Quantum pumping in closed systems, adiabatic transport, and the Kubo formula

Quantum pumping in closed systems is considered. We explain that the Kubo formula contains all the physically relevant ingredients for the calculation of the pumped charge ($Q$) within the framework of linear response theory. The relation to the common formulations of adiabatic transport and ``geometric magnetism" is clarified. We distinguish between adiabatic and dissipative contributions to $Q$. On the one hand we observe that adiabatic pumping does not have to be quantized. On the other hand we define circumstances in which quantized adiabatic pumping holds as an approximation. The deviation from exact quantization is related to the Thouless conductance. As an application we discuss the following examples: classical dissipative pumping by conductance control, classical adiabatic (non dissipative) pumping by translation, and quantum pumping in the double barrier model. In the latter context we analyze a 3 site lattice Hamiltonian, which represents the simplest pumping device. We remark on the connection with the popular $S$ matrix formalism which has been used to calculate pumping in open systems.