Anisotropic potential immersed in a dipolar Bose-Einstein condensate

We study a three-dimensional Gross-Pitaevskii equation that describes a static impurity in a dipolar Bose-Einstein condensate. Our focus is on the interplay between the shape of the impurity and the anisotropy of the medium manifested in the energy and the density of the system. Without external confinement, properties of the system are derived with basic analytical approaches. For a system in a harmonic trap, the model is investigated numerically, using the split-step Crank-Nicolson method. Our results demonstrate that the impurity self-energy is minimized when its shape more closely aligns with the anisotropic character of the bath; in particular a prolate deformed impurity aligned with the direction of the dipoles has the smallest self-energy for a repulsive impurity. Our work complements studies of impurities in Bose gases with zero-range interactions and paves the way for studies of dipolar polarons with a Gross-Pitaevskii equation.