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

arXiv:quant-ph/9706047 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Partial factorization of wave functions for a quantum dissipative system

C. P. Sun, Yibo Gao, Hang Dong, S. R. Zhao

Published Apr 1, 1998• 20 citations
arXivHTMLPDF

Abstract

In this paper, a microscopic approach treating the quantum dissipation process presented by Yu and Sun [Phys. Rev. A 49, 592 (1994); 51, 1845 (1995)] is developed to analyze the wave function structure for the dynamic evolution of a typical dissipative system---a single-mode boson soaked in a bath of many bosons. The wave function of the total system is explicitly obtained as a product of two components of the system and bath in a coherent state representation. It not only describes the influence of the bath on the variable of the system through the Brownian motion, but also manifests the back-action of the system on the bath and the mutual interaction among the bosons of bath. Due to the back-action, the total wave function can only be partially factorizable even if the Brownian motion can be completely ignored in certain cases, e.g., weak coupling and large detuning. The semiclassical implication of the back-action, the mutual interaction, and the Brownian motion in the present model are also discussed by considering the wave packet evolution of the dissipative system in the coordinate representation.

DOI
10.1103/physreve.57.3900

Related papers

Brownian Motion of the Quantum Dissipation in the RWA

Gao yibo

A microscopic approach treating the quantum dissipation process presented by Sun and Yu (Phys. Rev. A49 (1994) 592; A51 (1995) 1845) is invoked to construct the wavefunction of the composite system—the model for a harmonic oscillator interacting with a many-oscillator bath under the rotating wave approximation. It shows the back-action of the system on the bath. In particular, the dynamic evolution of the wavefunction for the composite system maintains a factorized form in its wavefunction. In the limited temperature, the reduced density matrix for the system is also calculated to clarify the influence of Brownian motion on the system.

Communications in Theoretical Physics•Jul 30, 2000•0 citations

Exact dynamics of a quantum dissipative system in a constant external field

Chang-Pu Sun, Li-Hua Yu

We study the quantum dynamics of the simplest dissipative system, a particle moving in a constant external field and interacting with a bath of harmonic oscillators with Ohmic spectral density. Applying the main idea and methods developed in our recent work [L. H. Yu and C. P. Sun, Phys. Rev. A 49, 592 (1994)] to this system, we obtain the simple and exact solutions for the coordinate operator of the system in the Heisenberg picture and the wave function of the composite system of the system and the bath in the Schr\"odinger picture. An effective Hamiltonian for the dissipative system is explicitly derived from these solutions. The meaning of the wave function described by this effective Hamiltonian is clarified by analyzing the effect of the Brownian motion. In particular, the general effective Hamiltonian for an arbitrary potential is directly derived with this method for the case when the Brownian motion can be ignored. Using this effective Hamiltonian, we show an interesting result that the dissipation suppresses the wave-packet spreading.

Physical Review A•Mar 1, 1995•87 citations

Exact Dynamics of Quantum Dissipative System in Constant External Field

Sun, Cesar E.P., Li Hua Yu

The quantum dynamics of a simplest dissipative system, a particle moving in a constant external field , is exactly studied by taking into account its interaction with a bath of Ohmic spectral density. We apply the main idea and methods developed in our recent work [1] to quantum dissipative system with constant external field. Quantizing the dissipative system we obtain the simple and exact solutions for the coordinate operator of the system in Heisenberg picture and the wave function of the composite system of system and bath in Schroedinger picture. An effective Hamiltonian for the dissipative system is explicitly derived from these solutions with Heisenberg picture method and thereby the meaning of the wavefunction governed by it is clarified by analyzing the effect of the Brownian motion. Especially, the general effective Hamiltonian for the case with arbitrary potential is directly derived with this method for the case when the Brownian motion can be ignored. Using this effective Hamiltonian, we show an interesting fact that the dissipation suppresses the wave packet spreading.

arXiv:hep-th/9306124•Jun 23, 1993•0 citations

Evolution of the wave function in a dissipative system

Li Hua Yu, Chang-Pu Sun

For a dissipative system with Ohmic friction, we obtain a simple and exact solution for the wave function of the system plus the bath. It is described by the direct product in two independent Hilbert spaces. One of them is described by an effective Hamiltonian, the other represents the effect of the bath, i.e., the Brownian motion, thus clarifying the structure of the wave function of the system whose energy is dissipated by its interaction with the bath. No path-integral technology is needed in this treatment. The derivation of the Weisskopf-Wigner linewidth theory follows easily.

arXiv:hep-th/9307044•Physical Review A•Jan 1, 1994•124 citations

Path integral formulation of dissipative quantum dynamics

Alexey Novikov

In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.

•Jan 20, 2005•0 citations

Binary collision model for quantum Brownian motion

Stefan I. Tsonchev, Philip Pechukas

A binary collision model for phenomena of quantum dissipation is developed. Unlike the harmonic oscillator model, widely used for over thirty years, this model assumes nonlinear coupling between system and environment, and is applicable to both bosonic and fermionic baths. The system interacts with an ideal bath through binary collisions only. Solutions for the classical and quantum-mechanical problems in the case of free Brownian motion are presented, and the quantum-classical correspondence for nonequilibrium processes is established. It is shown that in the Brownian motion limit the two models lead to identical dynamical behavior, provided the coupling coefficients in the harmonic oscillator Hamiltonian are temperature dependent. For cases of bath particles of finite mass and number the two models lead to different results. Linear response theory for the model is developed, and the results are compared with those for the harmonic oscillator model. At the end, possible applications of the model are suggested.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jun 1, 2000•10 citations
In this paper, a microscopic approach treating the quantum dissipation process presented by Yu and Sun [Phys. Rev. A 49, 592 (1994); 51, 1845 (1995)] is developed to analyze the wave function structure for the dynamic evolution of a typical dissipative system---a single-mode boson soaked in a bath of many bosons. The wave function of the total system is explicitly obtained as a product of two components of the system and bath in a coherent state representation. It not only describes the influence of the bath on the variable of the system through the Brownian motion, but also manifests the back-action of the system on the bath and the mutual interaction among the bosons of bath. Due to the back-action, the total wave function can only be partially factorizable even if the Brownian motion can be completely ignored in certain cases, e.g., weak coupling and large detuning. The semiclassical implication of the back-action, the mutual interaction, and the Brownian motion in the present model are also discussed by considering the wave packet evolution of the dissipative system in the coordinate representation.
A microscopic approach treating the quantum dissipation process presented by Sun and Yu (Phys. Rev. A49 (1994) 592; A51 (1995) 1845) is invoked to construct the wavefunction of the composite system—the model for a harmonic oscillator interacting with a many-oscillator bath under the rotating wave approximation. It shows the back-action of the system on the bath. In particular, the dynamic evolution of the wavefunction for the composite system maintains a factorized form in its wavefunction. In the limited temperature, the reduced density matrix for the system is also calculated to clarify the influence of Brownian motion on the system.
We study the quantum dynamics of the simplest dissipative system, a particle moving in a constant external field and interacting with a bath of harmonic oscillators with Ohmic spectral density. Applying the main idea and methods developed in our recent work [L. H. Yu and C. P. Sun, Phys. Rev. A 49, 592 (1994)] to this system, we obtain the simple and exact solutions for the coordinate operator of the system in the Heisenberg picture and the wave function of the composite system of the system and the bath in the Schr\"odinger picture. An effective Hamiltonian for the dissipative system is explicitly derived from these solutions. The meaning of the wave function described by this effective Hamiltonian is clarified by analyzing the effect of the Brownian motion. In particular, the general effective Hamiltonian for an arbitrary potential is directly derived with this method for the case when the Brownian motion can be ignored. Using this effective Hamiltonian, we show an interesting result that the dissipation suppresses the wave-packet spreading.
The quantum dynamics of a simplest dissipative system, a particle moving in a constant external field , is exactly studied by taking into account its interaction with a bath of Ohmic spectral density. We apply the main idea and methods developed in our recent work [1] to quantum dissipative system with constant external field. Quantizing the dissipative system we obtain the simple and exact solutions for the coordinate operator of the system in Heisenberg picture and the wave function of the composite system of system and bath in Schroedinger picture. An effective Hamiltonian for the dissipative system is explicitly derived from these solutions with Heisenberg picture method and thereby the meaning of the wavefunction governed by it is clarified by analyzing the effect of the Brownian motion. Especially, the general effective Hamiltonian for the case with arbitrary potential is directly derived with this method for the case when the Brownian motion can be ignored. Using this effective Hamiltonian, we show an interesting fact that the dissipation suppresses the wave packet spreading.
For a dissipative system with Ohmic friction, we obtain a simple and exact solution for the wave function of the system plus the bath. It is described by the direct product in two independent Hilbert spaces. One of them is described by an effective Hamiltonian, the other represents the effect of the bath, i.e., the Brownian motion, thus clarifying the structure of the wave function of the system whose energy is dissipated by its interaction with the bath. No path-integral technology is needed in this treatment. The derivation of the Weisskopf-Wigner linewidth theory follows easily.
In this thesis the path integral formalism is applied to the calculation of the dynamics of dissipative quantum systems. The time evolution of a system of bilinearly coupled bosonic modes is treated using the real-time path integral technique in coherent-state representation. This method is applied to a damped harmonic oscillator within the Caldeira-Leggett model. In order to get the stationary trajectories the corresponding Lagrangian function is diagonalized and then the path integrals are evaluated by means of the stationary-phase method. The time evolution of the reduced density matrix in the basis of coherent states is given in simple analytic form for weak system-bath coupling, i.e. the so-called rotating-wave terms can be evaluated exactly but the non-rotating-wave terms only in a perturbative manner. The validity range of the rotating-wave approximation is discussed from the viewpoint of spectral equations. In addition, it is shown that systems without initial system-bath correlations can exhibit initial jumps in the population dynamics even for rather weak dissipation. Only with initial correlations the classical trajectories for the system coordinate can be recovered. The path integral formalism in a combined phase-space and coherent-state representation is applied to the problem of curve-crossing dynamics. The system of interest is described by two coupled one-dimensional harmonic potential energy surfaces interacting with a heat bath. The mapping approach is used to rewrite the Lagrangian function of the electronic part of the system. Using the Feynman-Vernon influence-functional method the bath is eliminated whereas the non-Gaussian part of the path integral is treated using the perturbation theory in the small coordinate shift between potential energy surfaces. The vibrational and the population dynamics is considered in a lowest order of the perturbation. The dynamics of a Gaussian wave packet is analyzed along a one-dimensional reaction coordinate. Also the damping rate of coherence in the electronic part of the relevant system is evaluated within the ordinary and variational perturbation theory. The analytic expressions for the rate functions are obtained in the low and high temperature regimes.
A binary collision model for phenomena of quantum dissipation is developed. Unlike the harmonic oscillator model, widely used for over thirty years, this model assumes nonlinear coupling between system and environment, and is applicable to both bosonic and fermionic baths. The system interacts with an ideal bath through binary collisions only. Solutions for the classical and quantum-mechanical problems in the case of free Brownian motion are presented, and the quantum-classical correspondence for nonequilibrium processes is established. It is shown that in the Brownian motion limit the two models lead to identical dynamical behavior, provided the coupling coefficients in the harmonic oscillator Hamiltonian are temperature dependent. For cases of bath particles of finite mass and number the two models lead to different results. Linear response theory for the model is developed, and the results are compared with those for the harmonic oscillator model. At the end, possible applications of the model are suggested.