# Three-Level Landau-Zener Dynamics

## TL;DR

This paper analyzes three-level Landau-Zener dynamics, providing analytic solutions for su(2) Hamiltonians, numerical solutions for su(3), and studying open systems with noise, revealing population transfer, oscillations, and decoherence effects.

## Contribution

It offers the first analytic solution for 3-level Landau-Zener problems in su(2) and numerical analysis for su(3), including open system effects with noise.

## Key findings

- Full population transfer at large times in adiabatic regime
- St"uckelberg oscillations occur at intermediate times
- Noise suppresses oscillations and alters decay laws

## Abstract

We compute Landau-Zener probabilities for 3-level systems with a linear sweep of the uncoupled energy levels of the 3$\times$3 Hamiltonian $H(t)$. Two symmetry classes of Hamiltonians are studied: For $H(t) \in$ su(2) (expressible as a linear combination of the three spin 1 matrices), an analytic solution to the problem is obtained in terms of the parabolic cylinder $D$ functions. For $H(t) \in$ su(3) (expressible as a linear combination of the eight Gell-Mann matrices), numerical solutions are obtained. In the adiabatic regime, full population transfer is obtained asymptotically at large time, but at intermediate times, all three levels are populated and St\"uckelberg oscillations are typically manifest. For the open system, (wherein interaction with a reservoir occurs), we numerically solve a Markovian quantum master equation for the density matrix with Lindblad operators that models interaction with isotropic white Gaussian noise. We find that St\"uckelberg oscillations are suppressed and that the temporal decay law of the population probabilities is not a simple exponential.