Semiclassical theory of energy diffusive escape

Thermal escape out of a metastable well is considered in the weak friction regime, where the bottleneck for decay is energy diffusion, and at lower temperatures, where quantum tunneling becomes relevant. Within a systematic semiclassical formalism an extension of the classical diffusion equation is derived starting from a quantum mechanical master equation. In contrast to previous approaches finite barrier transmission also affects transition probabilities. The decay rate is obtained from the stationary nonequilibrium solution and captures the intimate interplay between thermal and quantum fluctuations above the crossover to the deep quantum regime.

Figure 1

(Color online) Typical metastable well potential discussed in the text with quasistationary states deep in the well and a continuum of states near the barrier top, around which a parabolic barrier approximation applies.

Figure 2

Escape rate normalized to the classical rate (8) versus the dimensionless inverse temperature $\theta =\hslash {\omega }_b\beta$. Thin lines refer to the result from [21] where $C\left(E\right)=1$ [cf. Eq. (56)], thick lines to the new expression (71) for $\gamma ∕{\omega }_0=0.0015$ (solid), 0.005 (dashed), 0.01 (dotted).

Figure 3

Escape rate normalized to the classical TST rate $({\Gamma }_{\mathrm{TST}}={\omega }_0∕2\pi e^{-\beta \Delta V})$ as a function of the dimensionless friction strength for various values of the dimensionless inverse temperature $\theta =\hslash {\omega }_b\beta ∕2\pi$ ($\theta =0$ denotes the classical rate).