  We explore a few-fermion mixture consisting of two components which are
repulsively interacting and confined in a one-dimensional harmonic trap.
Different scenarios of population imbalance ranging from the completely
imbalanced case where the physics of a single impurity in the Fermi-sea is
discussed to the partially imbalanced and equal population configurations are
investigated. For the numerical calculations the multi-configurational
time-dependent Hartree (MCTDH) method is employed, extending its application to
few-fermion systems. Apart from numerical calculations we generalize our Ansatz
for a correlated pair wave-function proposed in [1] for bosons to mixtures of
fermions. From weak to strong coupling between the components the energies, the
densities and the correlation properties of one-dimensional systems change
vastly with an upper limit set by fermionization where for infinite repulsion
all fermions can be mapped to identical ones. The numerical and analytical
treatments are in good agreement with respect to the description of this
crossover. We show that for equal populations each pair of different component
atoms splits into two single peaks in the density while for partial imbalance
additional peaks and plateaus arise for very strong interaction strengths. The
case of a single impurity atom shows rich behaviour of the energy and density
as we approach fermionization, and is directly connected to recent experiments
[2-4].
