Exact WKB analysis for adiabatic discrete-level Hamiltonians

TL;DR

This paper applies exact WKB analysis to adiabatic quantum systems, deriving transition probability formulas for both two-level and multilevel systems, expanding non-perturbative understanding in quantum dynamics.

Contribution

It introduces a novel application of exact WKB analysis to multilevel adiabatic systems, providing explicit transition probability formulas and demonstrating the method's broader applicability.

Findings

Derived a transition probability formula similar to known two-level results

Extended the analysis to multilevel systems, showing applicability beyond two levels

Provided a concrete example illustrating the method's effectiveness

Abstract

The dynamics of quantum systems under the adiabatic Hamiltonian has attracted attention not only in quantum control but also in a wide range of fields from condensed matter physics to high-energy physics because of its non-perturbative behavior. Here we analyze the adiabatic dynamics in the two-level systems and the multilevel systems using the exact WKB analysis, which is one of the non-perturbative analysis methods. As a result, we obtain a formula for the transition probability, which is similar to the known formula in the two-level system. Although non-perturbative analysis in the adiabatic limit has rarely been studied for multilevel systems, we show that the same analysis can be applied and also provide a concrete example. The results will serve as a basis for the application of the exact WKB analysis in various fields of physics.