  The description of an impurity atom in a Bose-Einstein condensate can be cast
in the form of Frohlich's polaron Hamiltonian, where the Bogoliubov excitations
play the role of the phonons. An expression for the corresponding polaronic
coupling strength is derived, relating the coupling strength to the scattering
lengths, the trap size and the number of Bose condensed atoms. This allows to
identify several approaches to reach the strong-coupling limit for the quantum
gas polarons, whereas this limit was hitherto experimentally inaccessible in
solids. We apply Feynman's path-integral method to calculate for all coupling
strengths the polaronic shift in the free energy and the increase in the
effective mass. The effect of temperature on these quantities is included in
the description. We find similarities to the acoustic polaron results and
indications of a transition between free polarons and self-trapped polarons.
The prospects, based on the current theory, of investigating the polaron
physics with ultracold gases are discussed for lithium atoms in a sodium
condensate.
