Quantum coherence and tunnelling in a double-well potential in a thermal environment: Dynamics of the weakly coupled spin-boson system

The real-time dynamics of the asymmetric two-state system weakly interacting with a thermal environment is derived in a rather simple yet novel manner in the Heisenberg picture. Through second order in the coupling coefficients general formulae are obtained (i) for the quantum coherence frequency shift and damping-important in the zero bias limit-and (ii) for the incoherent tunnelling rate-significant in the strong bias limit. The approach to equilibrium is investigated and confirmed in general but is noted to be complicated in particular for subohmic spectra. For the unbiased ohmic case a new-inverse quartic-power law is found for the long time relaxation at zero temperature. A validity analysis of the formulae is also presented and provides a constraint on the coupling strength.