathbf{k}^{\prime 2}}{2\mathbf{m}_{B}}+a\hbar\omega\right)t}. \tag{1}\]  
where \(C\) is a constant common to all the rates, \(\vec{k}\) and \(\vec{k}^{\prime}\) are the wavevectors of the bath atom before and the after the collision, \(m_{B}\) is the mass of the bath atom, \(\omega\) is the tweezer radial trapping frequency, \(\alpha=(n_{x}-m_{x})+\eta_{y}(n_{y}-m_{y})+\eta_{z}(n_{z}-m_{z})\), \(\eta_{y}/\eta_{z}\) is the aspect ratio of the trap, \(n(\vec{k})\) is the number of bath atoms with wave-vector \(\vec{k}\), and \(T(\mathbf{n},\mathbf{m},k,k^{\prime})\) is the overlap between the initial and the final wavefunctions of both the trapped atom and the bath atom. In Equation (1), it is assumed that the bath atom is a free particle during the collision. The above integrals for an isotropic trap were solved in References [11; 12]. Including the trap anisotropy, we find:  
\[\begin{split}\gamma_{\mathbf{n},\mathbf{m}}=&\tilde{C}e^{ 1/2\pi^{\prime}\delta}\sum_{l_{x}=0,K_{x}=0}^{p_{x}}\sum_{l_{y}=0,K_{y}=0}^{p_{ y}}\sum_{l_{z}=0,K_{z}=0}^{p_{z}}\left[\prod_{j=x,y,z}C_{m_{j},m_{j},l_{j  
\(K_{i}(x)\) denotes the modified Bessel function and \(i\) is a positive integer. Once we calculate the above rates, at each collision, the probability to jump from state \(n\) to \(m\) can be calculated as  
\[p_{n\to m}=\frac{\gamma_{n,m}}{\sum_{w}\gamma_{n,w}}. \tag{6}\]  
In the above calculations, we assume that the trapped atom interacts with a uniform density bath. This is a valid assumption for systems where the trapped atom is confined to a smaller volume compared to the trap of the bath atoms.  
To account for the finite trap depth, we restrict our calculations to a suitably chosen \((n_{x}^{\rm max},n_{y}^{\rm max},n_{z}^{\rm max})\) such that \(n_{x}^{\rm max},\eta_{y}n_{y}^{\rm max},\eta_{z}n_{z}^{\rm max}\gg k_{B}T_{A} /(\hbar\omega)\). Here, \(T_{A}\) is the initial temperature of the single atom. For the simulations in this article, we first calculate the \(p_{n\to m}\) for all the level combinations for each bath temperature \(T_{B}\). The second step is to perform Monte Carlo simulations where the atom jumps between the levels according to the probabilities \(p_{n\to m}\).