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arXiv:0809.4403

Introduction to dissipation and decoherence in quantum systems

Florian Marquardt, Annett Püttmann

Published Sep 25, 2008• 12 citations
arXivHTMLPDF

Abstract

These lecture notes address an audience of physicists or mathematicians who have been exposed to a first course in quantum mechanics. We start with a brief discussion of the general "system-bath" paradigm of quantum dissipative systems, analyze in some detail the simplest example of "pure dephasing" of a two-level system, and review the basic concept of the density matrix. We then treat the general dissipative time-evolution, introducing completely positive maps, their relation to entanglement theory, and their Kraus decomposition. Restricting ourselves to Markovian evolution, we discuss the Lindblad form of master equations. The notes conclude with an overview of topics of current interest that go beyond Lindblad Markov master equations. (These notes were prepared for lectures delivered by F. Marquardt in October 2007 at the Langeoog workshop of the SFB/TR 12, "Symmetries and Universality in Mesoscopic Systems")

DOI
10.48550/arxiv.0809.4403

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These lecture notes address an audience of physicists or mathematicians who have been exposed to a first course in quantum mechanics. We start with a brief discussion of the general "system-bath" paradigm of quantum dissipative systems, analyze in some detail the simplest example of "pure dephasing" of a two-level system, and review the basic concept of the density matrix. We then treat the general dissipative time-evolution, introducing completely positive maps, their relation to entanglement theory, and their Kraus decomposition. Restricting ourselves to Markovian evolution, we discuss the Lindblad form of master equations. The notes conclude with an overview of topics of current interest that go beyond Lindblad Markov master equations. (These notes were prepared for lectures delivered by F. Marquardt in October 2007 at the Langeoog workshop of the SFB/TR 12, "Symmetries and Universality in Mesoscopic Systems")
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.
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.
This text is a short introduction to the physics of driven-dissipative many-body systems, focusing on a few selected topics. Beyond its more ``historical'' interest in the study of atomic physics and quantum optics, presently the modeling and studying dissipative phenomena in open quantum systems is pivotal to understanding quantum hardware platforms. While the lack of a thermodynamic potential for these out-of-equilibrium open systems makes it theoretically challenging to investigate their physics, at the same time it allows going beyond the thermodynamic paradigms and investigating new and exotic phenomena. We will focus on one of the simplest, yet most effective, descriptions of open quantum systems, namely the (Gorini-Kossakowski-Sudarshan-) Lindblad master equation. This phenomenological approach describes quantum systems that weakly interact with their surrounding environment. Although many of the results derived below will apply to any quantum system, we will focus in particular on bosonic/spin systems.
1. First Lecture: Basics 1.1 Physical Derivation of the Master Equation 1.2 Some Simple Implications 1.3 Steady State 1.4 Action to the Left 2. Second Lecture: Eigenvalues and Eigenvectors of L 2.1 A Simple Case First 2.2 The General Case 3 Third Lecture: Completeness of the Damping Bases 3.1 Phase Space Functions 3.2 Completeness of the Eigenvectors of L 3.3 Positivity Conservation 3.4 Lindblad Form of Liouville Operators 4 Fourth Lecture: Quantum-Optical Applications 4.1 Periodically Driven Damped Oscillator 4.2 Conditional and Unconditional Evolution 4.3 Physical Signicance of Statistical Operators 5 Fifth Lecture: Statistics of Detected Atoms 5.1 Correlation Functions 5.2 Waiting Time Statistics 5.3 Counting Statistics
Dissipation and decoherence, and the evolution from pure to mixed states in quantum physics, are handled through master equations for the density matrix. Master equations such as the Lindblad equation preserve the trace of this matrix. Viewing them as first-order time-dependent operator equations for the elements of the density matrix, a unitary integration procedure can be adapted to solve for these elements. A simple model for decoherence preserves the Hermiticity of the density matrix. A single, classical Riccati equation is the only one requiring numerical handling to obtain a full solution of the quantum evolution. The procedure is general, valid for any number of levels, but is illustrated here for a three-level system with two driving fields. For various choices of the initial state, we study the evolution of the system as a function of the amplitudes, relative frequencies, and phases of the driven fields and of the strength of the decoherence. The monotonic growth of the entropy is followed as the system evolves from a pure to a mixed state. An example is provided by the n=3n=3n=3 states of the hydrogen atom in a time-dependent electric field, such degenerate manifolds affording an analytical solution.