Boundary time crystals are a class of exotic dissipative quantum phases that spontaneously break continuous time-translation symmetry in the thermodynamic limit of open quantum systems. In finite-size systems, the long-time evolution of boundary time crystals exhibits decaying oscillations that cannot be captured by widely used mean-field theory. To address this issue, we develop an effective approach called the stroboscopic rotating-wave approximation, which provides a well approximated state for the long-time evolution of boundary time crystals in the strongly driven regime. In this approach, the order parameter exhibits both a long-time decaying envelope governed by an effective Lindblad superoperator and short-time oscillations dominated by a reduced quantum dynamical semigroup. Our results reveal that the competition between dephasing processes in three directions induces persistent oscillations, marking the emergence of the boundary-time-crystal phase. We obtain analytical expressions for the steady-state density operator, the oscillation period, and the decay rate of the order parameter when the coherent energy splitting exceeds the dissipation rate. Our work provides a beyond-mean-field tool for studying the dynamics of periodically driven open quantum systems and understanding the formation of time crystals.

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• Open quantum systems & decoherence
• Time crystals