The Franke-Gorini-Kossakowski-Lindblad-Sudarshan (FGKLS) Equation for Two-Dimensional Systems

TL;DR

This paper thoroughly analyzes the FGKLS equation for two-dimensional quantum systems, deriving fixed states, solutions, and physical conditions, and clarifies the equation's applicability and the absence of oscillatory solutions.

Contribution

It provides a comprehensive solution to the FGKLS equation in 2D systems, including fixed points, physical validity, and behavior under weak interactions, filling gaps in understanding open quantum system dynamics.

Findings

Identified fixed states (pointers) of the evolution.

Derived a general solution to the FGKLS equation.

Proved the non-existence of oscillating solutions.

Abstract

Open quantum systems are, in general, described by a density matrix that is evolving under transformations belonging to a dynamical semigroup. They can obey the Franke-Gorini-Kossakowski-Lindblad-Sudarshan (FGKLS) equation. We exhaustively study the case of a Hilbert space of dimension $2$. First, we find final fixed states (called pointers) of an evolution of an open system, and we then obtain a general solution to the FGKLS equation and confirm that it converges to a pointer. After this, we check that the solution has physical meaning, i.e., it is Hermitian, positive and has trace equal to $1$, and find a moment of time starting from which the FGKLS equation can be used - the range of applicability of the semigroup symmetry. Next, we study the behavior of a solution for a weak interaction with an environment and make a distinction between interacting and non-interacting cases. Finally, we prove that there cannot exist oscillating solutions to the FGKLS equation, which would resemble the behavior of a closed quantum system.