Fermionization of a strongly interacting Bose-Fermi mixture in a one-dimensional harmonic trap

TL;DR

This paper provides an exact analysis of a strongly interacting 1D Bose-Fermi mixture in a harmonic trap, revealing how interactions influence density and momentum profiles and cause fermionization upon expansion.

Contribution

It introduces a generalized Bose-Fermi mapping to exactly determine the properties of a 1D Bose-Fermi mixture with hard-core interactions, including density and momentum distributions.

Findings

Bosons and fermions do not phase separate in real space.

Momentum distributions decay as $C p^{-4}$ at large momenta.

Expansion causes momentum distributions to fermionize, resembling a Fermi gas.

Abstract

We consider a strongly interacting one-dimensional (1D) Bose-Fermi mixture confined in a harmonic trap. It consists of a Tonks-Girardeau (TG) gas (1D Bose gas with repulsive hard-core interactions) and of a non-interacting Fermi gas (1D spin-aligned Fermi gas), both species interacting through hard-core repulsive interactions. Using a generalized Bose-Fermi mapping, we determine the exact particle density profiles, momentum distributions and behaviour of the mixture under 1D expansion when opening the trap. In real space, bosons and fermions do not display any phase separation: the respective density profiles extend over the same region and they both present a number of peaks equal to the total number of particles in the trap. In momentum space the bosonic component has the typical narrow TG profile, while the fermionic component shows a broad distribution with fermionic oscillations at small momenta. Due to the large boson-fermion repulsive interactions, both the bosonic and the fermionic momentum distributions decay as $C p^{-4}$ at large momenta, like in the case of a pure bosonic TG gas. The coefficient $C$ is related to the two-body density matrix and to the bosonic concentration in the mixture. When opening the trap, both momentum distributions "fermionize" under expansion and turn into that of a Fermi gas with a particle number equal to the total number of particles in the mixture.