Variational approach to the ground state of an impurity in a Bose-Einstein condensate

In this paper, we consider the effect of self-localization of a quantum impurity in a Bose-Einstein condensate. The space correlation function of the impurity is evaluated with the help of the imaginary-time path integral approach. Employing Feynman's variational method, we calculate the impurity correlation function as well as the energy of the system associated with the impurity. The effect of self-localization predicted before within the Gross-Pitaevskii approach is recovered by our treatment. The strong coupling regime with negative ground state energy is reached by the variational method, and the corresponding correlation function is calculated.