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We analyze the impact of trap states in the oxide layer of a superconductingtunnel junctions, on the fluctuation of the Josephson critical current, thus oncoherence in superconducting qubits. Two mechanisms are usually considered: thecurrent blockage due to repulsion at the occupied trap states, and the noisefrom electrons hopping across a trap. We extend previous studies ofnoninteracting traps to the case where the traps have on-site electronrepulsion inside one ballistic channel. The repulsion not only allows theappropriate temperature dependence of 1/f noise, but also is a control to thecoupling between the computational qubit and the spurious two-level systemsinside the oxide dielectric. We use second order perturbation theory whichallows to obtain analytical formulae for the interacting bound states andspectral weights, limited to small and intermediate repulsions. Remarkably, itstill reproduces the main features of the model as identified from theNumerical Renormalization Group. We present analytical formulations for thesubgap bound state energies, the singlet-doublet phase boundary, and thespectral weights. We show that interactions can reverse the supercurrent acrossthe trap. We finally work out the spectrum of junction resonators for qubits inthe presence of on-site repulsive electrons and analyze its dependence onmicroscopic parameters that may be controlled by fabrication.

Noise and microresonance of critical current in Josephson junction induced by Kondo trap states