Dynamics of a tracer particle interacting with excitations of a Bose-Einstein condensate

TL;DR

This paper analyzes the quantum dynamics of a large bosonic system with a tracer particle, showing that in the mean-field limit, the system's evolution is governed by the Bogoliubov-Fröhlich Hamiltonian, capturing the interaction with excitations.

Contribution

It rigorously derives the effective dynamics of a bosonic system with a tracer particle as the Bogoliubov-Fröhlich Hamiltonian in the mean-field limit.

Findings

Effective dynamics described by Bogoliubov-Fröhlich Hamiltonian.

Validates mean-field approximation for large N.

Connects microscopic interactions to macroscopic effective theory.

Abstract

We consider the quantum dynamics of a large number $N$ of interacting bosons coupled a tracer particle, i.e. a particle of another kind, on a torus. We assume that in the initial state the bosons essentially form a homogeneous Bose-Einstein condensate, with some excitations. With an appropriate mean-field scaling of the interactions, we prove that the effective dynamics for $N\to \infty$ is generated by the Bogoliubov-Fr\"ohlich Hamiltonian, which couples the tracer particle linearly to the excitation field.