Exactly solvable model for thermal activation and quantum tunneling in Ohmic systems

Combining Kramers’s energy diffusion (at temperature T ) with dissipative quantum tunneling (through a parabolic potential-energy barrier with curvature frequency ${\omega }_b$ and height $U_b$), a model for a localized metastable state (with harmonic frequency ${\omega }_0$) is formulated and solved exactly for its quasiequilibrium distribution ρ(E) and its decay rate Γ. It is shown that ρ remains locally extremely close to (although it essentially differs from) the Boltzmann distribution, unless the Ohmic friction coefficient λ is extremely small (of order Γ). Excluding this latter possibility, the ensuing decay rate is discussed for various temperature and friction regimes. It is shown to comprise several known results as special cases. First, an extended version of Bell’s formula is shown to be valid in the strong-to-moderate friction regime. It involves the recently discovered crossover between thermal hopping and quantum tunneling at temperature $T_0$=ħκ${\omega }_b$/2π$k_B$ (where κ is Kramers’s viscosity correction factor). At high temperatures and very strong damping the result neatly reduces to Kramers’s Smoluchovsky-limit formula. At zero temperature friction strongly suppresses the decay. Second, in the very weak friction regime Γ is shown to reduce to an extended version of Melnikov’s formula. At high temperatures it equals Kramers’s small-viscosity result, whereas at zero temperature and zero damping it attains the expected quantum value, the crossover again being at $T_0$. Third, in the classical limit (ħ→0) an extended version of Büttiker, Harris, and Landauer’s [Phys. Rev. B 28, 1268 (1983)] result is recovered, now valid for any value of the damping. Finally, some remarks are made in relation to recent cryogenic measurements on metastable flux states.