BCS models of Josephson qubits I. Energy spectra

There exists a large number of experimental and theoretical results supporting the picture of macroscopic qubits implemented by nanoscopic Josephson junctions of three different types - charge qubit, flux qubit and phase qubit. The standard unified description of such systems is based on the formal quantization of the phenomenological Kirchhoff equations for the corresponding circuits. In this paper a simplified version of the BCS theory for superconductors is used to derive microscopic models for all types of small Josephson junctions. For these models the state-dependent individual tunneling of Cooper pairs couples ground pair states with excited pair states what leads to a more complicated structure of the lowest lying energy levels. In particular, the highly degenerate levels emerge, which act as probability sinks for the qubit. These models allow also for the coupling to phonons as an efficient mechanism of relaxation for all types of junctions. The alternative formulas concerning basic spectral parameters of superconducting qubits are presented and compared with the experimental data. Finally, the question whether small Josephson junctions can be treated as macroscopic quantum systems is briefly discussed.