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.
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