Universality in Ionic Three-body Systems Near an Ion-atom Feshbach Resonance

TL;DR

The study addresses universality in ionic three-body systems near an ion-atom Feshbach resonance, focusing on a Li-Li-Ba$^+$ mixture. It employs a fully quantum hyperspherical framework with pairwise Lennard-Jones and polarization potentials to compute bound and scattering properties, emphasizing the long-range $1/r^4$ ion-atom interaction. The results reveal a distinct universality class for ionic systems: the three-body recombination rate $L_3$ is strongly suppressed (by about a factor of $250$) compared with neutral counterparts, and Efimov ground-state lifetimes in LiLiBa$^+$ can reach up to $\sim 10^{-1}$ s, five orders of magnitude longer than in the neutral case, due to weaker nonadiabatic couplings. The product-state distribution follows a universal $1/E_b$ propensity, and the spectrum of weakly bound triatomic molecular ions is notably dense, reflecting the influence of long-range interactions. These findings underscore that ion-atom-atom systems inhabit a different low-energy universality class from neutral atoms and motivate generalized effective-range treatments to describe their few- and many-body dynamics.

Abstract

We calculate bound and scattering properties of a system of two neutral atoms and an ion near an atom-ion Feshbach resonance. Our results indicate that long-range atom-ion interactions lead to significant deviations from universal behavior derived from contact or van der Waals potentials. We find that ionic systems display an overall suppression of inelastic transitions leading to recombination rates and lifetimes of Efimov state orders of magnitude smaller with respect to those for neutral atoms. We further characterize the dense spectra of triatomic molecular ions with extended lifetimes. Our results provide a deeper insight on the universality and structure of three-body ionic systems and establishing them as a promising platform for exploring novel few- and many-body phenomena with long-range interactions.

Figures (4)

• Figure 1: The three-body hyperspherical potentials $U_\nu(R)$ for the LiLiBa$^+$ (black) and LiLiBa (red) systems calculated at $a_{BX} = 0.1 \ (r_* \ \text{or} \ r_\text{vdW})$ for our interaction model supporting six $BX$$s$-wave bound states and two $BB$$s$-wave bound states. For large $R$ ($R/r_{\rm vdW} \gg 1$ or $R/r_* \gg 1$), potentials $U_\nu(R) > 0$ correspond to three-body continuum channels, describing collisions between three free atoms, while potentials $U_\nu(R) \simeq -E_b(v,l) < 0$ are atom-molecule channels describing collisions between an atom and a molecule. Here, $E_b(v,l)$ denote the diatomic molecular binding energies of the rovibrational states of the $BX$ and $BB$ interactions. According to the number of the atom-dimer channels in the figure and considering values of $E_{\rm vdW}$ and $E_{*}$ in absolute units, we estimate that the density of diatomic states for the ionic systems to be a 100 times larger than for the neutral counterpart.
• Figure 2: (a), (b): Three-body recombination rate $L_3$ and partial rates $L_3^{\rm w}$ and $L_3^{\rm d}$ for LiLiBa (red) and LiLiBa$^+$ (black). The results for LiLiBa$^+$ recombination are multiplied by $(r_\text{vdW}/r_*)^4$ in order to properly compare that with recombination of LiLiBa systems. For small values of $|a_{BX}|\lesssim r_*$ (or $r_{\rm vdW}$), $L_3$ display resonant effects associated with high-partial waves diatomic molecular states wang2012pra, some of which are not resolved in the figure. Dotted lines represent the amplitude for $L_3$, $A(L_3)$, from the universal theory Helfrich-Hammer:2010. The insets of panels (a) and (b) display the product state distribution of LiLiBa$^+$ recombination, $L_{3f}/L_3$ [see Eq. (\ref{['L3']})], in terms of the molecular final state binding energy, $E_b$, displaying the $1/E_b$ propensity rule of Ref. Haze2023. (c) Energy of lowest Efimov state for LiLiBa (red) and LiLiBa$^+$ (black) and corresponding width $\Gamma$ expressed as error bars. Two additional trimer states associated with two-body rotational states (pink) wang2012pra are also presented, together with their energies and widths. The inset in (c) shows the lifetime $\tau=\hbar/\Gamma$ of the Efimov states for both LiLiBa and LiLiBa$^+$.
• Figure 3: The time delay for $^7\text{Li}^7\text{Li}^{138}\text{Ba}$ (red) and $^7\text{Li}^7\text{Li}^{138}\text{Ba}^+$ (black) calculated for $a_{BX} = 0.1 \ (r_* \ \text{or} \ r_\text{vdW})$. The results for the neutral system have been multiplied by a factor of 10. The vertical dashed lines are the energies of two-body molecular states to which the effective potentials $W_\nu(R)$ converge at large values of $R$. The peaks and width of the time-delay parameter desrcibe the energies and lifetimes of the three-body bound states supported by $W_\nu(R)$.
• Figure 4: The (generalized) effective ranges for the ion-atom (black solid line) and neutral (red solid line) two-body interactions. The coefficient $\sim$${k}$ obtained for the ionic expansion is marked as $c_1^*$ (blue solid line). Dashed lines represent the analytical expressions given by Eq. (34) of Ref. Chin:2010 for the neutral interaction (red), the expression for $c_1^*(a)$ (blue), and the expression given by Eq. \ref{['eq:coeffk2']}, in which we set $r_{\rm eff}^* = 0$ (black) -- for the ion-atom interaction. The gray dot-dashed line shows the generalized ionic effective range plus the estimated logarithmic term at $k = 0.003 \ r_{*}^{-1}$.