Quasiparticle Properties of Long-range Impurities in a Bose Condensate

Atomic impurities inside of a Bose condensate facilitated the study of Bose polarons in ultracold atomic settings. The tunability of interactions enabled the exploration of attractive and repulsive polaron regimes across weak to strong couplings. In the weak coupling regime, the phonon-impurity interactions can be accounted up to the linear order, as represented in the conventional Fröhlich Hamiltonian. However, for the strong couplings, the inclusion of beyond Fröhlich interaction terms is necessary to account for both quantitative corrections to the polaron properties and the formation of impurity-bath bound states. Consequently, the characterization of the impurity as a quasiparticle across various regimes and the determination of its quasiparticle properties have attracted significant interest.



Here we employ two complementary methods to compute the quasiparticle properties of the contact, ion, and Rydberg impurities in the beyond-Fröhlich model. The first method uses an ansatz in the form of a coherent state of the condensate excitations. The zero-momentum equations for the coherent-state amplitudes are solved to determine the energy and quasiparticle weights, followed by solving the implicit equation for a moving impurity to obtain the effective mass for any finite momentum. The second method treats the impurity as a slowly moving external potential and solves the Gross-Pitaevskii (GP) equation, assuming small density perturbations around a uniform density. By expanding the GP energy in powers of the impurity velocity, we derive an analytical expression for the beyond-Fröhlich effective mass of short-range impurities, in excellent agreement with the former approach. For the long-range impurities, the effective masses are numerically calculated. Finally, the subsonic-to-supersonic transition of the impurity is determined by examining the finite momentum values at which the implicit equation for the effective mass yields no solution.