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:gr-qc/9403046 · Europhysics Letters (EPL)

Quantum Master Equation of a Particle in a Gas Environment

Lajos Diósi

Published Apr 10, 1995• 108 citations
arXivHTMLPDF

Abstract

The reduced dynamics of a Brownian particle is discussed in single collision approach valid typically in dilute-gas environments. Our main purpose is a consistent account of quantum friction caused by local environmental interactions. We derive a Lindblad master equation whose generators are calculated from the differential cross-section of single collisions between the Brownian and the gas particles, respectively. The existence of thermal equilibrium for the Brownian particle's state ρ is proved. Master equations proposed earlier are shown to be particular cases of our one.

DOI
10.1209/0295-5075/30/2/001

Related papers

Test particle in a quantum gas

Bassano Vacchini

A master equation with a Lindblad structure is derived, which describes the interaction of a test particle with a macroscopic system and is expressed in terms of the operator valued dynamic structure factor of the system. In the case of a free Fermi or Bose gas the result is evaluated in the Brownian limit, thus obtaining a single generator master equation for the description of quantum Brownian motion in which the correction due to quantum statistics is explicitly calculated. The friction coefficients for Boltzmann and Bose or Fermi statistics are compared.

arXiv:quant-ph/0012129•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•May 22, 2001•53 citations

Completely Positive Quantum Dissipation

Bassano Vacchini

A completely positive master equation describing quantum dissipation for a Brownian particle is derived starting from microphysical collisions, exploiting a recently introduced approach to subdynamics of a macrosystem. The obtained equation can be cast into Lindblad form with a single generator for each Cartesian direction. Temperature dependent friction and diffusion coefficients for both position and momentum are expressed in terms of the collision cross section.

arXiv:quant-ph/0002094•Physical Review Letters•Feb 14, 2000•129 citations

Quantum master equations for a fast particle in a gas

David Gaspard

The propagation of a fast particle in a low-density gas at thermal equilibrium is studied in the context of quantum mechanics. A quantum master equation in the Redfield form governing the reduced density matrix of the particle is derived explicitly from first principles. Under some approximations, this equation reduces to the linear Boltzmann equation. The issue of the positivity of the time evolution is also discussed by means of a Lindblad form. The Born and Markov assumptions underlying these equations, as well as other approximations regarding the bath correlation function, are discussed in detail. Furthermore, all these master equations are shown to be equivalent with each other if the density matrix of the particle is diagonal in the momentum basis, or if the collision rate is independent of the particle momentum.

arXiv:2209.02056•Physical Review A•Dec 16, 2022•2 citations

Master Equation for a Quantum Particle in a Gas

Klaus Hornberger

The equation for the quantum motion of a Brownian particle in a gaseous environment is derived by means of S-matrix theory. This quantum version of the linear Boltzmann equation accounts nonperturbatively for the quantum effects of the scattering dynamics and describes decoherence and dissipation in a unified framework. As a completely positive master equation it incorporates both the known equation for an infinitely massive Brownian particle and the classical linear Boltzmann equation as limiting cases.

arXiv:quant-ph/0607085•Physical Review Letters•Aug 7, 2006•92 citations

Quantum master equations for a system interacting with a quantum gas in the low-density limit and for the semiclassical collision model

Sergey N. Filippov, Georgii Semin, Alexander Nikolaevich Pechen

A quantum system interacting with a dilute gas experiences irreversible dynamics. The corresponding master equation can be derived within two different approaches: The fully quantum description in the low-density limit and the semiclassical collision model, where the motion of gas particles is classical whereas their internal degrees of freedom are quantum. The two approaches have been extensively studied in the literature, but their predictions have not been compared. This is mainly due to the fact that the low-density limit was extensively studied for mathematical physics purposes, whereas the collision models have been essentially developed for quantum information tasks such as a tractable description of the open quantum dynamics. Here we develop and compare both approaches for a spin system interacting with a gas of spin particles. Using some approximations, we explicitly find the corresponding master equations including the Lamb shifts and the dissipators. The low-density limit in the Born approximation for fast particles is shown to be equivalent to the semiclassical collision model in the stroboscopic approximation. We reveal that both approaches give exactly the same master equation if the gas temperature is high enough. This allows to interchangeably use complicated calculations in the low-density limit and rather simple calculations in the collision model.

arXiv:1908.11202•Physical Review A•Jan 16, 2020•14 citations

Exact master equation for generalized quantum Brownian motion with momentum-dependent system-environment couplings

Yu-Wei Huang, Wei-Min Zhang

In this paper we generalize the quantum Brownian motion (QBM) to include the momentum-dependent system-environment couplings. The resulting Hamiltonian in the particle number representation is given by ${H}_{t}ot=\ensuremath{\hbar}{\ensuremath{\omega}}_{\mathrm{S}}{a}^{\ifmmode\dagger\else\textdagger\fi{}}a+{\ensuremath{\sum}}_{k}\ensuremath{\hbar}{\ensuremath{\omega}}_{k}{b}_{k}^{\ifmmode\dagger\else\textdagger\fi{}}{b}_{k}+{\ensuremath{\sum}}_{k}\ensuremath{\hbar}({V}_{k}{a}^{\ifmmode\dagger\else\textdagger\fi{}}{b}_{k}+{V}_{k}^{*}{b}_{k}^{\ifmmode\dagger\else\textdagger\fi{}}a)+{\ensuremath{\sum}}_{k}\ensuremath{\hbar}({W}_{k}{a}^{\ifmmode\dagger\else\textdagger\fi{}}{b}_{k}^{\ifmmode\dagger\else\textdagger\fi{}}+{W}_{k}^{*}{b}_{k}a)$. The conventional QBM model corresponds to the special case ${W}_{k}={V}_{k}$. The generalized QBM can capture more physical applications because the single-particle transition and the two-particle pair production between the system and the environment represent two very different physical processes that usually do not have the same coupling strengths. We discuss the physical realizations of the generalized QBM in different physical systems, and derive its exact master equation for both the initial decoupled states and initial correlated states. The Hu-Paz-Zhang master equation of the conventional QBM model is reproduced as a special case. We find that the renormalized system Hamiltonian after tracing out all the environmental states induced naturally a momentum-dependent potential, which shows the consistency of including the momentum-dependent coupling in the QBM Hamiltonian. In the Hu-Paz-Zhang master equation, such a renormalized potential is misplaced so that the correct renormalization Hamiltonian has not been found. With the exact master equation for both the initial decoupled and initial correlated states, the issues about the initial jolt are also reexamined. We find that the so-called ``initial jolt,'' which has been thought to be an artificial effect due to the use of the initial decoupled system-environment states, has nothing to do with the initial decoupled state. The exact master equation for the generalized QBM also has potential applications to photonics quantum computing.

arXiv:2204.09965•Physical Review Research•Aug 26, 2022•12 citations
The reduced dynamics of a Brownian particle is discussed in single collision approach valid typically in dilute-gas environments. Our main purpose is a consistent account of quantum friction caused by local environmental interactions. We derive a Lindblad master equation whose generators are calculated from the differential cross-section of single collisions between the Brownian and the gas particles, respectively. The existence of thermal equilibrium for the Brownian particle's state ρ is proved. Master equations proposed earlier are shown to be particular cases of our one.
A master equation with a Lindblad structure is derived, which describes the interaction of a test particle with a macroscopic system and is expressed in terms of the operator valued dynamic structure factor of the system. In the case of a free Fermi or Bose gas the result is evaluated in the Brownian limit, thus obtaining a single generator master equation for the description of quantum Brownian motion in which the correction due to quantum statistics is explicitly calculated. The friction coefficients for Boltzmann and Bose or Fermi statistics are compared.
The propagation of a fast particle in a low-density gas at thermal equilibrium is studied in the context of quantum mechanics. A quantum master equation in the Redfield form governing the reduced density matrix of the particle is derived explicitly from first principles. Under some approximations, this equation reduces to the linear Boltzmann equation. The issue of the positivity of the time evolution is also discussed by means of a Lindblad form. The Born and Markov assumptions underlying these equations, as well as other approximations regarding the bath correlation function, are discussed in detail. Furthermore, all these master equations are shown to be equivalent with each other if the density matrix of the particle is diagonal in the momentum basis, or if the collision rate is independent of the particle momentum.
The equation for the quantum motion of a Brownian particle in a gaseous environment is derived by means of S-matrix theory. This quantum version of the linear Boltzmann equation accounts nonperturbatively for the quantum effects of the scattering dynamics and describes decoherence and dissipation in a unified framework. As a completely positive master equation it incorporates both the known equation for an infinitely massive Brownian particle and the classical linear Boltzmann equation as limiting cases.
A quantum system interacting with a dilute gas experiences irreversible dynamics. The corresponding master equation can be derived within two different approaches: The fully quantum description in the low-density limit and the semiclassical collision model, where the motion of gas particles is classical whereas their internal degrees of freedom are quantum. The two approaches have been extensively studied in the literature, but their predictions have not been compared. This is mainly due to the fact that the low-density limit was extensively studied for mathematical physics purposes, whereas the collision models have been essentially developed for quantum information tasks such as a tractable description of the open quantum dynamics. Here we develop and compare both approaches for a spin system interacting with a gas of spin particles. Using some approximations, we explicitly find the corresponding master equations including the Lamb shifts and the dissipators. The low-density limit in the Born approximation for fast particles is shown to be equivalent to the semiclassical collision model in the stroboscopic approximation. We reveal that both approaches give exactly the same master equation if the gas temperature is high enough. This allows to interchangeably use complicated calculations in the low-density limit and rather simple calculations in the collision model.
In this paper we generalize the quantum Brownian motion (QBM) to include the momentum-dependent system-environment couplings. The resulting Hamiltonian in the particle number representation is given by ${H}_{t}ot=\ensuremath{\hbar}{\ensuremath{\omega}}_{\mathrm{S}}{a}^{\ifmmode\dagger\else\textdagger\fi{}}a+{\ensuremath{\sum}}_{k}\ensuremath{\hbar}{\ensuremath{\omega}}_{k}{b}_{k}^{\ifmmode\dagger\else\textdagger\fi{}}{b}_{k}+{\ensuremath{\sum}}_{k}\ensuremath{\hbar}({V}_{k}{a}^{\ifmmode\dagger\else\textdagger\fi{}}{b}_{k}+{V}_{k}^{*}{b}_{k}^{\ifmmode\dagger\else\textdagger\fi{}}a)+{\ensuremath{\sum}}_{k}\ensuremath{\hbar}({W}_{k}{a}^{\ifmmode\dagger\else\textdagger\fi{}}{b}_{k}^{\ifmmode\dagger\else\textdagger\fi{}}+{W}_{k}^{*}{b}_{k}a)$. The conventional QBM model corresponds to the special case Wk=Vk{W}_{k}={V}_{k}Wk​=Vk​. The generalized QBM can capture more physical applications because the single-particle transition and the two-particle pair production between the system and the environment represent two very different physical processes that usually do not have the same coupling strengths. We discuss the physical realizations of the generalized QBM in different physical systems, and derive its exact master equation for both the initial decoupled states and initial correlated states. The Hu-Paz-Zhang master equation of the conventional QBM model is reproduced as a special case. We find that the renormalized system Hamiltonian after tracing out all the environmental states induced naturally a momentum-dependent potential, which shows the consistency of including the momentum-dependent coupling in the QBM Hamiltonian. In the Hu-Paz-Zhang master equation, such a renormalized potential is misplaced so that the correct renormalization Hamiltonian has not been found. With the exact master equation for both the initial decoupled and initial correlated states, the issues about the initial jolt are also reexamined. We find that the so-called ``initial jolt,'' which has been thought to be an artificial effect due to the use of the initial decoupled system-environment states, has nothing to do with the initial decoupled state. The exact master equation for the generalized QBM also has potential applications to photonics quantum computing.