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Proceedings of the Steklov Institute of Mathematics

Kinetic State and Emergence of Markovian Dynamics in Exactly Solvable Models of Open Quantum Systems

Anton Sergeevich Trushechkin

Published Mar 1, 2024• 2 citations
Publisher

Abstract

Abstract Typically, the theory of open quantum systems studies the dynamics of the reduced state (density operator) of the system. However, in the early stages of evolution, it is impossible to separate the reservoir dynamics from the system dynamics. Among the consequences of this fact is the violation of the positivity of solutions of some quantum master equations for the reduced density operator. In this paper we study the joint dynamics of the system and reservoir at an early stage of evolution and the pre-relaxation of the joint state to a so-called kinetic state. A kinetic state of the system and reservoir is characterized by the fact that it is completely determined by the reduced density operator of the system alone. Only after the formation of a kinetic state, it becomes possible to describe the evolution of the reduced density operator of the system in terms of a semigroup.

DOI
10.1134/s0081543824010188

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Abstract Typically, the theory of open quantum systems studies the dynamics of the reduced state (density operator) of the system. However, in the early stages of evolution, it is impossible to separate the reservoir dynamics from the system dynamics. Among the consequences of this fact is the violation of the positivity of solutions of some quantum master equations for the reduced density operator. In this paper we study the joint dynamics of the system and reservoir at an early stage of evolution and the pre-relaxation of the joint state to a so-called kinetic state. A kinetic state of the system and reservoir is characterized by the fact that it is completely determined by the reduced density operator of the system alone. Only after the formation of a kinetic state, it becomes possible to describe the evolution of the reduced density operator of the system in terms of a semigroup.
We investigate the mathematical structure of entropy-weighted quantum evolution within the framework of open quantum systems. Starting from a system–reservoir interaction model with weak entropy-coupled dynamics, we derive the reduced evolution of the system density matrix under the Born–Markov approximation. The resulting generatoris shown to be of Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) form, ensuring complete positivity and trace preservation. We further demonstrate that the associated dynamical semigroup admits both Stinespring dilation and Kraus operator representations, establishing that entropy-weighted reduced dynamics can be embedded into a fully unitary system environment framework. The entropy operator appearing in the effective description is identified with the positive-semidefinite dissipative structure induced by reservoir corre lation functions.The construction does not modify standard quantum mechanics but provides a parameterization of dissipative generators in terms of entropy-weighted couplings. This establishes a mathematically consis tent bridge between entropy-based dynamical formulations and conventional open quantum system theory.
Non-Markovian corrections to the Markovian quantum master equation of an open quantum system are investigated up to the second order of the interaction between the system of interest and a thermal reservoir. The concept of ``natural correlation'' is discussed. When the system is naturally correlated with a thermal reservoir, the time evolution of the reduced density matrix looks Markovian even in a short-time regime. If the total system was initially in an ``unnatural'' state, the natural correlation is established during the time evolution, and after that the time evolution becomes Markovian in a long-time regime. It is also shown that for a certain set of reduced density matrices, the naturally correlated state does not exist. If the initial reduced density matrix has no naturally correlated state, the time evolution is inevitably non-Markovian in a short-time regime.
These notes are a short introduction to the mathematical theory of open quantum systems. They are meant to serve as an entry point into a broad research area which has applications across the quantum sciences dealing with systems subjected to external noise. The guiding idea is to let the key structures of the theory emerge from a concrete model. By working through the dissipative Jaynes-Cummings model the reader will dis- cover explicitly how irreversible dynamics arises from a unitary system-reservoir evolution. The notions of the continuous mode limit, correlation functions, spectral density appear in a natural manner and lead to the evolution equation of the open system in form of a master equation. This sets the stage for the more general analysis of completely positive, trace preserving (CPTP) maps and the study of quantum dynamical semigroups. We motivate and prove the Kraus representation theorem, the dilation theorem and the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) theorem. Working through the exercises (for which full solutions are supplied) will reinforce the ideas introduced in the main text.
The non-Markovian master equation for open quantum systems is obtained by generalization of the standard Zwanzig-Nakajima (ZN) projection technique. To this end, a coupled chain of equations is written for the reduced density matrices of the bath ϱB(t) and of the system ϱS(t). A formal solution of the equation for ϱB(t) in the 2nd approximation in interaction yields a specific extra term related to the intrinsic bath dynamics. This term is nonlinear in the reduced density matrix ϱS(t), and vanishes in the Markovian limit. To verify the consistence and robustness of our approach, we apply the generalized ZN projection scheme to a simple dephasing model. We study the obtained kinetic equation both in the Markovian approximation and beyond it (for the term related to the intrinsic bath dynamics) and compare the results with the exact ones.
These notes are a short introduction to the mathematical theory of open quantum systems. They are meant to serve as an entry point into a broad research area that has applications across the quantum sciences dealing with systems subjected to external noise. The idea is to let the key structures of the theory emerge from a concrete model. By working through the dissipative Jaynes–Cummings model the reader will discover explicitly how irreversible dynamics arises from a unitary system-reservoir evolution. The notions of the continuous mode limit, correlation functions, spectral density appear in a natural manner and lead to the evolution equation of the open system in form of a master equation. This sets the stage for the more general analysis of completely positive, trace preserving maps and the study of quantum dynamical semigroups. We motivate and prove the Kraus representation theorem, the dilation theorem and the Gorini–Kossakowski–Sudarshan–Lindblad theorem. Working through the exercises (for which full solutions are supplied) will reinforce the ideas introduced in the main text.