Exact Study of Finite-Well Impurity in One-Dimensional Bose Condensates

The quantum impurity immersed inside a Bose condensate forms a Bose polaron. The conventional formalism of the polarons often treats the impurity-bath interactions through the Fröhlich-Bogoliubov Hamiltonian, which accounts of the collective excitations of the condensate over a uniform condensate field. This approach neglects the back-action of the impurity on the condensate, which causes density deformations around the impurity for the strong couplings. The deformation becomes notably significant for impurities in quasi-one-dimensional Bose condensates, an early experimental realization of Bose polarons.

Here we explore the energy and the effective mass of a short-range polaron, modeled with a finite square well potential, in a one-dimensional condensate. We obtain the semi-analytical solution of the Gross-Pitaevskii equation —accounts for the density deformation around the impurity already at the mean-field level— for both attractive and repulsive finite well potentials. We calculate the energy and effective mass of the impurity as a function of the scattering length of the finite well potential, where the well width is much smaller than the condensate coherence length, thereby approximating a short-range polaron. For the repulsive potential, our energy results are in agreement with previous studies that model the impurity using a Dirac delta potential, both in the weak and strong impurity-bath coupling regimes. For the attractive potential, the polaron energy and effective mass converge to a finite value in the limit of infinite impurity-bath coupling, in contrast to the divergent behavior with the Dirac delta potential. This is attributed to the fact that the delta potential always supports a bound state, independent of the strength of effective repulsion of the cloud around the impurity, whereas the cloud effectively cancels the attractive finite well potential.