Exact solution of qubit decoherence models by a transfer matrix method

We present a method for the solution of the behavior of an ensemble of qubits in a random time-dependent external field. The forward evolution in time is governed by a transfer matrix. The elements of this matrix determine the various decoherence times. The method provides an exact solution in cases where the noise is piecewise constant in time. We show that it applies, for example, to a realistic model of decoherence of electron spins in semiconductors. Results are obtained for the nonperturbative regimes of the models, and we see a transition from weak relaxation to overdamped behavior as a function of noise anisotropy.