  We generalize the adiabatic approximation to the case of open quantum
systems, in the joint limit of slow change and weak open system disturbances.
We show that the approximation is ``physically reasonable'' as under wide
conditions it leads to a completely positive evolution, if the original master
equation can be written on a time-dependent Lindblad form. We demonstrate the
approximation for a non-Abelian holonomic implementation of the Hadamard gate,
disturbed by a decoherence process. We compare the resulting approximate
evolution with numerical simulations of the exact equation.

Quantum Physics