Dynamics of a dissipative two level system. II

The problem of quantum coherence in a symmetric double well potential with a dimensionless damping coefficient α for classical motion is studied within a spin 1/2-boson model. The experimentally measured probability p(t) of refinding a definite initial state after time t is approximately expressed by the transverse spin relaxation function φ{symbol}(t), which is determined from a three-pole approximation, that incorporates both the correct long and short time behaviour. For a bare tunnelsplitting δ small compared to the heat bath cutoff ω_c we find, that the oscillating component of φ{symbol}(t) is negligible compared to the relaxational one unless α is of order Δ/ω_c. Thus p(t)≃(1+exp(-νt))/2 with a mean tunneling rate ν proportional to {Mathematical expression} for α < 2 and T ≪ {Mathematical expression} and proportional to Δ²/ω_c·(T/ω_c)^α-1 otherwise. The results apply directly to recent measurements of the dynamics of flux states in a SQUID.