Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

CRC Press eBooks

Particle in Oscillator Environment

Jørgen Rammer

Published May 4, 2018• 0 citations
Publisher

Abstract

This chapter considers the interaction between a single degree of freedom and an environment. It assumes that the environment can be represented by a set of harmonic oscillators, and that the interaction between the system of interest and the environment is linear in the coupling to the oscillator degrees of freedom. There are systems where the linear coupling to the environment degrees of freedom is exact, such as the fundamental interaction of a charged particle interacting with the electromagnetic field, electron-photon interaction. Furthermore, many situations are adequately described by such an interaction, because it is sufficient to consider each environment degree of freedom only weakly perturbed. The chapter examines the time evolution of the reduced density matrix on integral form, and described the diagrammatic expansion in the spatial representation. It explores the kinetic approach for studying the motion of a particle interacting with an oscillator environment.

DOI
10.1201/9780429502835-6

Related papers

Reduced density matrices of oscillator systems

V. V. Dodonov, Olga V. Man’ko, Vladimir I. Man’ko

We study the evolution of an oscillator interacting via the most general bilinear coupling (with time-independent coefficients) with an ``environment'' consisting of a set of other harmonic oscillators. We are mainly interested in a possibility of using the Fokker-Planck equation to describe this evolution. Studying different interaction Hamiltonians, we show that unambiguous reduction to the Fokker-Planck equation is possible only within the framework of the so called rotating-wave approximation. As special cases we consider in detail the evolution of two coupled oscillators and relaxation of a charged oscillator in a uniform magnetic field.

arXiv:quant-ph/0210030•Oct 4, 2002•2 citations

Dissipation in Lagrangian Formalism

Yoon Ayanajee Peiris Thelge

This thesis examines dissipative forces in physical systems in both microscopic and quantum level. The system-environment interaction is modeled by assuming that the environment is composed of simple harmonic oscillators and that there is a linear interaction between these oscillators and the system. The classical case is solved by determining the retarded Green’s function, which yields the equation of motion of the system. For quantum dissipation, the Lindblad master equation is used to describe the time evolution of the reduced density matrix of the system, and the propagator is investigated to identify important parameters, including the term of Newton's friction force.

•Apr 25, 2023•0 citations

Open Quantum Electrodynamics

Heinz‐Peter Breuer, Francesco Petruccione

Abstract In quantum electrodynamics, the matter degrees of freedom are coupled to the radiation field through a local, gauge-invariant interaction density. Due to the linear structure of this coupling, the problem of constructing a complete formal representation of the reduced matter dynamics can be solved exactly if the electromagnetic radiation field is initially in a Gaussian state. This chapter combines field theoretical methods with superoperator and influence functional techniques to derive an exact, relativistic representation for the reduced density matrix of the matter degrees of freedom which completely describes the influence of the electromagnetic radiation field on the matter dynamics. Several applications are treated, such as the suppression of the quantum coherence of charged particles caused by the emission of bremsstrahlung and the decoherence of many-particle states.

•Jan 25, 2007•4 citations

Minimal irreversible quantum mechanics. The decay of unstable states

Arbó, Diego G., Castagnino, Mario A., Gaioli, Fabián H. +1 more

Brownian motion is modelled by a harmonic oscillator (Brownian particle) interacting with a continuous set of uncoupled harmonic oscillators. The interaction is linear in the coordinates and the momenta. The model has an analytical solution that is used to study the time evolution of the reduced density operator. It is derived in a closed form, in the one-particle sector of the model. The irreversible behavior of the Brownian particle is described by a reduced density matrix.

•Jan 1, 2000•15 citations

Quantum Dynamics of a Particle in a Fermionic Environment

Per Hedegård, A. O. Caldeira

The reduced density operator of a particle coupled to a fermionic environment (metal electrons) is written in terms of a Feynman-Vernon influence functional. Two different approaches are used in order to achieve this goal. Firstly, the environmental interacting electron gas is treated in RPA and the coupling is correctly accounted for up to second order in perturbation theory. Then the local and tight-binding models are studied. For the latter a second approach for obtaining the influence functional is used in the narrow-band limit. It is found that the popular representation of the environment in terms of a single set of harmonic oscillators is not feasible unless the particle is confined to move in a local region of radial extension r ≃ k F -1 .

Physica Scripta•May 1, 1987•47 citations

The Role of the Environment in Molecular Systems

Dmitri S. Kilin

The work is devoted to the investigation of the influence of a heat bath on the physical processes in a quantum system. We use the density matrix theory as one of the most powerfool tool for investigation of quantum relaxation. In the beginning of the work (chapter 2) we mention and recall the most important steps of derivation of the equation of motion for the reduced density matrix (master equation) for an arbitrary quantum system in diabatic representation interacting with the environment modeled by a set of independent harmonic oscillators. Chapter 3 deals with the question of the border between classical and quantum effects and reports on a study of the environmental influence on the time evolution of a coherent state or the superposition of two coherent states of a harmonic oscillator as a simple system displaying the peculiarities of the transition from quantum to classical regime. Chapters 4 and 5 concern the electron transfer (ET) problem, namely the mathematical description of the ET in molecular zinc-porphyrin-quinone complexes modeling artificial photosynthesis (chapter 4) and photoinduced processes in the porphyrin triad (chapter 5). Each chapter starts with an introduction and ends with a brief summary. The main achievements of the present work are summarized in the Conclusions.

arXiv:quant-ph/0001004•Jan 3, 2000•2 citations
This chapter considers the interaction between a single degree of freedom and an environment. It assumes that the environment can be represented by a set of harmonic oscillators, and that the interaction between the system of interest and the environment is linear in the coupling to the oscillator degrees of freedom. There are systems where the linear coupling to the environment degrees of freedom is exact, such as the fundamental interaction of a charged particle interacting with the electromagnetic field, electron-photon interaction. Furthermore, many situations are adequately described by such an interaction, because it is sufficient to consider each environment degree of freedom only weakly perturbed. The chapter examines the time evolution of the reduced density matrix on integral form, and described the diagrammatic expansion in the spatial representation. It explores the kinetic approach for studying the motion of a particle interacting with an oscillator environment.
We study the evolution of an oscillator interacting via the most general bilinear coupling (with time-independent coefficients) with an ``environment'' consisting of a set of other harmonic oscillators. We are mainly interested in a possibility of using the Fokker-Planck equation to describe this evolution. Studying different interaction Hamiltonians, we show that unambiguous reduction to the Fokker-Planck equation is possible only within the framework of the so called rotating-wave approximation. As special cases we consider in detail the evolution of two coupled oscillators and relaxation of a charged oscillator in a uniform magnetic field.
This thesis examines dissipative forces in physical systems in both microscopic and quantum level. The system-environment interaction is modeled by assuming that the environment is composed of simple harmonic oscillators and that there is a linear interaction between these oscillators and the system. The classical case is solved by determining the retarded Green’s function, which yields the equation of motion of the system. For quantum dissipation, the Lindblad master equation is used to describe the time evolution of the reduced density matrix of the system, and the propagator is investigated to identify important parameters, including the term of Newton's friction force.
Abstract In quantum electrodynamics, the matter degrees of freedom are coupled to the radiation field through a local, gauge-invariant interaction density. Due to the linear structure of this coupling, the problem of constructing a complete formal representation of the reduced matter dynamics can be solved exactly if the electromagnetic radiation field is initially in a Gaussian state. This chapter combines field theoretical methods with superoperator and influence functional techniques to derive an exact, relativistic representation for the reduced density matrix of the matter degrees of freedom which completely describes the influence of the electromagnetic radiation field on the matter dynamics. Several applications are treated, such as the suppression of the quantum coherence of charged particles caused by the emission of bremsstrahlung and the decoherence of many-particle states.
The reduced density operator of a particle coupled to a fermionic environment (metal electrons) is written in terms of a Feynman-Vernon influence functional. Two different approaches are used in order to achieve this goal. Firstly, the environmental interacting electron gas is treated in RPA and the coupling is correctly accounted for up to second order in perturbation theory. Then the local and tight-binding models are studied. For the latter a second approach for obtaining the influence functional is used in the narrow-band limit. It is found that the popular representation of the environment in terms of a single set of harmonic oscillators is not feasible unless the particle is confined to move in a local region of radial extension r ≃ k F -1 .
The work is devoted to the investigation of the influence of a heat bath on the physical processes in a quantum system. We use the density matrix theory as one of the most powerfool tool for investigation of quantum relaxation. In the beginning of the work (chapter 2) we mention and recall the most important steps of derivation of the equation of motion for the reduced density matrix (master equation) for an arbitrary quantum system in diabatic representation interacting with the environment modeled by a set of independent harmonic oscillators. Chapter 3 deals with the question of the border between classical and quantum effects and reports on a study of the environmental influence on the time evolution of a coherent state or the superposition of two coherent states of a harmonic oscillator as a simple system displaying the peculiarities of the transition from quantum to classical regime. Chapters 4 and 5 concern the electron transfer (ET) problem, namely the mathematical description of the ET in molecular zinc-porphyrin-quinone complexes modeling artificial photosynthesis (chapter 4) and photoinduced processes in the porphyrin triad (chapter 5). Each chapter starts with an introduction and ends with a brief summary. The main achievements of the present work are summarized in the Conclusions.