Stochastic Schr\"odinger-Lohe model

TL;DR

This paper introduces a stochastic version of the Schr"odinger-Lohe model, demonstrating that even with randomness, two wave functions tend to synchronize over time under certain conditions.

Contribution

It extends the deterministic Schr"odinger-Lohe model by incorporating stochastic perturbations and proves a weak synchronization result for two oscillators.

Findings

Weak synchronization observed in the stochastic model

Synchronization persists despite stochastic perturbations

Results specific to the case of two oscillators

Abstract

The Schr\"odinger-Lohe model consists of wave functions interacting with each other, according to a system of Schr\"odinger equations with a specific coupling such that all wave functions evolve on the $L^2$ unit ball. This model has been extensively studied over the last decade and it was shown that under suitable assumptions on the initial state, if one waits long enough all the wave functions become arbitrarily close to each other, which we call a synchronization. In this paper, we consider a stochastic perturbation of the Schr\"odinger-Lohe model and show a weak version of synchronization for this perturbed model in the case of two oscillators.