Phase-dependent thermal transport in Josephson junctions

We calculated the energy current through a Josephson junction and found it to consist of three contributions: a quasiparticle current, an interference current, and a pair current, similar to the total electrical current in a Josephson junction. The quasiparticle part satisfies Onsager relations, and represents the normal dissipative heat current. The other two parts depend on the phase drop across the junction {δ}{θ} and are related to pair tunneling. We show that the pair energy current, like the Josephson current, is nondissipative. It appears only when there is a voltage across the junction, and therefore oscillates in time. The interference energy current appears when either a voltage or a temperature drop is present. In the latter case, the interference current can flow in either direction, depending on the sign of cos ({δ}{θ}). Thus, this part of the energy current can flow in the opposite direction to the temperature drop, causing a reduction in the dissipation as compared to what would occur without the interference current. Nevertheless, the second law of thermodynamics is not violated with respect to the total current. This effect is related to the coupling of quasiparticles and pairs in the superconducting electrodes comprising the junction.