Interplay between Markovianity and progressive quenching

Progressive quenching (PQ) is a process in which we sequentially fix a system's degrees of freedom, which would otherwise evolve according to their stochastic dynamics. Previous studies have discovered what we refer to as the hidden martingale property in PQ. Here we first attribute this martingale property to the canonicity of the two-layer ensemble comprising quenched and thermal ensembles and demonstrate that the Markovian property, coupled with the detailed balance (DB) of the evolution dynamics, underpins this canonicity. We then expand the PQ to the Markovian dynamics on the transition network where the DB is not required. Additionally, we examine the PQ of the systems that evolve through non-Markovian dynamics between consecutive quenching. When non-Markovian dynamics ensure a trajectory-wise DB, such as in an equilibrium spin system with a hidden part, the PQ can occasionally maintain the canonical structure of the overall statistical ensemble but not always. Last, we analytically and numerically investigate the PQ of a non-Markovian spin system with delayed interaction and illustrate how the reduction of spin correlations due to the delay can be compensated by the PQ.