Trapped one-dimensional ideal Fermi gas with a single impurity

Ground-state properties of a single impurity in a one-dimensional Fermi gas are investigated in uniform and trapped geometries. The energy of a trapped system is obtained (i) by generalizing the McGuire expression from a uniform to trapped system (ii) within the local density approximation (iii) using the perturbative approach in the case of a weakly interacting impurity and (iv) diffusion Monte Carlo method. We demonstrate that there is a closed formula based on the exact solution of the homogeneous case which provides a precise estimation for the energy of a trapped system even for a small number of fermions and arbitrary coupling constant of the impurity. Using this expression, we analyze energy contributions from kinetic, interaction, and potential components, as well as spatial properties such as the system size and the pair-correlation function. Finally, we calculate the frequency of the breathing mode. Our analysis is directly connected and applicable to the recent experiments in microtraps.