  We implement the proof of principle for the quantum walk of one ion in a
linear ion trap. With a single-step fidelity exceeding 0.99, we perform three
steps of an asymmetric walk on the line. We clearly reveal the differences to
its classical counterpart if we allow the walker/ion to take all classical
paths simultaneously. Quantum interferences enforce asymmetric, non-classical
distributions in the highly entangled degrees of freedom (of coin and position
states). We theoretically study and experimentally observe the limitation in
the number of steps of our approach, that is imposed by motional squeezing. We
propose an altered protocol based on methods of impulsive steps to overcome
these restrictions, in principal allowing to scale the quantum walk to several
hundreds of steps.
