The classical and quantum synchronization between two nonlinear mechanical modes of Bose–Einstein condensates is investigated by different types of measures in order to reveal macroscopic and microscopic properties of synchronized behaviors in a closed quantum system. The classical measure synchronization (CMS) is studied by Pearson correlation coefficient, the orbital overlapping and covering areas in the phase space based on mean-value dynamical equations. The dynamical transitions of CMS are analyzed with phase diagrams in the parametric plane of population imbalance and phase difference between two modes in a wide range of mode coupling rate. Based on Husimi Q functions, the synchronized behaviors of quantum measure synchronization (QMS) are displayed by density overlapping and correlated probability dynamics in phase space, and further investigated by two quantum measures: Mari measure and mutual information. These results demonstrate that the “revival and collapse” of quantum fluctuations beyond mean-value dynamics discriminates QMS from CMS. The overwhelming dynamics of error fluctuations not only excludes complete CMS and perfect phase overlap in QMS, but also leads to upper bound to Mari measure and unceasing oscillations of mutual information. We reveal that the correlation between Mari measure and mutual information for QMS is derived from the similar dynamics of error fluctuations with respect to their opposite mean-value behaviors.
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We are thankful to Keye Zhang for the useful discussions and suggestions. This work is supported by the National Natural Science Foundation of China (Grants Nos. 11447025 and 11234003) and the National Basic Research Program of China (973 Program) under Grant No. 2011CB921604.
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Zhang, L., Xu, X. & Zhang, W. The classical and quantum synchronization between two scattering modes in Bose–Einstein condensates. Eur. Phys. J. Plus 135, 202 (2020). https://doi.org/10.1140/epjp/s13360-020-00179-0