Photonic heat conduction in Josephson-coupled Bardeen-Cooper-Schrieffer superconductors

We investigate the photon-mediated heat flow between two Josephson-coupled Bardeen-Cooper-Schrieffer (BCS) superconductors. We demonstrate that in standard low temperature experiments involving temperature-biased superconducting quantum interference devices (SQUIDs), this radiative contribution is negligible if compared to the direct galvanic one, but it largely exceeds the heat exchanged between electrons and the lattice phonons. The corresponding thermal conductance is found to be several orders of magnitude smaller, for real experiments setup parameters, than the universal quantum of thermal conductance, ?0(T)=?kB2T/6.