The description of an impurity immersed in an ensemble of particles is one of the simplest though non-trivial problems in quantum many-body physics. In this kind of situation, the impurity is dressed by a cloud of excitations of the many-body background leading to the This scenario was first proposed by Landau and Pekar to describe the interaction of an electron with the excitations of a crystal and has since then been generalized to a large number of physical situations, from condensed matter to nuclear physics.

Over the past 10 years, ultracold atoms have emerged as a new platform for the study of impurity problems. We have developped a variational approach allowing its properties to be captured quantitatively and that we have first applied to the so-called Fermi polaron problem describing an atom immersed in a Fermi sea of spin-polarized non-interacting fermions.

We have then extended this approach to the case where the background is an ensemble of attractive spin 1/2 fermions. Contrary to the previous example, the zero-range limit characterizing ultracold collisions is singular and we had to cure the associated divergences by a proper treatment of three-body interactions.

Initially introduced by R. Feynman to describe elementary processes in quantum electrodynamics, diagrammatic techniques have been extended to the study of the many-body problem and now form a very powerful tool in quantum statistical physics.