No equivalence between hydrodynamic and dispersive mass of the charged polaron

We consider the problem of a charged impurity exerting a weak, slowly decaying force on its surroundings, treating the latter as an ideal compressible fluid. In the semiclassical approximation, the ion is described by the Newton equation coupled to the Euler equation for the medium. After linearization, we obtain a simple closed formula for the effective mass of the impurity, depending on the interaction potential, the mean medium density, and sound velocity. Thus, once the interaction and the equation of state of the fluid is known, an estimate of the hydrodynamic effective mass can be quickly provided. Going beyond the classical case, we show that replacing the Newton with Schr\"{o}dinger equation can drastically change the behavior of the impurity. In particular, the scaling of the Fermi polaron effective mass with the medium density is opposite in quantum and classical scenario. Our results are relevant for experimental systems featuring low energy impurities in Fermi or Bose systems, such as ions immersed in neutral atomic gases.