Quantum phase synchronization and non-Markovian relaxation dynamics of a driven two-level system in a hierarchical environment

Abstract

We investigate forced quantum phase synchronization and transient relaxation dynamics of a driven two-level system (TLS) coupled to a hierarchical structured environment arranged in a wheel-graph, where a primary cavity directly couples to the TLS and interacts uniformly with a closed ring of N nearest-neighbor-coupled auxiliary cavities. Employing the dressed-state secular representation and conditional non-Hermitian Hamiltonian in the dressed-state subspace, we characterize the phase-locking behavior using the Husimi Q-function and the synchronization measure S. In the steady-state regime, the TLS exhibits anti-phase locking relative to the coherent drive, governed by the dynamic competition between detuning-induced phase drift and drive-induced phase torque, which naturally delineates a quantum analogue of the classical Arnold tongue. Crucially, we identify an environmental dimensionality decoupling effect, wherein the steady-state phase synchronization remains invariant to the auxiliary reservoir size N. In contrast, the transient relaxation time toward synchronization displays a non-monotonic dependence on the primary–auxiliary coupling strength κ. Below a critical threshold κcrit, the auxiliary array acts as a broadband Markovian reservoir, accelerating phase locking via Purcell-like dissipation. Above this threshold, mode hybridization triggers non-Markovian information backflow, which prolongs transient oscillations and delays synchronization. Finally, we discuss an experimentally feasible implementation based on superconducting circuit quantum electrodynamics. These results provide insights into the mechanism governing steady-state quantum phase synchronization in structured environments and offer practical guidance for reservoir engineering in quantum technologies.