© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2604.12976

Hamiltonian Chaos

Steven Tomsovic

quant-phnlin.CDphysics.class-phSubmitted Apr 14, 2026• Announced Apr 15, 2026• Revised Oct 27, 2026 (v2)

v1 · Apr 14, 2026 v2 · Oct 27, 2026

Abstract

Through semiclassical methods the subject of quantum chaos motivates and depends on Hamiltonian chaos research. Presented here is a selection of some of the most important Hamiltonian chaos topics that in this way get directly related to any of a variety of quantum chaos research problems. The overall emphasis is on intuitive explanations and illustrations of various ideas with the references containing more mathematically rigorous expositions. The chapter begins with a description of various useful theoretical and computational tools of chaos research, e.g.~surfaces of section, paradigms of chaos, stability analysis, and symbolic dynamics... This is followed by discussions regarding the geometry of chaos, how chaotic systems respond to perturbations, and the complexification of Hamiltonian dynamics. Most of the contents concern topics that are long established and well known by now. The exceptions involve complications that arise from the complexification of phase space variables and time. There are relatively recent results summarized regarding the appropriate branches of determinant phases and how to distinguish complex trajectories that are associated with classically allowed processes versus those associated with non-allowed ones.

Comments: Chapter for the Quantum Chaos volume in 'Comprehensive Quantum Mechanics', to be published by Elsevier (Main editor: R. B. Mann; volume editors: S. Gnutzmann and K. Życzkowski),

Related papers

What We Talk About When We Talk About Dissipative Quantum Chaos

Lucas Sá, Pedro Ribeiro, Sergey Denisov

Dissipative quantum chaos is an emerging theory that is expected to extend the ideas, concepts, and methodology of conventional Hamiltonian quantum chaos from coherent evolution to open quantum dynamics. The new theory should provide a set of tools to distinguish chaotic open quantum systems from integrable ones, as well as quantitative measures of their chaoticity (or, conversely, integrability). The foundations of this theory were laid in the late 1980s, and from the very start it was clear that, like its Hamiltonian predecessor, it had to be based on the spectral properties of the operators governing open quantum evolution. After these first steps, the field remained relatively quiet for many years and it is only over the last decade that the development of dissipative quantum chaos has received a strong boost, as confirmed by a large number of publications on this topic and, very recently, the first experiments performed to test its theoretical predictions. In this chapter, we review these recent developments and outline the basic foundations of dissipative quantum chaos.

quant-pharXiv:2605.21628•May 22, 2026•0 citations

Chaos and Quantum Tunneling

Akira Shudo

In generic Hamiltonian systems that are neither completely integrable nor fully chaotic, phase space consists of a mixture of regular and chaotic components. In classical dynamics, transitions between different invariant sets in phase space are strictly forbidden, and these sets act as dynamical barriers to one another. In quantum mechanics, in contrast, wave effects allow transitions through such dynamical barriers. This process, known as dynamical tunneling, refers to penetration through dynamical barriers in phase space and was first recognized in the early 1980s. Since then, various aspects of dynamical tunneling have been elucidated, significantly advancing our understanding of such a novel quantum phenomenon. In this article, we provide an overview of several phenomenological perspectives of dynamical tunneling, including chaos-assisted and resonance-assisted tunneling, and also introduce approaches based on classical mechanics extended into the complex domain. In particular, we seek to clarify what is meant by the common claim that "chaos leads to an enhancement of the tunneling probability", which is often made when dynamical tunneling is dressed. We discuss what regime this refers to and, if such an enhancement occurs, what its likely origin is.

nlin.CDCross-listedarXiv:2604.12926•Apr 15, 2026

Synopsis

Belal Ehsan Baaquie

This book studies the mathematical aspect of path integrals and Hamiltonians – which emerge from the formulation of quantum mechanics. The theoretical framework of quantum mechanics provides the mathematical tools for studying both quantum indeterminacy and classical randomness . Many problems arising in quantum mechanics as well as in vastly different fields such as finance and economics can be addressed by the mathematics of quantum mechanics, or quantum mathematics in short. All the topics and subjects in the various chapters have been specifically chosen to illustrate the structure of quantum mathematics, and are not tied to any specific discipline, be it quantum mechanics or stochastic systems. The book is divided into the following six parts, in accordance with the Chapter dependency flowchart given below. Part one addresses the Fundamental principles of path integrals and (Hamiltonian) operators and consists of five chapters. Chapter 2 is on the Mathematical structure of quantum mechanics and introduces the mathematical framework that emerges from the quantum principle. Chapters 3 to 6 discuss the mathematical pillars of quantum mathematics, starting from the Feynman path integral, summarizing Hamiltonian mechanics and introducing path integral quantization. Part two is on Stochastic processes . Stochastic systems are dissipative and are shown to be effectively modeled by the path integral. Chapter 7 is focused on the application of quantum mathematics to classical random systems and to stochastic processes.

Cambridge University Press eBooks•Mar 27, 2014•0 citations

Entanglement dynamics and chaos in long-range quantum systems

Silvia Pappalardi

Over the past twenty years, experimental and technological progresses have motivated a renewed attention to the study of non-equilibrium isolated many-body systems, leading to a relatively well-established paradigm in the case of local Hamiltonians. In the present thesis, I have used quantum information theoretical tools to study out-of-equilibrium dynamics, with particular attention on long-range interacting many-body systems. I have explored the dynamics of bipartite and multipartite entanglement in connection to chaos and scrambling in various long-range (clean and disordered) models. The results contained in this thesis contribute to establishing semi-classical tools as powerful techniques for the description of the quantum information spreading in long-range systems. I have further considered a different, yet connected question, concerning the multipartite entanglement structure of chaotic eigenstates and its generic evolution.

•Sep 29, 2020•2 citations

Chaotic Dynamics and Quantum Transport

Andrey R. Kolovsky

This chapter gives an overview of transport problems where chaotic dynamics of the system plays a crucial role. We begin with single-particle transport problems and then come to conservative and then dissipative systems of identical particles, which follows the historical way of developing the theory of Quantum Chaos over the past 40 years. We also include brief descriptions of key laboratory experiments on the discussed transport problems.

nlin.CDCross-listedarXiv:2604.12409•Apr 15, 2026•0 citations

1 Dissipative Chaos in Quantum Distributions

T. V. Gevorgyan, Suren B. Manvelyan

We discuss some problems of dissipative chaos for open quantum systems in the framework of semiclassical and quantum distributions.For this goal, we propose a driven nonlinear oscillator with time-dependent coefficients, i.e. with time-dependent Kerr-nonlinearity and time-modulated driving field.This model showing both regular and chaotic dynamics in the classical limit is realized in several experimental schemes.Quantum dissipative chaos is analyzed on the base of numerical method of quantum trajectories.Three quantities are studied: the Wigner function of oscillatory mode from the point of view of quantum-assemble theory and both semiclassical Poincaré section and quantum Poincaré section calculated on a single quantum trajectory.The comparatively analysis of these distributions for various operational chaotic regimes of the models is performed, as well as scaling invariance in dissipative chaos and quantum interference effects assisted by chaos are discussed.

•Aug 14, 2016•2 citations

Through semiclassical methods the subject of quantum chaos motivates and depends on Hamiltonian chaos research. Presented here is a selection of some of the most important Hamiltonian chaos topics that in this way get directly related to any of a variety of quantum chaos research problems. The overall emphasis is on intuitive explanations and illustrations of various ideas with the references containing more mathematically rigorous expositions. The chapter begins with a description of various useful theoretical and computational tools of chaos research, e.g.~surfaces of section, paradigms of chaos, stability analysis, and symbolic dynamics... This is followed by discussions regarding the geometry of chaos, how chaotic systems respond to perturbations, and the complexification of Hamiltonian dynamics. Most of the contents concern topics that are long established and well known by now. The exceptions involve complications that arise from the complexification of phase space variables and time. There are relatively recent results summarized regarding the appropriate branches of determinant phases and how to distinguish complex trajectories that are associated with classically allowed processes versus those associated with non-allowed ones.

Dissipative quantum chaos is an emerging theory that is expected to extend the ideas, concepts, and methodology of conventional Hamiltonian quantum chaos from coherent evolution to open quantum dynamics. The new theory should provide a set of tools to distinguish chaotic open quantum systems from integrable ones, as well as quantitative measures of their chaoticity (or, conversely, integrability). The foundations of this theory were laid in the late 1980s, and from the very start it was clear that, like its Hamiltonian predecessor, it had to be based on the spectral properties of the operators governing open quantum evolution. After these first steps, the field remained relatively quiet for many years and it is only over the last decade that the development of dissipative quantum chaos has received a strong boost, as confirmed by a large number of publications on this topic and, very recently, the first experiments performed to test its theoretical predictions. In this chapter, we review these recent developments and outline the basic foundations of dissipative quantum chaos.

In generic Hamiltonian systems that are neither completely integrable nor fully chaotic, phase space consists of a mixture of regular and chaotic components. In classical dynamics, transitions between different invariant sets in phase space are strictly forbidden, and these sets act as dynamical barriers to one another. In quantum mechanics, in contrast, wave effects allow transitions through such dynamical barriers. This process, known as dynamical tunneling, refers to penetration through dynamical barriers in phase space and was first recognized in the early 1980s. Since then, various aspects of dynamical tunneling have been elucidated, significantly advancing our understanding of such a novel quantum phenomenon. In this article, we provide an overview of several phenomenological perspectives of dynamical tunneling, including chaos-assisted and resonance-assisted tunneling, and also introduce approaches based on classical mechanics extended into the complex domain. In particular, we seek to clarify what is meant by the common claim that "chaos leads to an enhancement of the tunneling probability", which is often made when dynamical tunneling is dressed. We discuss what regime this refers to and, if such an enhancement occurs, what its likely origin is.

This book studies the mathematical aspect of path integrals and Hamiltonians – which emerge from the formulation of quantum mechanics. The theoretical framework of quantum mechanics provides the mathematical tools for studying both quantum indeterminacy and classical randomness . Many problems arising in quantum mechanics as well as in vastly different fields such as finance and economics can be addressed by the mathematics of quantum mechanics, or quantum mathematics in short. All the topics and subjects in the various chapters have been specifically chosen to illustrate the structure of quantum mathematics, and are not tied to any specific discipline, be it quantum mechanics or stochastic systems. The book is divided into the following six parts, in accordance with the Chapter dependency flowchart given below. Part one addresses the Fundamental principles of path integrals and (Hamiltonian) operators and consists of five chapters. Chapter 2 is on the Mathematical structure of quantum mechanics and introduces the mathematical framework that emerges from the quantum principle. Chapters 3 to 6 discuss the mathematical pillars of quantum mathematics, starting from the Feynman path integral, summarizing Hamiltonian mechanics and introducing path integral quantization. Part two is on Stochastic processes . Stochastic systems are dissipative and are shown to be effectively modeled by the path integral. Chapter 7 is focused on the application of quantum mathematics to classical random systems and to stochastic processes.

Over the past twenty years, experimental and technological progresses have motivated a renewed attention to the study of non-equilibrium isolated many-body systems, leading to a relatively well-established paradigm in the case of local Hamiltonians. In the present thesis, I have used quantum information theoretical tools to study out-of-equilibrium dynamics, with particular attention on long-range interacting many-body systems. I have explored the dynamics of bipartite and multipartite entanglement in connection to chaos and scrambling in various long-range (clean and disordered) models. The results contained in this thesis contribute to establishing semi-classical tools as powerful techniques for the description of the quantum information spreading in long-range systems. I have further considered a different, yet connected question, concerning the multipartite entanglement structure of chaotic eigenstates and its generic evolution.

We discuss some problems of dissipative chaos for open quantum systems in the framework of semiclassical and quantum distributions.For this goal, we propose a driven nonlinear oscillator with time-dependent coefficients, i.e. with time-dependent Kerr-nonlinearity and time-modulated driving field.This model showing both regular and chaotic dynamics in the classical limit is realized in several experimental schemes.Quantum dissipative chaos is analyzed on the base of numerical method of quantum trajectories.Three quantities are studied: the Wigner function of oscillatory mode from the point of view of quantum-assemble theory and both semiclassical Poincaré section and quantum Poincaré section calculated on a single quantum trajectory.The comparatively analysis of these distributions for various operational chaotic regimes of the models is performed, as well as scaling invariance in dissipative chaos and quantum interference effects assisted by chaos are discussed.