Unraveling the Structure of Ultracold Mesoscopic Collinear Molecular Ions

We present an in-depth many-body investigation of the so-called mesoscopic molecular ions that can buildup when an ion is immersed into an atomic Bose-Einstein condensate in one dimension. To this end, we employ the multilayer multiconfiguration time-dependent Hartree method for mixtures of ultracold bosonic species for solving the underlying many-body Schrödinger equation. This enables us to unravel the actual structure of such massive charged molecules from a microscopic perspective. Laying out their phase diagram with respect to atom number and interatomic interaction strength, we determine the maximal number of atoms bound to the ion and reveal spatial densities and molecular properties. Interestingly, we observe a strong interaction-induced localization, especially for the ion, that we explain by the generation of a large effective mass, similarly to ions in liquid Helium. Finally, we predict the dynamical response of the ion to small perturbations. Our results provide clear evidence for the importance of quantum correlations, as we demonstrate by benchmarking them with wave function ansatz classes employed in the literature.

Figure 1

Setup and atom-ion interaction. (Left) Atoms (blue) and ions (red) in a quasi one-dimensional harmonic trap. (Right) Bare and regularized [26] atom-ion interaction potential (solid and dashed-dotted black lines) together with the two most weakly bound states and their energies ${\epsilon }_i$.

Figure 2

Phase diagram. Chemical potential $\mu$ from the Gross ansatz as a function of $N$ and $g$. Black circle (crosses) mark $\mu =0$ from ML-MCTDHB (Gross). The black dashed line presents the estimation $g_c\approx (\omega -{\epsilon }_1)/(N_c-1)$. (Inset) Total energy $E(N)$ as a function of $N$ for $g=3E^{*}{R}^{*}$.

Figure 3

Molecular structure. [(a)–(c)] Atomic (shaded) and ionic (solid line) density profiles. In (c) also a Thomas-Fermi profile with $N-N_c$ particles is shown (dashed line). (d) Atom-ion correlation function $g_2(z)$ for $g=3R^{*}{E}^{*}$. (e) Population of the bound states $f_j/N$. (i) Delocalized versus (ii) localized ion and its impact on the atomic density.

Figure 4

Self-localization. Variance of the atomic (blue) and the ionic (red) variance normalized by the noninteracting variance ${\sigma }_0$. Ionic variance from the Gross (dashed) and the mean-field (dashed dotted) ansatz for $g=0$ are shown, too. The dark (light) thick lines represent ${\sigma }_I$ (${\sigma }_A$) solely including the increase of $M$ [26]. The dark (light) gray area represents the spatial extent of the bound states (${\sigma }_0$).

Figure 5

Effective ion behavior. (a) Effective mass $m^{*}/m$ and (b) effective trapping frequency ${\omega }^{*}/\omega$. Note that the Gross ansatz gives ${\omega }_G=\omega \sqrt{1+N}$ and $m_G=m$.