  The most sensitive haloscopes that search for axion dark matter through the
two photon electromagnetic anomaly, convert axions into photons through the
mixing of axions with a large DC magnetic field. In this work we apply Poynting
theorem to the resulting axion modified electrodynamics and identify two
possible Poynting vectors, one similar to the Abraham Poynting vector and the
other to the Minkowski Poynting vector in electrodynamics. The latter picks up
the extra non-conservative terms while the former does not. To understand the
source of energy conversion and power flow in the detection systems, we apply
the two Poynting theorems to axion modified electrodynamics, for both the
resonant cavity and broadband low-mass axion detectors. We show that both
Poynting theorems give the same sensitivity for a resonant cavity axion
haloscope, but predict markedly different sensitivity for a low-mass broadband
capacitive haloscope. Hence we ask the question, can understanding which one is
the correct one for axion dark matter detection, be considered under the
framework of the Abraham-Minkowski controversy? In reality, this should be
confirmed by experiment when the axion is detected. However, many
electrodynamic experiments have ruled in favour of the Minkowski Poynting
vector when considering the canonical momentum in dielectric media. In light of
this, we show that the axion modified Minkowski Poynting vector should indeed
be taken seriously for sensitivity calculation for low-mass axion haloscope
detectors in the quasi static limit, and predict orders of magnitude better
sensitivity than the Abraham Poynting vector equivalent.
