Few particles with an impurity in a one-dimensional harmonic trap

We present a comprehensive study of the static properties of a mobile impurity interacting with a bath with a few particles trapped in a one-dimensional harmonic trap. We consider baths with either identical bosons or distinguishable particles and we focus on the limiting case where the bath is non-interacting. We provide numerical results for the energy spectra and density profiles by means of the exact diagonalization of the Hamiltonian, and find that these systems show non-trivial solutions, even in the limit of infinite repulsion. A detailed physical interpretation is provided for the lowest energy states. In particular, we find a seemingly universal transition from the impurity being localized in the center of the trap to being expelled outside the majority cloud. We also develop an analytical ansatz and a mean-field solution to compare them with our numerical results in limiting configurations.