  We discuss decoherence in discrete-time quantum walks in terms of a
phenomenological model that distinguishes spin and spatial decoherence. We
identify the dominating mechanisms that affect quantum walk experiments
realized with neutral atoms walking in an optical lattice.
  From the measured spatial distributions, we determine with good precision the
amount of decoherence per step, which provides a quantitative indication of the
quality of our quantum walks. In particular, we find that spin decoherence is
the main mechanism responsible for the loss of coherence in our experiment. We
also find that the sole observation of ballistic instead of diffusive expansion
in position space is not a good indicator for the range of coherent
delocalization.
  We provide further physical insight by distinguishing the effects of short
and long time spin dephasing mechanisms. We introduce the concept of coherence
length in the discrete-time quantum walk, which quantifies the range of spatial
coherences. Unexpectedly, we find that quasi-stationary dephasing does not
modify the local properties of the quantum walk, but instead affects spatial
coherences.
  For a visual representation of decoherence phenomena in phase space, we have
developed a formalism based on a discrete analogue of the Wigner function. We
show that the effects of spin and spatial decoherence differ dramatically in
momentum space.
