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:0905.1019

Weak-Coupling Limit. II On the Quantum Fokker-Planck Equation

David Taj

Published May 7, 2009• 1 citation
arXivHTMLPDF

Abstract

In a recent work we have found a contraction semigroup able to correctly approximate a projected and perturbed one-parameter group of isometries in a generic Banach space, in the limit of weak-coupling. Here we study its generator by specializing to $W^*$-algebras: after defining a Physical Subsystem in terms of a completely positive projecting conditional expectation, we find that it generates a Quantum Dynamical Semigroup. As a consequence of uniqueness and strong generality (well defined dynamics, irrespective of the Physical Subsystem spectral properties or dimensions), its generator deserves to be referred as "the" Quantum Fokker-Planck Equation. We then provide important examples of the limit dynamics, one of which constitutes a new Quantum generalization of the celebrated Fermi Golden Rule.

DOI
10.48550/arxiv.0905.1019

Related papers

The classical limit of quantum dissipative generators

Joseph V. Pulé, André F. Verbeure

For finite quantum systems the classical limit of the general dissipative generator of a semigroup of completely positive maps is obtained, yielding a generalized Fokker–Planck generator.

Journal of Mathematical Physics•Apr 1, 1979•7 citations

QUANTUM FOKKER-PLANCK MODELS: THE LINDBLAD AND WIGNER APPROACHES

Anton Arnold, Franco Fagnola, Lukas Neumann

In this article we try to bridge the gap between the quantum dynamical semigroup and Wigner function approaches to quantum open systems. In particular we study stationary states and the long time asymptotics for the quantum Fokker-Planck equation. Our new results apply to open quantum systems in a harmonic confinement potential, perturbed by a (large) sub-quadratic term.

arXiv:0806.2984•Oct 1, 2008•10 citations

The classical limit of a class of quantum dynamical semigroups

Fabio Benatti

A class of quantum dynamical semigroups is proved to possess a residual phase-space stochasticity, if a weak-coupling limit is performed, whereas the usual limit in which h→0 is completely deterministic. As a consequence, solutions of the corresponding Chapman-Kolmogorov differential equation, with linear Liouville term, are constructed

Annales De L Institut Henri Poincare-physique Theorique•Jan 1, 1993•0 citations

Quantum Fokker-Planck Models: Limiting Case in the Lindblad Condition

Franco Fagnola, Lukas Neumann

In this article we study stationary states and the long time asymptotics for the quantum Fokker–Planck equation. We continue the investigation of an earlier \nwork in which we derived convergence to a steady state if the Lindblad condition $D_{pp}D_{qq} −D_{pq} \n\\ge \\gamma^2/4$ is satisfied with strict inequality. Here we extend our results to the limiting case that turns out to be more difficult because irreducibility \nof the quantum Markov semigroup does not follow from triviality of the generalized commutator with position and momentum operators.

Quantum Probability and Infinite Dimensional Analysis•Feb 1, 2010•0 citations

Quantum Markov Semigroups of Low Density Limit: the Generic Case

Luigi Accardi, F. Guerrero-Poblete

We investigate the structure of quantum Markov generators that describe the reduced dynamics of a test particle interacting with a dilute Bose gas in the low density limit. These generators, called low density limit type generators (LDL), differ from the weak coupling limit type generators (WCL) studied in [4, 5] because of the presence of the T-operator that describes the change in momentum of the gas particle due to collisions with the test particle. We propose a general definition of Markov generator of stochastic limit type that includes (almost) all generators arising in the stochastic limit approach and we prove that the associated semigroups as well as their invariant states (or weights) have a very special structure which extends the explicit representation for the generic quantum Markov semigroups of weak coupling limit type, due to Accardi, Hachicha and Ouerdiane [7]. We also investigate the structure of invariant states and give conditions on the T - operator for the existence of a unique invariant state and of equilibrium states.

Open Systems & Information Dynamics•Dec 1, 2019•2 citations

Extended weak coupling limit for Friedrichs Hamiltonians

Jan Dereziński, Wojciech De Roeck

We study a class of self-adjoint operators defined on the direct sum of two Hilbert spaces: a finite dimensional one called sometimes a “small subsystem” and an infinite dimensional one called a “reservoir.” The operator, which we call a “Friedrichs Hamiltonian,” has a small coupling constant in front of its off-diagonal term. It is well known that under some conditions in the weak coupling limit the appropriately rescaled evolution in the interaction picture converges to a contractive semigroup when restricted to the subsystem. We show that in this model, the properly renormalized and rescaled evolution converges on the whole space to a new unitary evolution, which is a dilation of the above mentioned semigroup. Similar results have been studied before (Accardi et al., 1990) in more complicated models under the name of “stochastic limit.”

arXiv:math-ph/0604058•Journal of Mathematical Physics•Jan 1, 2007•6 citations
In a recent work we have found a contraction semigroup able to correctly approximate a projected and perturbed one-parameter group of isometries in a generic Banach space, in the limit of weak-coupling. Here we study its generator by specializing to W∗W^*W∗-algebras: after defining a Physical Subsystem in terms of a completely positive projecting conditional expectation, we find that it generates a Quantum Dynamical Semigroup. As a consequence of uniqueness and strong generality (well defined dynamics, irrespective of the Physical Subsystem spectral properties or dimensions), its generator deserves to be referred as "the" Quantum Fokker-Planck Equation. We then provide important examples of the limit dynamics, one of which constitutes a new Quantum generalization of the celebrated Fermi Golden Rule.
In this article we study stationary states and the long time asymptotics for the quantum Fokker–Planck equation. We continue the investigation of an earlier \nwork in which we derived convergence to a steady state if the Lindblad condition $D_{pp}D_{qq} −D_{pq} \n\\ge \\gamma^2/4$ is satisfied with strict inequality. Here we extend our results to the limiting case that turns out to be more difficult because irreducibility \nof the quantum Markov semigroup does not follow from triviality of the generalized commutator with position and momentum operators.
We investigate the structure of quantum Markov generators that describe the reduced dynamics of a test particle interacting with a dilute Bose gas in the low density limit. These generators, called low density limit type generators (LDL), differ from the weak coupling limit type generators (WCL) studied in [4, 5] because of the presence of the T-operator that describes the change in momentum of the gas particle due to collisions with the test particle. We propose a general definition of Markov generator of stochastic limit type that includes (almost) all generators arising in the stochastic limit approach and we prove that the associated semigroups as well as their invariant states (or weights) have a very special structure which extends the explicit representation for the generic quantum Markov semigroups of weak coupling limit type, due to Accardi, Hachicha and Ouerdiane [7]. We also investigate the structure of invariant states and give conditions on the T - operator for the existence of a unique invariant state and of equilibrium states.
We study a class of self-adjoint operators defined on the direct sum of two Hilbert spaces: a finite dimensional one called sometimes a “small subsystem” and an infinite dimensional one called a “reservoir.” The operator, which we call a “Friedrichs Hamiltonian,” has a small coupling constant in front of its off-diagonal term. It is well known that under some conditions in the weak coupling limit the appropriately rescaled evolution in the interaction picture converges to a contractive semigroup when restricted to the subsystem. We show that in this model, the properly renormalized and rescaled evolution converges on the whole space to a new unitary evolution, which is a dilation of the above mentioned semigroup. Similar results have been studied before (Accardi et al., 1990) in more complicated models under the name of “stochastic limit.”