Contact and Structure Factor for Bosonic and Fermionic Mixtures

We study measurable quantities of bosonic and fermionic mixtures on a one-dimensional ring. These few-body ensembles consist of majority atoms obeying certain statistics (Fermi or Bose) and an impurity atom in a different hyperfine state. The repulsive interactions between majority-impurity and majority-majority are varied from weak to strong. We show that the majority-impurity repulsion is mainly responsible for the loss of coherence in the strongly interacting regime. The momentum distribution follows the $\mathcal{C}/p^4$ universal behaviour for the high momentum tail, but the contact $\mathcal{C}$ is strongly dependent on the strength of the majority-impurity and in a different way on the majority-majority interactions. The static structure factor of the majority atoms exposes a low-momentum peak for strong majority-impurity repulsion, which is attributed to an effective attraction not expected for purely repulsive forces.