  Recently it has been demonstrated that it is possible to harvest quantum
resources other than entanglement from a coherent scalar field. Employing
time-dependent perturbation theory, we present a complete analysis of the
conditions under which a spatially extended Unruh-DeWitt detector coupled to
the proper time derivative of the field can harvest coherence for any initial
state of the field, as well as the energy cost that is required for each
harvest. By studying harvesting under repeatable extractions it is proven that
when the detector interacts with the field through a delta coupling coherence
is catalytic. For a Gaussian smeared detector it is shown that harvesting from
a coherent field depends on the phase of its amplitude distribution and its
initial energy as well as on the mean radius of the detector and the mean
interaction duration between the two. For a detector moving at a constant
velocity and with a mean radius of the same order as its transition wavelength,
we observe that, for relativistic speeds, coherence swelling effects are
present the intensity of which depends on the dimension of the underlying
Minkowski spacetime.
