Quantum pumping and dissipation: From closed to open systems

Current can be pumped through a closed system by changing parameters (or fields) in time. The Kubo formula allows one to distinguish between dissipative and nondissipative contributions to the current. We obtain a Green function expression and an S-matrix formula for the associated terms in the generalized conductance matrix: the geometric magnetism term that corresponds to adiabatic transport, and the Fermi golden rule term which is responsible for the irreversible absorption of energy. We explain the subtle limit of an infinite system, and demonstrate the consistency with the formulas by Landauer and Buttiker, Pretre and Thomas. We also discuss the generalization of the fluctuation-dissipation relation, and the implications of the Onsager reciprocity.