Adiabatic-Impulse approximation for avoided level crossings: from phase transition dynamics to Landau-Zener evolutions and back again

TL;DR

This paper introduces an adiabatic-impulse approximation inspired by Kibble-Zurek theory to analyze avoided level crossings, linking classical and quantum phase transition dynamics with applications to the Ising model.

Contribution

It presents a novel approximation method for quantum avoided level crossings, connecting classical phase transition concepts with quantum dynamics, and provides exact solutions for Landau-Zener problems.

Findings

The approximation offers intuitive insights into two-level quantum systems.

Application to the Ising model illustrates the method's effectiveness.

Exact solutions for Landau-Zener problems are provided.

Abstract

We show that a simple approximation based on concepts underlying the Kibble-Zurek theory of second order phase transition dynamics can be used to treat avoided level crossing problems. The approach discussed in this paper provides an intuitive insight into quantum dynamics of two level systems, and may serve as a link between the theory of dynamics of classical and quantum phase transitions. To illustrate these ideas we analyze dynamics of a paramagnet-ferromagnet quantum phase transition in the Ising model. We also present exact unpublished solutions of the Landau-Zener like problems.