On the Floquet analysis of commutative periodic Lindbladians in finite dimension

We consider the Markovian Master Equation over matrix algebra $\mathbb{M}_d$, governed by periodic Lindbladian in standard (Kossakowski-Lindblad-Gorini-Sudarshan) form. It is shown that under simplifying assumption of commutativity, the Floquet normal form of resulting completely positive dynamical map is not guaranteed to be given by simultaneously globally Markovian maps. In fact, the periodic part of the solution is even shown to be necessarily non-Markovian. Two examples in algebra $\mathbb{M}_2$ are explicitly calculated: a periodically modulated random qubit dynamics, being a generalization of pure decoherence scheme, and a classically perturbed two-level system, coupled to reservoir via standard ladder operators.