Validity of the Fr\"ohlich model for a mobile impurity in a Bose-Einstein condensate

TL;DR

This paper rigorously validates the Bogoliubov-Fr"ohlich model for a mobile impurity in a dilute Bose-Einstein condensate under certain interaction conditions, and derives a universal energy correction.

Contribution

It provides a rigorous mathematical proof of the model's validity and analytically computes a universal energy correction due to phonon interactions.

Findings

Validation of the Bogoliubov-Fr"ohlich Hamiltonian in specific regimes

Analytical derivation of a universal logarithmic energy correction

Quantitative estimates of error bounds for the model's applicability

Abstract

We analyze the many-body Hamiltonian describing a mobile impurity immersed in a Bose-Einstein condensate (BEC). Using exact unitary transformations and rigorous error estimates, we show the validity of the Bogoliubov-Fr\"ohlich Hamiltonian for the Bose polaron in the regime of moderately strong, repulsive interactions with a dilute BEC. Moreover, we calculate analytically the universal logarithmic correction to the ground state energy that arises from an impurity mediated phonon-phonon interaction.