Quasiparticle properties of long-range impurities in a Bose condensate

An impurity immersed in a Bose condensate can form a quasiparticle known as a Bose polaron. When the impurity-boson interaction is short-ranged, the quasiparticle properties can be characterized in terms of the impurity-boson scattering length $a_{\mathrm{IB}}$ and the condensate coherence length $\xi$, a universal description that remains valid irrespective of the bath density $n_0$. Long-ranged interactions -- such as provided by Rydberg or ionic impurities -- introduce an effective interaction range $r_{\mathrm{eff}}$ as the third length scale. These competing length scales raise the question of whether a universal description remains valid across different bath densities. In this study, we discuss the quasiparticle nature of long-range impurities and its dependence on the length scales $n_0^{-1/3}$, $r_\mathrm{eff}$, and $\xi$. We employ two complementary theories -- the coherent state Ansatz and the perturbative Gross-Pitaevskii theory -- which incorporate beyond-Fr\"ohlich interactions. We derive an analytical expression for the beyond-Fr\"ohlich effective mass for a contact interaction and numerically compute the effective mass for long-range impurities. We argue that the coupling parameter $|a_{\mathrm{IB}}|n_0^{1/3}$ remains the principal parameter governing the properties of the polaron. For weak ($|a_\mathrm{IB}|n_0^{1/3}\ll 1$) and intermediate ($|a_\mathrm{IB}|n_0^{1/3}\simeq 1$) values of the coupling parameter, long-range impurities in a Bose condensate are well-described as quasiparticles with a finite quasiparticle weight and a well-defined effective mass. However, the quasiparticle weight becomes significantly suppressed as the effective impurity volume is occupied by an increasing number of bath particles ($r_{\mathrm{eff}}n_0^{1/3} \gg 1$).