  Bloch oscillations appear when an electric field is superimposed on a quantum
particle that evolves on a lattice with a tight-binding Hamiltonian (TBH),
i.e., evolves via what we will call an electric TBH; this phenomenon will be
referred to as TBH Bloch oscillations. A similar phenomenon is known to show up
in so-called electric discrete-time quantum walks (DQWs); this phenomenon will
be referred to as DQW Bloch oscillations. This similarity is particularly
salient when the electric field of the DQW is weak. For a wide, i.e., spatially
extended initial condition, one numerically observes semi-classical
oscillations, i.e., oscillations of a localized particle, both for the electric
TBH and the electric DQW. More precisely: The numerical simulations strongly
suggest that the semi-classical DQW Bloch oscillations correspond to two
counter-propagating semi-classical TBH Bloch oscillations. In this work it is
shown that, under certain assumptions, the solution of the electric DQW for a
weak electric field and a wide initial condition is well approximated by the
superposition of two continuous-time expressions, which are counter-propagating
solutions of an electric TBH whose hopping amplitude is the cosine of the
arbitrary coin-operator mixing angle. In contrast, if one wishes the
continuous-time approximation to hold for spatially localized initial
conditions, one needs at least the DQW to be lazy, as suggested by numerical
simulations and by the fact that this has been proven in the case of a
vanishing electric field.
