# Lefschetz-thimble inspired analysis of the Dykhne-Davis-Pechukas method and an application for the Schwinger Mechanism

## TL;DR

This paper critically examines the Dykhne-Davis-Pechukas (DDP) method for quantum transition probabilities, introduces a Lefschetz-thimble inspired alternative, and applies both to the Schwinger Mechanism, highlighting the new method's robustness in complex scenarios.

## Contribution

The paper develops a new complex analysis-based formula for quantum transition probabilities, improving upon the DDP method, especially for short-lived electric pulses in the Schwinger Mechanism.

## Key findings

- The new method outperforms DDP for Sauter pulses.
- Both methods are applied to the Schwinger Mechanism.
- The new approach remains accurate where DDP fails.

## Abstract

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.