# The classical and quantum synchronization between two scattering modes in Bose-Einstein condensates generated by the standing-wave laser beams

## TL;DR

This paper compares classical and quantum synchronization in two nonlinear modes of Bose-Einstein condensates driven by standing-wave lasers, revealing quantum-specific behaviors like revival and collapse of fluctuations.

## Contribution

It introduces a comprehensive analysis of both classical and quantum synchronization in BECs, highlighting quantum effects such as fluctuation revival and their impact on synchronization measures.

## Key findings

- Quantum fluctuations exhibit revival and collapse, distinguishing quantum from classical synchronization.
- Quantum synchronization is limited by fluctuations, preventing perfect phase space overlap.
- A dynamic connection between Mari measure and mutual information is demonstrated in BEC modes.

## Abstract

Both classical and quantum dynamics of the synchronization between two nonlinear mechanical modes scattered from Bose-Einstein condensates (BECs) by the standing-wave laser beam are comparatively investigated. As the ultra-low dissipations of the momentum modes in the atomic BECs, the synchronized dynamics are studied in a framework of closed-system theory in order to track down both the classical and the quantum synchronizations from an angle of quantum control. The classical synchronization and the relevant dynamics of measure synchronization, the quantum synchronization and two different types of measures proposed by Mari and estimated by mutual information based on $Q$-function are studied respectively in order to reveal both the macroscopic and the microscopic signatures of synchronized behaviors in a closed quantum system. The results demonstrate that the "revival and collapse" of the quantum fluctuations beyond the classical mean-value dynamics due to long-lasting mode coherence discriminates the quantum synchronization from the classical one, which not only excludes the possibilities of an exact synchronization and a perfect density overlap in phase space, but also leads to upper limitations to Mari measure and large unceasing fluctuations to mutual information between two scattering modes. We reveal a close dynamic connection between Mari measure and the mutual information of two nonlinear momentum modes in closed BEC systems by demonstrating an opposite mean-value behavior but a similar fluctuation variation with respect to their respective evolutionary scales.