Role of impurity statistics and medium constraints in polaron-polaron interactions

TL;DR

This paper develops a unified theoretical framework to understand how impurity statistics and medium constraints influence polaron-polaron interactions across various quantum gases.

Contribution

It introduces a comprehensive approach to analyze polaron interactions considering impurity statistics and medium constraints, including exact relationships between different fixed parameters.

Findings

Derived rigorous results for weak impurity-medium coupling.

Established an exact relationship between interactions at fixed density and fixed chemical potential.

Provided a foundation for future strong-coupling theories of polaron interactions.

Abstract

We consider the behavior of a small density of mobile impurities (polarons) immersed in a quantum gas, a generic scenario that can be realized in cold atomic gases, liquid helium mixtures, and doped semiconductors. We present a unified theoretical framework for understanding polaron quasiparticles beyond the single-impurity limit, and we identify two key factors that control the polaron-polaron interactions: (i) the statistics of the impurities, including whether or not they are degenerate, and (ii) the constraints on the medium response, i.e., whether the medium density or chemical potential is held fixed. By constructing wave functions for two bosonic, fermionic, or distinguishable impurities immersed in a Bose or Fermi gas, we derive rigorous results for the polaron interactions in the limit of weak impurity-medium coupling. We furthermore obtain an exact relationship between the polaron interactions at fixed medium density and at fixed chemical potential, a result which is valid for arbitrary interaction strength. Our work provides an important guide for understanding experiments, and it acts as a starting point for future strong-coupling theories of polaron interactions that capture all of the effects identified in this work.