Analytic approach to the Landau-Zener problem in bounded parameter space

TL;DR

This paper presents three exact analytic solutions for the Landau-Zener problem in a bounded parameter space, exploring different finite-time paths and their effects on excitation probabilities in a two-level quantum system.

Contribution

It introduces three new analytic solutions for the Landau-Zener problem with finite-time driving paths, extending understanding beyond the classic infinite-time approximation.

Findings

Exact time-dependent excitation probabilities derived for each path.

Landau-Zener formula as an approximation within specific time intervals.

Long-time behavior aligns with adiabatic perturbation theory.

Abstract

Three analytic solutions to the Schr\"{o}dinger equation for the time-dependent Landau-Zener Hamiltonian are presented. They correspond to specific finite-time driving paths in a bounded parameter space of a two-level system. Two of these paths go through the avoided crossing of levels, either with a constant speed or with variable speed that decreases in the region of reduced energy gap, the third path bypasses the crossing such that the energy gap remains constant. The solutions yield exact time dependencies of the excitation probability for the system evolving from the ground state of the initial Hamiltonian. The Landau-Zener formula emerges as an approximation valid within a certain interval of driving times for the constant-speed driving through the avoided crossing. For long driving times, all solutions converge to the prediction of the adiabatic perturbation theory. The excitation probability vanishes at some discrete time instants.