Charged Polarons and Molecules in a Bose-Einstein Condensate

Ultracold hybrid ion-atom gases represent an exciting frontier for quantum simulation offering a new set of functionalities and control. Here, we study a mobile ion immersed in a Bose-Einstein condensate and show that the long-range nature of the ion-atom interaction gives rise to an intricate interplay between few- and many-body physics. This leads to the existence of several polaronic and molecular states due to the binding of an increasing number of bosons to the ion, which is well beyond what can be described by a short-range pseudopotential. We use a complementary set of techniques including a variational ansatz and field theory to describe this rich physics and calculate the full spectral response of the ion. It follows from thermodynamic arguments that the ion-atom interaction leads to a mesoscopic dressing cloud of the polarons, and a simplified model demonstrates that the spectral weight of the molecules scale with increasing powers of the density. We finally calculate the quantum dynamics of the ion after a quench experiment.

Figure 1

Atom-ion $s$-wave scattering length $a$ as a function of $b$. Inset: atom-ion potential for $b/r_{\mathrm{ion}}=0.3$ and 0.35.

Figure 2

Zero momentum ion spectral function $A(\omega )$ as a function of the potential parameter $b$ and the corresponding scattering length $a$ for $n_0{r}_{\mathrm{ion}}^3=1$ (top) and $n_0{r}_{\mathrm{ion}}^3=0.01$ (middle). The black line is the mean-field energy, the red line is the ladder approximation for the attractive polaron present for $b/r_{\mathrm{ion}}\gtrsim 0.5$, the red dashed line the repulsive polaron present for $b/r_{\mathrm{ion}}\lesssim 0.34$, and the white lines are the molecular states obtained from the Bethe-Salpeter equation. The stars $\star$ signify the breakdown of the Bogoliubov approximation. Bottom: number of atoms in the dressing cloud with the same color coding as the top panel, except the white molecular lines, which are represented as green lines. In the top panel, we take the logarithm to the spectral function, which visually makes the width appear larger. Middle panel is in linear scale.

Figure 3

Left: $|S(t)|$ is shown for $b/r_{\mathrm{ion}}=2$ (orange) and $b/r_{\mathrm{ion}}=0.5$ (blue). Right: $|S(t)|$ for $b/r_{\mathrm{ion}}=0.3$ (orange) and $b/r_{\mathrm{ion}}=0.25$ (blue). We take $n_0{r}_{\mathrm{ion}}^3=1$.