Continuous time quasi-crystal and its transition to continuous time crystal in coupled Rydberg atomic chains

We investigate the nonlinear dynamics of two coupled driven–dissipative Rydberg atomic chains, each of which formed a previously reported continuous time crystal phase. By tuning the inter-chain interaction, the two time-crystalline subsystems with distinct intrinsic oscillation frequencies will naturally mix, and support luxuriant stable and collective oscillatory behavior. We identify a dynamical phase characterized by quasi-periodic oscillations with multiple incommensurate frequencies, corresponding to a continuous-time quasi-crystalline order. In addition, we identify a phase transition from a time quasi-crystalline phase to a time-crystalline phase at sufficiently strong inter-chain coupling. This transition can be understood as the evolution from partially synchronized oscillations between the two chains to fully synchronized dynamics. From a mathematical perspective, we confirm that the transition is governed by either a saddle-node or a torus bifurcation, depending on the separation between the two original oscillation frequencies. These theoretical results provide guidance for exploring complex periodic dynamics in Rydberg atomic platforms.