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

TL;DR

This paper clarifies the role of the Kubo formula in quantum pumping within closed systems, distinguishing between adiabatic and dissipative contributions, and discusses conditions for quantized pumping with practical examples.

Contribution

It provides a comprehensive analysis of quantum pumping in closed systems, connecting the Kubo formula to adiabatic transport and quantized pumping conditions, with illustrative models.

Findings

Kubo formula captures all relevant physics for pumped charge in linear response.

Adiabatic pumping is not necessarily quantized, but can be approximated as such under certain conditions.

Deviations from quantization relate to Thouless conductance.

Abstract

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.