Trapped one-dimensional ideal Fermi gas with a single impurity

TL;DR

This paper investigates the properties of a single impurity in a one-dimensional Fermi gas, providing new analytical and numerical methods to estimate energies and dynamics in both homogeneous and trapped systems, relevant to recent experiments.

Contribution

The study introduces a closed-form formula for trapped system energy based on the homogeneous solution, applicable across various coupling strengths and small particle numbers.

Findings

Accurate energy estimation for trapped systems using the homogeneous solution

Analysis of kinetic, interaction, and potential energy contributions

Calculation of the breathing mode frequency

Abstract

Properties of a single impurity in a one-dimensional Fermi gas are investigated in homogeneous and trapped geometries. In a homogeneous system we use McGuire's expression [J. B. McGuire, J. Math. Phys. 6, 432 (1965)] to obtain interaction and kinetic energies, as well as the local pair correlation function. The energy of a trapped system is obtained (i) by generalizing McGuire expression (ii) within local density approximation (iii) using perturbative approach in the case of a weakly interacting impurity and (iv) diffusion Monte Carlo method. We demonstrate that a closed formula based on the exact solution of the homogeneous case provides a precise estimation for the energy of a trapped system for arbitrary coupling constant of the impurity even for a small number of fermions. We analyze energy contributions from kinetic, interaction and potential components, as well as spatial properties such as the system size. Finally, we calculate the frequency of the breathing mode. Our analysis is directly connected and applicable to the recent experiments in microtraps.