Ion-induced density bubble in a strongly correlated one-dimensional gas

We consider a harmonically trapped Tonks-Girardeau gas of impenetrable bosons in the presence of a single embedded ion, which is assumed to be tightly confined in an RF trap. In an ultracold ion-atom collision the ion’s charge induces an electric dipole moment in the atoms which leads to an attractive $r^{-4}$ potential asymptotically. We treat the ion as a static deformation of the harmonic trap potential and model its short range interaction with the gas in the framework of quantum defect theory. The molecular bound states of the ionic potential are not populated due to the lack of any possible relaxation process in the Tonks-Girardeau regime. Armed with this knowledge we calculate the density profile of the gas in the presence of a central ionic impurity and show that a density bubble of the order of $1\mu$m occurs around the ion for typical experimental parameters. From these exact results we show that an ionic impurity in a Tonks gas can be described using a pseudopotential approximation, allowing for significantly easier treatment.

Figure 1

Schematic showing (a) an positively charged ion embedded in a harmonically trapped Tonks-Girardeau gas and (b) the interaction potential for the ion-atom interaction under the Born-Oppenheimer approximation (dashed). The potential minimumis at $r_{min}$ and the polarization length of the ion is indicated by $R^{*}$. The potential curve goes as $r^{-4}$ at large distances.

Figure 2

The first two trap states of a harmonic trap with a centrally embedded ion for a ${}^{135}\mathrm{Ba}$${}^{+}$-${}^{87}\mathrm{Rb}$ system for axial trapping frequency, $\omega =70$ Hz. The inset for the $n=0$ state shows a zoom in of the region around zero where the wave function is rapidly oscillating. The short-range phases are chosen to be ${\varphi }_e=\frac{\pi }{4}$ and ${\varphi }_o=-\frac{\pi }{4}$.

Figure 3

The single-particle densities of a TG gas of 20 particles in the presence of a central ion for ${}^{135}\mathrm{Ba}$${}^{+}$-${}^{87}\mathrm{Rb}$ and ${}^{174}\mathrm{Yb}$${}^{+}$-${}^{87}\mathrm{Rb}$ systems with a typical trapping frequency, $\omega =70$ Hz (thick black line). The short range phases in each case are chosen to be ${\varphi }_e=\frac{\pi }{4}$ and ${\varphi }_o=-\frac{\pi }{4}$. The thin red line in the plots represents the result of a pseudopotential approximation for the ion, using a large value for the scattering length.