We study decoherence due to low frequency noise in Josephson qubits. Non-Markovian classical noise due to switching impurities determines inhomogeneous broadening of the signal. The theory is extended to include effects of high-frequency quantum noise, due to impurities or to the electromagnetic environment. The interplay of slow noise with intrinsically non-Gaussian noise sources may explain the rich physics observed in the spectroscopy and in the dynamics of charge based devices.

By means of the Lanczos method we analyze superconducting correlations in ultrasmall grains at fixed particle number. We compute the ground state properties and the excitation gap of the pairing Hamiltonian as a function of the level spacing δ. Both quantities turn out to be parity dependent and universal functions of the ratio δ/∆ (∆ is the BCS gap). We then characterize superconductivity in the canonical ensemble from the scaling behavior of correlation functions in energy space.

We study dynamical decoupling of a qubit from non gaussian quantum noise due to discrete sources, as bistable fluctuators and 1/f noise. We obtain analytic and numerical results for generic operating point. For very large pulse frequency, where dynamic decoupling compensates decoherence, we found universal behavior. At intermediate frequencies noise can be compensated or enhanced, depending on the nature of the fluctuators and on the operating point. Our technique can be applied to a larger class of non-gaussian environments.

We study the spin susceptibility chi of a small, isolated superconducting grain. Because of the interplay between parity effects and pairing correlations, the dependence of chi on temperature T is qualitatively different from the standard BCS result valid in the bulk limit. If the number of electrons on the grain is odd, chi shows a re-entrant behavior as a function of temperature. This behavior persists even in the case of ultrasmall grains where the mean level spacing is much larger than the BCS gap. If the number of electrons is even, chi(T) is exponentially small at low temperatures.