Fig. 2

Dynamical evolution of quantum synchronization \(S_{qij}(t)\) and \(S_{q}(t)\) under driving field modulation for (a) the open configuration and (b) the closed configuration. In the unmodulated scenario, quantum synchronization between subsystems are depicted by red dashed line (\(S_{q12}^{0}(t)\)), blue dashed line (\(S_{q23}^{0}(t)\)), and green dashed line (\(S_{q13}^{0}(t)\)), along with the magenta dashed line for the system-wide synchronization (\(S_{q}^{0}(t)\)). In the driving-field modulated scenario, they are represented by red solid line (\(S_{q12}(t)\)), blue solid line (\(S_{q23}(t)\)), and green solid line (\(S_{q13}(t)\)), with the system-wide synchronization (\(S_{q}(t)\)) shown by the magenta solid line. Relevant parameters are: \(\Delta _1 = \omega _{m1} = 0.995\), \(\Delta _2 = \omega _{m2} = 1.000\), \(\Delta _3 = \omega _{m3} = 1.005\), \(\lambda _{1,2} = 0.05\), \(\lambda _{3} = 0 (0.05)\) for open (closed) configuration, \(g = 0.005\), \(\gamma _m = 0.005\), \(\kappa = 0.15\), \(E = 100\), \(\Omega _D = 0(4)\), \(\eta _D = 0(0.8)\) in the unmodulated (driving-field modulated) case, \(\Omega _{C}=0\) and \(\eta _{C}=0\).