Tarasov Vasily E
Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, 119991 Moscow, Russia.
Faculty "Information Technologies and Applied Mathematics", Moscow Aviation Institute (National Research University), 125993 Moscow, Russia.
Entropy (Basel). 2021 Apr 28;23(5):544. doi: 10.3390/e23050544.
In this paper, we proposed the exactly solvable model of non-Markovian dynamics of open quantum systems. This model describes open quantum systems with memory and periodic sequence of kicks by environment. To describe these systems, the Lindblad equation for quantum observable is generalized by taking into account power-law fading memory. Dynamics of open quantum systems with power-law memory are considered. The proposed generalized Lindblad equations describe non-Markovian quantum dynamics. The quantum dynamics with power-law memory are described by using integrations and differentiation of non-integer orders, as well as fractional calculus. An example of a quantum oscillator with linear friction and power-law memory is considered. In this paper, discrete-time quantum maps with memory, which are derived from generalized Lindblad equations without any approximations, are suggested. These maps exactly correspond to the generalized Lindblad equations, which are fractional differential equations with the Caputo derivatives of non-integer orders and periodic sequence of kicks that are represented by the Dirac delta-functions. The solution of these equations for coordinates and momenta are derived. The solutions of the generalized Lindblad equations for coordinate and momentum operators are obtained for open quantum systems with memory and kicks. Using these solutions, linear and nonlinear quantum discrete-time maps are derived.
在本文中，我们提出了开放量子系统非马尔可夫动力学的精确可解模型。该模型描述了具有记忆以及环境周期性脉冲序列的开放量子系统。为了描述这些系统，通过考虑幂律衰减记忆对量子可观测量的林德布拉德方程进行了推广。研究了具有幂律记忆的开放量子系统的动力学。所提出的广义林德布拉德方程描述了非马尔可夫量子动力学。具有幂律记忆的量子动力学通过非整数阶的积分和微分以及分数阶微积分来描述。考虑了一个具有线性摩擦和幂律记忆的量子振荡器的例子。在本文中，提出了从广义林德布拉德方程无任何近似推导出来的具有记忆的离散时间量子映射。这些映射精确对应于广义林德布拉德方程，它们是具有非整数阶卡普托导数的分数阶微分方程以及由狄拉克δ函数表示的周期性脉冲序列。推导了这些方程关于坐标和动量的解。对于具有记忆和脉冲的开放量子系统，得到了广义林德布拉德方程关于坐标和动量算符的解。利用这些解，推导了线性和非线性量子离散时间映射。
