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arXiv:1901.03848 · Journal of Physics A Mathematical and Theoretical

Supersymmetrization of the Franke–Gorini–Kossakowski–Lindblad–Sudarshan equation

A A Andrianov, M V Ioffe, O O Novikov

Published Sep 10, 2019• 6 citations
arXivHTMLPDF

Abstract

Abstract We investigate departures from the Hamiltonian dynamics such as the quantum decoherence in superpartner quantum systems due to their interaction with the environment. The effective description is given by the Franke–Gorini–Kossakowski–Lindblad–Sudarshan (FGKLS) equation. The prescription for supersymmetrization of this equation is elaborated upon and pairs of related isospectral quantum systems are built. We present some instructive examples of such a construction.

DOI
10.1088/1751-8121/ab4338

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Abstract We investigate departures from the Hamiltonian dynamics such as the quantum decoherence in superpartner quantum systems due to their interaction with the environment. The effective description is given by the Franke–Gorini–Kossakowski–Lindblad–Sudarshan (FGKLS) equation. The prescription for supersymmetrization of this equation is elaborated upon and pairs of related isospectral quantum systems are built. We present some instructive examples of such a construction.
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.
An open quantum system can be described by a master equation, of which one of the most popular is the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation. We revisit description of driven-dissipative quantum systems focusing on the appropriate choice of the system's basis and the respective transformations. We consider the GKSL equation in different bases and calculate the dynamics for a qubit and for a qudit. An appropriate choice of the basis is a fundamental problem for theoretical consideration of open quantum systems and provides an opportunity to obtain the desired evolution in practice.
The year 2026 marks the fiftieth anniversary of the two seminal papers in which Gorini, Kossakowski, and Sudarshan, and independently Lindblad, characterized the generators of completely positive quantum dynamical semigroups. We revisit the conceptual and historical route leading to the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) master equation and examine several of its early developments and consequences. Our aim is to show how several initially separate mathematical and physical ideas converged in the GKLS structure theorem, imme- diately opening the way to manifold consequences for the description of dissipative quantum phenomena and how that theorem, fifty years later, remains both a foundational result and an active framework for open quantum dynamics.
Abstract Chapter 22 investigates the dynamics of open quantum systems that interact with their environment. We discuss the time-evolution of the joint system in terms of a time-ordered exponential of an integral over the Hamiltonian and determine a similar formulation for a Liouvillian super-operator. Based on these ideas we develop the notion of dynamical maps and apply the Born-Markov approximation to arrive at a dynamical semi-group and general Markovian master equations. We then turn to a specific example of such an equation of motion for the density operator: the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) equation which captures both dissipative dynamics and decoherence using Lindblad operators. We employ the GKLS equation to model the absorption, and spontaneous as well as stimulated emission of radiation by a two-level atom. We then discuss the Jaynes-Cummings model that describes the interaction of such an atom with a single mode of the electromagnetic field in a cavity
A longstanding tool to characterize the evolution of open Markovian quantum systems is the GKSL (Gorini-Kossakowski-Sudarshan-Lindblad) master equation. However, in some cases, open quantum systems can be effectively described with non-Hermitian Hamiltonians, which have attracted great interest in the last twenty years due to a number of unconventional properties, such as the appearance of exceptional points. Here, we present a short review of these two different approaches aiming in particular to highlight their relation and illustrate different ways of connecting non-Hermitian Hamiltonian to a GKSL master equation for the full density matrix.