Non-linear Coulomb blockade microscopy of a correlated one-dimensional quantum dot

We evaluate the chemical potential of a one-dimensional quantum dot, coupled to an atomic force microscope tip. The dot is described within the Luttinger liquid framework and the conductance peaks positions as a function of the tip location are calculated in the linear and non-linear transport regimes for an arbitrary number of particles. The differences between the chemical potential oscillations induced by Friedel and Wigner terms are carefully analyzed in the whole range of interaction strength. It is shown that Friedel oscillations, differently from the Wigner ones, are sensitive probes to detect excited spin states and collective spin density waves involved in the transport.