Dephasing of Josephson qubits close to optimal points

Decoherence of Josephson qubits can be substantially reduced by tuning their parameters to optimal operation points, with only quadratic coupling to fluctuations. We analyze dephasing due to $1/f$ noise for a two-level system, detuned from an optimal point, i. e., the crossover to the linear-coupling regime, both for free induction decay and for spin-echo experiments. Influence of several noise sources is also discussed.