Dynamic response of interacting one-dimensional fermions in the harmonic atom trap: Phase response and the inhomogeneous mobility

The problem of the Kohn mode in bosonized theories of one-dimensional interacting fermions in the harmonic trap is investigated and a suitable modification of the interaction is proposed which preserves the Kohn mode. The modified theory is used to calculate exactly the inhomogeneous linear mobility μ(z, z0;ω) at position z in response to a spatial force pulse at position z0. It is found that the inhomogeneous particle mobility exhibits resonances not only at the trap frequency ωl but also at multiples mǫ̃, m = 2, 3, ... of a new renormalized collective mode frequency which depends on the strength of the interaction. In contrast, the homogeneous response obtained by an average over z0 remains that of the non-interacting system.