Chaotic scattering and heating in cold ion-atom collisions: two sides of the same coin

Abstract

We study the classical dynamics of a Paul-trapped ion in a low-density bath of atoms above 1 $μ\textrm{K}$. We find that lower energy collisions with more massive atoms, especially at energies less than the initial micromotion heating, are more likely to form atom-ion complexes. These complexes evolve in a fractal structure for every scattering observable, showing non-hyperbolic chaotic dynamics. To explore the chaotic dynamics, we use a GPU-accelerated methodology allowing us to run over $3\times 10^{8}$ trajectories of a $^{174}$Yb$^+$ and different atoms. As a result, after analyzing the dynamics as a function of the atom species, collision energy, trap parameters, and ion-atom potential depth, we find a link between heating and the onset of chaos in the first atom-ion interaction that occurs when a low-density atomic bath is merged with a trapped ion.

Figures (4)

• Figure 1: Chaotic structure of observables for head-on collisions of Yb$^+$Rb, for $D_e=1$K and $E=1.5\mu$K and $r_0=\textrm{5,000}a_0$, as a function of incoming atom's spherical angles $\theta$ and $\phi$. Panel (a) shows a schematic of the setup: the ion is placed at the center of the trap, and the atom in the surface of the octant of the sphere. The dashed line represents an ion-atom trajectory showing the formation of a complex. The heatmap of the scattering angle is projected on the surface of the sphere. Panel (b) shows the scattering angle, defined as $\arccos\frac{\vec{v}_i\cdot\vec{v}_f}{v_iv_f}$, where $\vec{v}_i$ and $\vec{v}_f$ are the initial and final atom velocities. Panel (c) shows the logarithm of the number of bounces in the atom-ion trajectory. Panel (d) shows the momentum transfer of the trajectory, defined as $q=|\vec{p}_\text{f}-\vec{p}_\text{i}|$, where $\vec{p}_\text{i}$ and $\vec{p}_\text{f}$ are the initial and final momentum of the atom. Due to outliers, $\textrm{min}(q,0.015\textrm{a.u.})$ is shown.
• Figure 2: Fraction of complexes still alive after time $t$, for head-on collisions of different atoms with Yb$^+$ for $D_e=1$K and $E=1.5\mu$K and $r_0=\textrm{5,000}a_0$.
• Figure 3: Probability of complex formation, as a function of bath temperature and atomic species, from thermal runs. The darker and lighter colors correspond to potential depths $D_e=100$K, $1$K respectively. Each bar represents the number of trajectories in which the atom and ion formed a complex, out of the $10^7$ computed trajectories. The ion starts near the center of the trap, and the atom starts from a uniformly chosen position on the surface of a sphere of radius 5,000$a_0$ centered at the trap center, with Cartesian velocity components $v_x,v_y,v_z$ sampled from a Gaussian distribution for the given bath temperature, negated if needed to enforce $v_r<0$.
• Figure 4: Probability of complex formation as a function of collision energy, for different atomic species. The four main curves show the fraction of collisions which formed a complex, for 10,000 head-on collisions of atoms with Yb$^+$, each for 100 different fixed collision energies, with $r_0=\textrm{20,000}a_0$ and $D_e=1$K. The simulation's initial rf phase $\Omega_\textrm{rf}t_0$ was uniformly randomly chosen from $[0,2\pi]$. The black vertical line is the lowest temperature used previously in the paper, $1\mu$K, corresponding to a collision energy of $1.5\mu$K via $\frac{1}{2}m_av_0^2=\frac{3}{2}k_BT$. The colored vertical lines are $W_0^\textrm{3D}$, the heating energy scale from Cetina2012. The inset shows the probability of complex formation versus collision energy for Yb$^+$Li, for four $q_x$ values, keeping $\Omega_\textrm{rf}$ fixed.