Trap-induced atom-ion complexes: a time-independent approach

TL;DR

This work addresses chaotic scattering in a trapped ion–atom system by replacing the ion with a charge distribution given by its ground-state wavefunction, enabling a fully time-independent treatment of ion–atom dynamics. The authors map the time-dependent problem to a static-charge model, compute the interaction via $V(r,\theta)=\frac{C_8}{r^8}-\frac{1}{2}\alpha|E(r,\theta)|^2$, and analyze delay times $t_d$ across trap configurations, atomic species, and collision energies. They show that chaotic scattering in this system is hyperbolic, evidenced by an exponential decay of complex lifetimes, and that the onset of chaos correlates with the anisotropy of the interaction, modulated by the trap frequencies and short-range parameter $C_8$. A key finding is that the probability of atom–ion complex formation, quantified as the uncertainty fraction $f(\epsilon^*)$, serves as an experimentally accessible metric of chaos and links a measurable observable to the chaotic dynamical structure. The results suggest that atomic polarizability dominates over mass in governing dynamics and open routes to controlled heating or stability in ion–atom hybrid platforms.

Abstract

A trapped ion immersed in a neutral bath shows long-lived atom-ion complexes that significantly alter its chemical properties, and, thus the ion stability. In this work, we present a general study of trapped ion-atom scattering with the ion modeled as a charge distribution defined by the spatial extent of its ground-state wavefunction. After mapping the time-dependent problem onto a time-independent framework, we investigate the role of the trap, the atomic species, atom-ion interaction, and collision energy in shaping the chaotic dynamics of the system. We find that the probability of atom-ion complex formation directly measures its chaoticity. Therefore, our results establish a clear relationship between the emergence of chaotic scattering and the presence of ion-atom complexes.

Figures (10)

• Figure 1: Trapped ion-atom interaction and scattering. Panel (a) displays a schematic representation of a trapped ion colliding with an atom. Panel (b) shows two distinct atomic trajectories for very similar values of the initial angle solved for the potential including the first two anisotropic terms. The zoom-in shows the atom-ion complex lifetime. Panel (c) displays the trapped ion-atom interaction potential. The different colors denote the three main terms of the potential expansion following Eq. (\ref{['potential_expansion']}) for $\omega_z$ = 1 MHz and $\omega_\rho$ = 50 kHz (solid lines), and for $\omega_z$ = 350 kHz and $\omega_\rho$ = 50 kHz (dashed lines).
• Figure 2: Delay time $t_d$ for trapped Ba$^+$-Li scattering as a function of initial angle $\theta_0$ and the ratio $V_{min}/E_0$ at $\omega_z$ = 350 kHz and $\omega_\rho$ = 50 kHz. The delay time $t_d$ is normalized against its smallest value $t_E$ at a given $E_0$ and its values are color-coded as demonstrated by the legends included in panel (c). Panel (a) has $C_8 = 10^4$ a.u., panel (b) corresponds to $C_8 = 10^5$ a.u., and $C_8 = 10^6$ a.u. for panel (c). Inside each panel, the subplot on the top solves $t_d$ with the potential only expanded to $l = 2$, whereas the one on the bottom considers the $l=4$ term as well. The horizontal red lines cut through the parameter space at $V_{min}/E_0 = 20, 40, 150, 2000$, corresponding to values for regular scattering, the onset of chaos, and deeply chaotic. A detailed look at the cuts is presented in Fig. \ref{['fig:cuts_350k50k']}.
• Figure 3: Uncertainty fraction $f(\epsilon^*)$ as a function of $V_{\text{min}}/E_0$ for a Li scattering off a Ba$^+$ inside an ion trap that has $\omega_z$ = 350 kHz and $\omega_\rho$ = 50 kHz for different values of $C_8$ (in a.u.). Panel (a) is calculated when only $V_0(r)$ and $V_2(r)$ are included, whereas panel (b) also takes into account $V_4(r)$.
• Figure 4: Delay time $t_d$ for trapped Ba$^+$-Li scattering as a function of incoming angle $\theta_0$ for $C_8 =$ (a) $10^4$ a.u., (b) $10^5$ a.u., (c) $10^6$ a.u. and $V_{\text{min}}/E_0 =$ (I) 20, (II) 40, (III) 150, (IV) 2000 at $\omega_z$ = 350 kHz and $\omega_\rho$ = 50 kH The color red is for $l= 2$ and the blue dots are for the interaction potential including terms up to $l = 4$. The uncertainty fraction $f(\epsilon)$ at $\epsilon^* = 10^{-5} \times \pi/2$ is then calculated. Their values are shown in Table \ref{['tab:uncertainty_frac_350k50k']}.
• Figure 5: Delay time $t_d$ as a function of initial angle $\theta_0$ and the ratio $V_{\text{min}}/E_0$ for a H [panel (a)] or a Rb [panel (b)] scattering off a Ba$^+$ inside a Paul trap of frequencies $\omega_z$ = 350 kHz and $\omega_\rho$ = 50 kHz. The first two anisotropic terms are included in the potential to produce this figure and $C_8=10^5$ a.u..