Anomalous Joule law in the adiabatic dynamics of a quantum dot in contact with normal-metal and superconducting reservoirs

We formulate a general theory to study the time-dependent charge and energy transport of an adiabatically driven quantum dot in contact with normal and superconducting reservoirs at zero temperature. This setup is a generalization of a quantum RC circuit, with capacitive components due to Andreev processes and induced pairing fluctuations, in addition to the conventional normal charge fluctuations. The dynamics for the dissipation of energy is ruled by a Joule law for four channels in parallel with the universal Büttiker resistance per channel. Two transport channels are associated with the two spin components of the usual charge fluctuations, while the other two are associated with electrons and holes due to pairing fluctuations. The latter leads to an “anomalous” component of the Joule law and takes place with a vanishing net current due to the opposite flows of electrons and holes.