Spectral representation of the heat current in a driven Josephson junction

We discuss thermal transport through a Josephson junction in a time-dependent situation. We write the spectral representation of the heat current pumped by a generic drive. This enables separation of the dissipative and reactive contributions, of which the latter does not contribute to long-time averages. We discuss the physical interpretation, and note that the condensate heat current identified by Maki and Griffin [Phys. Rev. Lett. 15, 921 (1965)] is purely reactive. The results enable a convenient description of heat exchanges in a Josephson system in the presence of an external drive, with possible applications for the implementation of cooling devices.