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Lectures on Open Quantum Systems

Marco Merkli, Ángel Neira

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.

Lectures on Open Quantum Systems

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.
Paper Structure (12 sections, 3 theorems, 114 equations, 2 figures)

This paper contains 12 sections, 3 theorems, 114 equations, 2 figures.

Table of Contents

  1. Formalism of quantum theory
  2. Open quantum systems by example: The open Jaynes-Cummings model
  3. CPTP maps
  4. Quantum dynamical semigroups
  5. Solutions to the exercises
  6. Solution to Exercise \ref{['ex1']}
  7. Solution to Exercise \ref{['ex2']}
  8. Solution to Exercise \ref{['exercise3']}
  9. Solution to Exercise \ref{['exercise4']}
  10. Solution to Exercise \ref{['exercise5']}
  11. Solution to Exercise \ref{['exercise6']}
  12. Solution to Exercise \ref{['exercise7']}

Key Result

Theorem 1

Suppose $\Phi$ is CPTP on $\mathcal{B}({\mathcal{H}})$, $\dim{\mathcal{H}}=d<\infty$. Then there are operators $K_\alpha\in\mathcal{B}({\mathcal{H}})$, $\alpha=1,\ldots,d^2$ such that $\sum_\alpha K^\dag_\alpha K_\alpha=\rm 1 l$ and for all $X\in\mathcal{B}({\mathcal{H}})$, Conversely, if $\Phi$ is a map on $\mathcal{B}({\mathcal{H}})$ having this form for some operators $K_\alpha$ with $\sum_\al

Figures (2)

  • Figure 1: Lorentzian spectral density \ref{['ny4']} for different values of $\Gamma$.
  • Figure 2: Contour of integration in the complex plane.

Theorems & Definitions (3)

  • Theorem 1: Kraus representation theorem
  • Theorem 2: Open systems representation of CPTP maps | dilation theorem
  • Theorem 3: Gorini-Kossakowski-Sudarshan-Lindblad