Trapped one-dimensional ideal Fermi gas with a single impurity

Properties of a single impurity in a one-dimensional Fermi gas are investigated in homogeneous and trapped geometries. In a homogeneous system we use McGuires expression [J. B. McGuire, J. Math. Phys. 6, 432 (1965)] to obtain interaction and kinetic energies, as well as the local pair correlation function. The energy of a trapped system is obtained (i) by generalizing McGuire expression (ii) within local density approximation (iii) using perturbative approach in the case of a weakly interacting impurity and (iv) diffusion Monte Carlo method. We demonstrate that a closed formula based on the exact solution of the homogeneous case provides a precise estimation for the energy of a trapped system for arbitrary coupling constant of the impurity even for a small number of fermions. We analyze energy contributions from kinetic, interaction and potential components, as well as spatial properties such as the system size. Finally, we calculate the frequency of the breathing mode. Our analysis is directly connected and applicable to the recent experiments in microtraps.

Scattering off an oscillating target: Basic mechanisms and their impact on cross sections

We investigate classical scattering off a harmonically oscillating target in two spatial dimensions. The shape of the scatterer is assumed to have a boundary which is locally convex at any point and does not support the presence of any periodic orbits in the corresponding dynamics. As a simple example we consider the scattering of a beam of non-interacting particles off a circular hard scatterer. The performed analysis is focused on experimentally accessible quantities, characterizing the system, like the differential cross sections in the outgoing angle and velocity. Despite the absence of periodic orbits and their manifolds in the dynamics, we show that the cross sections acquire rich and multiple structure when the velocity of the particles in the beam becomes of the same order of magnitude as the maximum velocity of the oscillating target. The underlying dynamical pattern is uniquely determined by the phase of the first collision between the beam particles and the scatterer and possesses a universal profile, dictated by the manifolds of the parabolic orbits, which can be understood both qualitatively as well as quantitatively in terms of scattering off a hard wall. We discuss also the inverse problem concerning the possibility to extract properties of the oscillating target from the differential cross sections.