The density of a one-dimensional Bose gas far from an impurity

We consider an impurity in a one-dimensional weakly-interacting Bose gas and analytically calculate the density profile of the Bose gas. Within the mean-field approximation, by increasing the distance from the impurity, the Bose gas density saturates exponentially fast to its mean thermodynamic-limit value at distances beyond the healing length. The effect of quantum fluctuations drastically changes this behavior, leading to a power law decay of the density deviation from the mean density. At distances longer than the healing length and shorter than a new length scale proportional to the impurity coupling strength, the power-law exponent is $2$, while at longest distances the corresponding exponent becomes $3$. The latter crossover does not exist in two special cases. The first one is realized for infinitely strongly coupled impurity; then the density deviation always decays with the exponent $2$. The second special case occurs when the new length scale is smaller than the healing length, i.e., at weak impurity coupling; then the density deviation always decays with the exponent $3$. The obtained results are exact in the impurity coupling strength and account for the leading order in the interaction between the particles of the Bose gas.