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arXiv:2609.13863

Open Quantum Systems

Erik Torrontegui, Ricardo Puebla

quant-phcond-mat.stat-mechSubmitted Sep 12, 2026• Announced Sep 15, 2026• 0 citations
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Abstract

Open quantum systems describe quantum systems that interact unavoidably with their environments. These notes introduce the physical origins of decoherence and dissipation in realistic settings, developing the theoretical framework of reduced dynamics and quantum master equations within the connection to thermodynamics in the quantum regime. Additionally, a set of relevant exercises are proposed to reinforce the acquisition of knowledge. The text equips readers with the tools to model and analyze open quantum dynamics.

Comments
Lecture Notes (MSc level); 105pages including solutions to exercises. Comments and feedback always welcome

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Open quantum systems describe quantum systems that interact unavoidably with their environments. These notes introduce the physical origins of decoherence and dissipation in realistic settings, developing the theoretical framework of reduced dynamics and quantum master equations within the connection to thermodynamics in the quantum regime. Additionally, a set of relevant exercises are proposed to reinforce the acquisition of knowledge. The text equips readers with the tools to model and analyze open quantum dynamics.
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 of dissipative phenomena in open quantum systems are pivotal to understanding quantum hardware platforms. Although 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 one to go beyond the thermodynamic paradigms and investigate new and exotic phenomena. We will focus on one of the simplest, yet most effective, descrip-tions of open quantum systems, namely the (Gorini-Kossakowski-Sudarshan-) Lindblad master equation. This phenomenological approach has been devel-oped to describe quantum systems that weakly interact (Born approximation) with a memoryless (Markovian) surrounding environment. However, it is often a remarkably simple and predictive tool to characterize the essential features of an open quantum system. Although many of the results derived below will ap-ply to any quantum system, we will focus in particular on bosonic/spin 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.
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.
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.
We develop a general theory describing the thermodynamical behavior of open quantum systems coupled to thermal baths beyond perturbation theory. Our approach is based on the exact time-local quantum master equation for the reduced open system states, and on a principle of minimal dissipation. This principle leads to a unique prescription for the decomposition of the master equation into a Hamiltonian part representing coherent time evolution and a dissipator part describing dissipation and decoherence. Employing this decomposition we demonstrate how to define work, heat and entropy production, formulate the first and second law of thermodynamics, and establish the connection between violations of the second law and quantum non-Markovianity.
We develop a general theory describing the thermodynamical behavior of open quantum systems coupled to thermal baths beyond perturbation theory. Our approach is based on the exact time-local quantum master equation for the reduced open-system states, and on a principle of minimal dissipation. This principle leads to a unique prescription for the decomposition of the master equation into a Hamiltonian part representing coherent time evolution and a dissipator part describing dissipation and decoherence. Employing this decomposition we demonstrate how to define work, heat, and entropy production, formulate the first and second laws of thermodynamics, and establish the connection between violations of the second law and quantum non-Markovianity.