From flux quantization to superconducting quantum bits

One of the important predictions of the early phenomenological theories of superconductivity such as the London and Ginzburg-Landau (GL) theory is the quantization of magnetic flux in multiply connected superconductors, which is one of the first demonstrations of a quantum effect on a macroscopic scale. In this paper, which is devoted to Vitalij Lazarevich Ginzburg on the occasion of his 90th birthday, we analyze a superconducting cylinder acting as a flux box as well as a superconducting disk acting as a Cooper pair box in the framework of GL theory. We extend this analysis to leaky flux and Cooper pair boxes which are obtained by introducing weak links allowing for the entry and exit of flux quanta and Cooper pairs from the respective boxes at finite rates. Flux and Cooper pair slippage processes by coherent quantum tunneling result in effective two-level quantum systems forming the basis for flux and charge quantum bits presently considered for the solid-state implementation of quantum information processing. We show that the corresponding Hamiltonians describing the leaky flux and Cooper pair box can be transformed into each other by a canonical transformation.