  We reveal an intriguing manifestation of topology, which appears in the
depletion rate of topological states of matter in response to an external
drive. This phenomenon is presented by analyzing the response of a generic 2D
Chern insulator subjected to a circular time-periodic perturbation: due to the
system's chiral nature, the depletion rate is shown to depend on the
orientation of the circular shake. Most importantly, taking the difference
between the rates obtained from two opposite orientations of the drive, and
integrating over a proper drive-frequency range, provides a direct measure of
the topological Chern number of the populated band ($\nu$): this "differential
integrated rate" is directly related to the strength of the driving field
through the quantized coefficient $\eta_0\!=\!\nu /\hbar^2$. Contrary to the
integer quantum Hall effect, this quantized response is found to be non-linear
with respect to the strength of the driving field and it explicitly involves
inter-band transitions. We investigate the possibility of probing this
phenomenon in ultracold gases and highlight the crucial role played by edge
states in this effect. We extend our results to 3D lattices, establishing a
link between depletion rates and the non-linear photogalvanic effect predicted
for Weyl semimetals. The quantized circular dichroism revealed in this work
designates depletion-rate measurements as a universal probe for topological
order in quantum matter.
