Analytical Control of Quantum Coherence: Markovian Revival via Basis Engineering and Exact Non-Markovian Criteria

The preservation of quantum coherence is besieged by a fundamental dogma: its revival necessitates non-Markovian memory effects from structured environments. This paradigm has constrained quantum control strategies and obscured simpler paths to coherence protection. Here, we shatter this belief by demonstrating unambiguous coherence revival even in strictly Markovian regimes, achieved solely through basis engineering in the $σ_x/σ_y$ bases. We establish a comprehensive analytical framework for predictive coherence control, delivering three universal design principles. First, we derive a minimum critical noise based frequency, $ω_{0}^{c} = 1.57/(0.4996 \cdot t_{\max})$, serving as a universal criterion for engineering non-Markovian dynamics over any interval $[0, t_{\max}]$. Crucially, we show that Markovian environments ($ω_0 < ω_0^c$) can exhibit coherence revival when the Zeeman energy satisfies $ω_k > π/(2t_{\max})$, decoupling revival from environmental memory. Furthermore, for non-Markovian environments, we provide exact conditions for periodic and complete revival: setting $ω_0 = n \cdot 6.285/t_{\max}$ guarantees revival in the $σ_z$ basis, while combining it with $ω_k = πω_0 / 6.285$ ensures perfect revival in the $σ_x/σ_y$ bases. Our results, validated by rigorous quantum simulations, provide a predictive toolkit for coherence control, offering immediate strategies for enhancing quantum memory, sensing, and error mitigation.