  Dykhne-Davis-Pechukas (DDP) method is a common approximation scheme for the
transition probability in two-level quantum systems, as realized in the
Landau-Zener effect, leading to an exponentially damping form comparable to the
Schwinger pair production rate. We analyze the foundation of the DDP method
using a modern complex technique inspired by the Lefschetz-thimble method. We
derive an alternative and more adaptive formula that is useful even when the
DDP method is inapplicable. As a benchmark, we study the modified Landau-Zener
model and compare results from the DDP and our methods. We then revisit a
derivation of the Schwinger Mechanism of particle production under electric
fields using the DDP and our methods. We find that the DDP method gets worse
for the Sauter type of short-lived electric pulse, while our method is still a
reasonable approximation. We also study the Dynamically Assisted Schwinger
Mechanism in two methods.
