Quantum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ϕ</mml:mi></mml:math> synchronization in a coupled optomechanical system with periodic modulation

Abstract

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 $\ensuremath{\phi}=\ensuremath{\pi}$.