有限维交换周期Lindbladian的Floquet分析
On the Floquet analysis of commutative periodic Lindbladians in finite dimension
摘要：
考虑矩阵代数$\mathbb{M}_d$上的马尔可夫主方程，该方程由标准（Kossakowski-Lindblad-Gorini-Sudarshan）形式的周期Lindbladian $L_t$控制.证明了在交换性的简化假设下，如果对任意时刻$t,t'\in\mathbb{R}_+$有$L_t L_{t'} = L_{t'}L_t$，则不保证所得到的完全正动力映射的Floquet规范形由同时全局马尔可夫映射给出.事实上，解的周期部分甚至被证明必然是非马尔可夫的。明确计算了代数$\mathbb{M}_2$中的两个例子：周期调制的随机量子比特动力学，是纯退相干方案的推广，和经典微扰的二能级系统，通过标准的梯形算子耦合到水库。
英文摘要：
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.
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