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 a₈₁ and the condensate coherence length ξ, a universal description that remains valid irrespective of the bath density n₀. Long-ranged interactions -- such as provided by Rydberg or ionic impurities -- introduce an effective interaction range r₄₅₅ 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 n₀^-1/3, rₑff, and ξ. We employ two complementary theories -- the coherent state Ansatz and the perturbative Gross-Pitaevskii theory -- which incorporate beyond-Fröhlich interactions. We derive an analytical expression for the beyond-Fröhlich effective mass for a contact interaction and numerically compute the effective mass for long-range impurities. We argue that the coupling parameter |a₈₁|n₀^1/3 remains the principal parameter governing the properties of the polaron. For weak (|aIB|n₀^1/3 1) and intermediate (|aIB|n₀^1/3 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 (r₄₅₅n₀^1/3 1).