Path integral derivation of Bloch-Redfield equations for a qubit weakly   coupled to a heat bath: Application to nonadiabatic transitions
  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. 
Nicholas Wynn Watkins and David Waxman