Analysis of the tunnelling-charging Hamiltonian of a Cooper pair pump

Abstract

This thesis consists of an introductory part and three research papers. The main body of the work can be summarised as the development of a model describing gated arrays of superconducting Josephson junctions or Cooper pair pumps. The pumping of single electrons in gated arrays of normal-state tunnelling junctions has reached an accuracy that can be considered for metrological applications. The aim of the present study is to investigate whether the quantum mechanical nature of superconductivity restricts the accuracy of Cooper pair pumps. The answer is affirmative, at least if the impedance of the environment is assumed to be vanishingly small and the phase difference over the array ɸ is fixed. By using the canonical Hamiltonian for N independent Josephson junctions and neglecting any coupling to the environment as well as quasiparticle degrees of freedom, the problem becomes tractable. Both the direct supercurrent through the array and the adiabatically pumped charge are examined in detail. The leading order pumped charge for the most feasible pumping cycle is found to behave as I≈ -2ef [1 - ɑN (EJ / Ec)N-2 cos ф], where -2e is the charge of a Cooper pair, f is the gating frequency, aN is a constant which only depends on N, EJ is the Josephson coupling energy, and Ee is the charging energy. The deviation is rather large for short arrays, and provided the other sources of inaccuracy can be suppressed, the adiabatic pumping can be tested experimentally when the direct supercurrent vanishes at ф = 0. The measurents also probe the coherence of the system, especially the effects due to the dephasing, in other words the fluctuations in value of ф. The adopted renormalisation approach is then used for obtaining quantitative predictions of the pumping inaccuracy when the array is inhomogeneous or the gating sequence is not ideal. The predictions have been confirmed numerically and they can be related to experimentally measurable quantities. The small size of these corrections means that they may be masked by other effects in experiments. Further studies using more realistic and sophisticated models should be performed, also. Finally, the symmetries of the model are examined in detail and the systematics of the pumped charge with respect to all of the model parameters are obtained, although not rigorously proven. This is done by using an efficient block-diagonalisation scheme combined with a Fourier expansion of the eigenstates.