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arXiv:1509.01956 · Physical review. E

Quantum synchronization in an optomechanical system based on Lyapunov control

Wenlin Li, Chong Li, He‐Shan Song

Published Jun 22, 2016• 86 citations
arXivHTMLPDF

Abstract

We extend the concepts of quantum complete synchronization and phase synchronization, which were proposed in A. Mari et al., Phys. Rev. Lett. 111, 103605 (2013)PRLTAO0031-900710.1103/PhysRevLett.111.103605, to more widespread quantum generalized synchronization. Generalized synchronization can be considered a necessary condition or a more flexible derivative of complete synchronization, and its criterion and synchronization measure are proposed and analyzed in this paper. As examples, we consider two typical generalized synchronizations in a designed optomechanical system. Unlike the effort to construct a special coupling synchronization system, we purposefully design extra control fields based on Lyapunov control theory. We find that the Lyapunov function can adapt to more flexible control objectives, which is more suitable for generalized synchronization control, and the control fields can be achieved simply with a time-variant voltage. Finally, the existence of quantum entanglement in different generalized synchronizations is also discussed.

DOI
10.1103/physreve.93.062221

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Based on the concepts of quantum synchronization and quantum phase synchronization proposed by A. Mari et al. [Phys. Rev. Lett. 111, 103605 (2013)], we introduce and characterize the measure of a more generalized quantum synchronization called quantum $\ensuremath{\phi}$ synchronization under which pairs of variables have the same amplitude and possess the same $\ensuremath{\phi}$ phase shift. Naturally, quantum synchronization and quantum antisynchronization become special cases of quantum $\ensuremath{\phi}$ synchronization. Their relations and differences are also discussed. To illustrate these theories, we investigate the quantum $\ensuremath{\phi}$ synchronization and quantum phase synchronization phenomena of two coupled optomechanical systems with periodic modulation and show that quantum $\ensuremath{\phi}$ synchronization is more general as a measure of synchronization. We also show the phenomenon of quantum antisynchronization when $\ensuremath{\phi}=\ensuremath{\pi}$.
Abstract We propose a quantitative criterion to determine whether the coupled quantum systems can achieve complete synchronization or phase synchronization in the process of analyzing quantum synchronization. Adopting the criterion, we discuss the quantum synchronization effects between optomechanical systems and find that the error between the systems and the fluctuation of error is sensitive to coupling intensity by calculating the largest Lyapunov exponent of the model and quantum fluctuation, respectively. By taking the appropriate coupling intensity, we can control quantum synchronization even under different logical relationships between switches. Finally, we simulate the dynamical evolution of the system to verify the quantum synchronization criterion and to show the ability of synchronization control.
Quantum synchronization and its connection with other quantum correlations have attracted considerable attention. Here we present a theoretical scheme to simultaneously represent and significantly enhance the level of quantum synchronization and entanglement between two indirectly coupled mechanical membranes, which are coupled to a common optical field within a cavity. By applying a two-tone driving laser with weighted amplitudes and specific frequencies, both synchronization gauged by Mari's criterion and entanglement estimated by logarithmic negativity can be greatly enhanced. We then clarify the relationship between quantum synchronization and entanglement in detail. Numerical simulation results show that the influence of the coupling asymmetry G2/G1{G}_{2}/{G}_{1}G2​/G1​ on quantum complete synchronization behaves similarly to that on the purity while the influence of the coupling asymmetry G2/G1{G}_{2}/{G}_{1}G2​/G1​ on quantum phase synchronization is more similar to that on quantum entanglement. Besides, we demonstrate that although quantum synchronization and quantum entanglement are not directly related, both of them are sensitive to the squeezing parameter and the cooling effect. Furthermore, it is also shown that detuning the frequencies of two mechanical oscillators can actually help quantum synchronization and entanglement, which is somewhat similar to the case of quantum synchronization blockade.
In this work, we investigate the dynamics of quantum synchronization in a four-mode optomechanical system, focusing on the influence of the Coulomb interaction between two mechanical resonators. We analyze the effect of the Coulomb coupling on three distinct synchronization regimes, i.e., complete quantum synchronization, φφφ-synchronization, and quantum phase synchronization. Our results show that while the Coulomb interaction plays a pivotal role in significantly enhancing quantum phase synchronization by facilitating energy exchange and phase coherence, it has little impact on complete and φφφ-synchronization. This indicates that amplitude and frequency locking are primarily determined by the optical driving, whereas phase alignment depends critically on inter-resonator coupling. We also demonstrate that the oscillations of the two optical cavities, which are indirectly coupled via the mechanical resonators, can become aligned over time, resulting in classical synchronization. These findings provide a robust mechanism for controlling collective quantum dynamics and offer a foundation for applications in quantum communication, precision sensing, and the development of synchronized quantum networks.
We introduce and characterize two different measures which quantify the level of synchronization of coupled continuous variable quantum systems. The two measures allow us to extend to the quantum domain the notions of complete and phase synchronization. The Heisenberg principle sets a universal bound to complete synchronization. The measure of phase synchronization is, in principle, unbounded; however, in the absence of quantum resources (e.g., squeezing) the synchronization level is bounded below a certain threshold. We elucidate some interesting connections between entanglement and synchronization and, finally, discuss an application based on quantum optomechanical systems.
Although quantum synchronization phenomena and corresponding measures have been widely discussed recently, it is still an open question how to characterize directly the influence of nonlocal correlation, which is the key distinction for identifying classical and quantum synchronizations. In this paper, we present basic postulates for quantifying quantum synchronization based on the related theory in Mari's work [Phys. Rev. Lett. 111, 103605 (2013)PRLTAO0031-900710.1103/PhysRevLett.111.103605], and we give a general formula of a quantum synchronization measure with clear physical interpretations. By introducing Pearson's parameter, we show that the obvious characteristics of our measure are the relativity and monotonicity. As an example, the measure is applied to describe synchronization among quantum optomechanical systems under a Markovian bath. We also show the potential by quantifying generalized synchronization and discrete variable synchronization with this measure.