The electromagnetic Casimir effect manifests as the interaction between uncharged conducting objects that are placed in a vacuum. More generally, the Casimir-like effect denotes an induced interaction between external bodies in a fluctuating medium. We study the Casimir-like interaction between two impurities embedded in a weakly interacting one-dimensional Bose gas. We develop a theory based on the Gross-Pitaevskii equation that accounts for the effect of quantum fluctuations. At small separations, the induced interaction between the impurities decays exponentially with the distance. This is a classical result that can be understood using the mean-field Gross-Pitaevskii equation. We find that at larger distances, the induced interaction crosses over into a power law dependence due to the quantum fluctuations. We obtain an analytic expression for the interaction that interpolates between the two limiting behaviors. The obtained result does not require any regularization.Thus Schecter and Kamenev [12] were able to describe the Casimir-like interaction only at very large distances, ℓ?ξ. The contact interaction between impurities obtained in [12] is an artifact of the abovementioned limitation.An impurity that interacts repulsively with the particles of the liquid produces the depletion of the liquid density. Such an effect occurs predominantly locally; its spatial extent is set by the healing length ξ. In the presence of two impurities, it becomes energetically more favorable that the two depletion regions overlap. Hence there is an induced attraction between the impurities at small distances, ℓξ. At separations above ξ, the two depletion clouds practically do not overlap and thus do not interact within such a classical picture. However, as a consequence of (quasi-)long-range correlations of fluctuations in the liquid, the two depletion regions indeed feel each other. This mechanism produces the long-range Casimir-like interaction between the impurities. Such contribution can be seen as a quantum fluctuation correction to the classical contribution that prevails at small distances. The described scenario is common in other physical problems. For example, the domain walls (solitons) in the system of atoms adsorbed on a periodic substrate interact exponentially at small separations [17]. Due to entropic effects, this dependence crosses over into a long-range power law once one accounts for the effect of thermal fluctuations [17]. We also note that the electromagnetic Casimir-Polder interaction [18] between neutral atoms, which are usually the realization of impurities in Bose liquids, decays faster than the induced interaction studied here, as we discuss below.

We consider an impurity in a semi-infinite one-dimensional system of weakly-interacting bosons. We calculate the interaction potential for the impurity due to the end of the system, i.e., the wall. For local repulsive (attractive) interaction between the impurity and the Bose gas, the interaction potential is attractive (repulsive). At short distances from the wall it decays exponentially crossing over into a universal 1/r 2 behavior at separations r above the healing length. Our results can be also interpreted as a Casimir-like interaction between two impurities, where one of them is infinitely strongly coupled to the Bose gas. We finally discuss the phenomenon of localization of the impurity near the wall.