# Chaos-preserving reduction of the spin-flip model for VCSELs: failure of the adiabatic elimination of the spin-population difference

## TL;DR

This paper investigates the spin-flip model for VCSELs, demonstrating that adiabatic elimination of the spin-population difference fails to preserve key chaotic dynamics, emphasizing the variable's essential role.

## Contribution

The study shows that the spin-population difference cannot be adiabatically eliminated without losing critical dynamical features like polarization chaos in the model.

## Key findings

- Adiabatic elimination fails to reproduce polarization chaos.
- The spin-population difference is essential for complex dynamics.
- Minimal embedding dimension analysis supports model reduction limitations.

## Abstract

When studying the dynamics of Vertical-Cavity Surface-Emitting Lasers, and their polarization properties, the spin-flip model appears to be the simplest model qualitatively reproducing all dynamical features that have been observed experimentally. Nonetheless, because of the fast time-scale of the spin-relaxation processes, the specific role and the importance of the spin-population difference - which is one of the specific feature of the spin-flip model - has been continuously questioned. In fact, the debate regarding the possible adiabatic elimination of the spin-population remains fully open. In this paper, our goal is to bring new light into this issue by demonstrating that this variable is essential to preserve the most complex dynamical features predicted by the spin-flip model, such as polarization chaos, and, therefore, needs to be conserved. To do so, we first perform a detailed analysis, focusing on the chaotic dynamics, to determine the minimal embedding dimension for the spin-flip model. As the latter confirms that a reduction of the model could be envisaged, we then consider the adiabatic elimination of the spin-population difference to highlight and explain its failure to reproduce essential dynamical features obtained in the original model.