We study a one-dimensional quasiperiodic tight-binding model with simultaneous off-diagonal (hopping) and diagonal (on-site) modulations. Using the inverse participation ratio and the wave-function centroid, we construct localization-delocalization phase diagrams for both equilibrium and nonequilibrium steady states. We analyze the robustness of the system's initial properties under dissipation and characterize dissipation-induced localization-delocalization transitions (and their reversals) in detail. Trace-distance dynamics provide evidence for a quantum Mpemba effect: states prepared farther from the steady state can relax faster than states initialized closer to it. We propose a starting-line hypothesis to explain the presence or absence of this effect in different parameter ranges of our selected system. Through the study of wave-packet transport under the same dissipative system, we analyze the applicable conditions of the starting-line hypothesis and summarize a way to design the Mpemba effect based on this. These results advance the understanding of steady-state phase transitions and relaxation dynamics in dissipatively driven quasiperiodic systems.

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• Anderson localization
• Cold atoms & matter waves
• Dissipative dynamics
• 1-dimensional systems
• Quasicrystals