Degenerate Landau-Zener model: Exact analytical solution

TL;DR

This paper provides an exact analytical solution to the degenerate Landau-Zener model using the Morris-Shore transformation, clarifying transition probabilities in systems with degenerate energy levels crossing over time.

Contribution

It introduces a novel exact solution for the degenerate Landau-Zener model, extending the understanding of transition dynamics in degenerate quantum systems.

Findings

Solution reduces the system to independent two-state models

Transition probabilities mainly exist within the same degenerate set

Illustration with magnetic sublevels of atomic levels J=2 and 1

Abstract

The exact analytical solution of the degenerate Landau-Zener model, wherein two bands of degenerate energies cross in time, is presented. The solution is derived by using the Morris-Shore transformation, which reduces the fully coupled system to a set of independent nondegenerate two-state systems and a set of decoupled states. Due to the divergence of the phase of the off-diagonal element of the propagator in the original Landau-Zener model, not all transition probabilities exist for infinite time duration. In general, apart from some special cases, only the transition probabilities between states within the same degenerate set exist, but not between states of different sets. An illustration is presented for the transition between the magnetic sublevels of two atomic levels with total angular momenta J=2 and 1.