Casimir forces between defects in one-dimensional quantum liquids

We discuss the effective interactions between two localized perturbations in one-dimensional quantum liquids. For noninteracting fermions, the interactions exhibit Friedel oscillations, giving rise to a Ruderman-Kittel-Kasuya-Yosida-type interaction familiar from impurity spins in metals. In the interacting case, at low energies, a Luttinger-liquid description applies. In the case of repulsive fermions, the Friedel oscillations of the interacting system are replaced, at long distances, by a universal Casimir-type interaction which depends only on the sound velocity and decays inversely with the separation. The Casimir-type interaction between localized perturbations embedded in a fermionic environment gives rise to a long-range coupling between quantum dots in ultracold Fermi gases, opening an alternative to couple qubits with neutral atoms. We also briefly discuss the case of bosonic quantum liquids in which the interaction between weak impurities turns out to be short ranged, decaying exponentially on the scale of the healing length.