Lectures on Open Quantum Systems

TL;DR

This paper provides an introductory overview of the mathematical framework for open quantum systems, illustrating key concepts through the dissipative Jaynes-Cummings model and foundational theorems.

Contribution

It introduces the core mathematical structures of open quantum systems, including master equations, CPTP maps, and the GKSL theorem, with detailed examples and exercises.

Findings

Explicit derivation of irreversible dynamics from unitary evolution

Introduction of the master equation and spectral density concepts

Proofs of the Kraus, dilation, and GKSL theorems

Abstract

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.