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 peak 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 the Friedel and Wigner terms are carefully analysed in the whole range of interaction strengths. It is shown that Friedel oscillations, unlike the Wigner ones, are sensitive probes for detecting excited spin states and collective spin density waves involved in the transport.