Electron Bubbles in Superfluid $^3$He-A: Exploring the Quasiparticle-Ion Interaction
When an electron is forced into liquid $^3$He it forms an "electron bubble", a heavy ion with radius, $R\simeq 1.5$ nm, and mass, $M\simeq 100\,m_3$, where $m_3$ is the mass of a $^3$He atom. These negative ions have proven to be powerful local probes of the physical properties of the host quantum fluid, especially the excitation spectra of the superfluid phases. We recently developed a theory for Bogoliubov quasiparticles scattering off electron bubbles embedded in a chiral superfluid that provides a detailed understanding of the spectrum of Weyl Fermions bound to the negative ion, as well as a theory for the forces on moving electron bubbles in superfluid $^3$He-A (Shevtsov et al. in arXiv:1606.06240). This theory is shown to provide quantitative agreement with measurements reported by the RIKEN group [Ikegami et al., Science 341:59, 2013] for the drag force and anomalous Hall effect of moving electron bubbles in superfluid $^3$He-A. In this report, we discuss the sensitivity of the forces on the moving ion to the effective interaction between normal-state quasiparticles and the ion. We consider models for the quasiparticle-ion (QP-ion) interaction, including the hard-sphere potential, constrained random-phase-shifts, and interactions with short-range repulsion and intermediate range attraction. Our results show that the transverse force responsible for the anomalous Hall effect is particularly sensitive to the structure of the QP-ion potential, and that strong short-range repulsion, captured by the hard-sphere potential, provides an accurate model for computing the forces acting on the moving electron bubble in superfluid $^3$He-A.