On the Floquet analysis of commutative periodic Lindbladians in finite dimension

TL;DR

This paper analyzes the Floquet properties of commutative periodic Lindbladians in finite dimensions, revealing that the periodic part of the solution can be inherently non-Markovian, with explicit examples provided.

Contribution

It demonstrates that commutative periodic Lindbladians can have non-Markovian Floquet components, challenging assumptions about Markovianity in such systems.

Findings

Floquet normal form may not be globally Markovian.

Periodic part of the solution is necessarily non-Markovian.

Explicit examples in two-level systems illustrate these phenomena.

Abstract

We consider the Markovian Master Equation over matrix algebra $\mathbb{M}_d$, governed by periodic Lindbladian $L_t$ in standard (Kossakowski-Lindblad-Gorini-Sudarshan) form. It is shown that under simplifying assumption of commutativity, i.e. if $L_t L_{t'} = L_{t'}L_t$ for any moments of time $t,t'\in\mathbb{R}_+$, 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.