  Quantum information processing has greatly increased interest in the
phenomenon of environmentally-induced decoherence. The spin boson model is
widely used to study the interaction between a spin-modelling a quantum
particle moving in a double well potential-and its environment-modelled by a
heat bath of harmonic oscillators. This paper extends a previous analysis of
the static spin boson study to the driven spin boson case, with the derivation
of an exact integro-differential equation for the time evolution of the
propagator of the reduced spin density matrix. This is the first main result.
By specializing to weak damping we then obtain the next result, a set of
Bloch-Redfield equations for the equilibrium fixed spin initial condition.
Finally we show that these equations can be used to solve the classic
dissipative Landau-Zener problem and illustrate these solutions for the weak
damping case. The effect of dissipation is seen to be minimised as the speed of
passage is increased, implying that qubits need to be switched as fast as
possible.
