Unstable Semiclassical Trajectories in Tunneling

Some tunneling phenomena are described, in the semiclassical approximation, by unstable complex trajectories. We develop a systematic procedure to stabilize the trajectories and to calculate the tunneling probability, including both the suppression exponent and prefactor. We find that the instability of tunneling solutions modifies the power-law dependence of the prefactor on $\hslash$ as compared to the case of stable solutions.

Figure 1

Comparison between the semiclassical (lines) and exact (points) results for the suppression exponent and prefactor in the cases of (a) direct and (b) sphaleron-driven tunneling. The comparison is performed in the model (2) with $\omega =0.5$, $E_y=0.05$. The error bars represent the uncertainty of the fit (19).