A Closed-Form Approach to Oscillatory Integrals in Level-Crossing Physics

TL;DR

This paper introduces a closed-form analytical solution for a class of highly oscillatory integrals related to level-crossing physics, validated by numerical data and supported by an accessible Mathematica package.

Contribution

The authors develop a novel exact analytical method for oscillatory integrals in level-crossing models, bridging a gap in analytical solutions for these complex integrals.

Findings

Excellent agreement with numerical simulations

Valid for finite times in level-crossing models

Provides a publicly available computational tool

Abstract

We present a closed-form, exact analytical solution, valid at finite times, to a class of multiple integrals with highly oscillatory kernels. Our approach leverages the intimate connection between these integrals and the minimal level-crossing model, namely the Landau-Zener model. Benchmarking against data from numerical simulations demonstrates excellent agreement validating our analytical method. Impacts of our results in level-crossing dynamics are also discussed. A dedicated Mathematica package named {\bf OscillatoryIntegralAnalytical.wl} publicly accessible in our \href{https://github.com/Kenmax15/Closed-Form-Approach-to-Oscillatory-Integrals/tree/main}{GitHub repository} allows generating the integrals for arbitrary order.