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/0108112 · Journal of Physics A Mathematical and General

Quantum stochastic equation for the low density limit

Luigi Accardi, Alexander Nikolaevich Pechen, Igor Vasil'evich Volovich

Published May 31, 2002• 27 citations
arXivHTMLPDF

Abstract

A new derivation of the quantum stochastic differential equation for the evolution operator in the low density limit is presented. We use the distribution approach and derive a new algebra for quadratic master fields in the low density limit by using the energy representation. We formulate the stochastic golden rule in the low density limit case for a system coupling with a Bose field via quadratic interaction. In particular, the vacuum expectation value of the evolution operator is computed and its exponential decay is shown.

DOI
10.1088/0305-4470/35/23/306

Related papers

A STOCHASTIC GOLDEN RULE AND QUANTUM LANGEVIN EQUATION FOR THE LOW DENSITY LIMIT

Luigi Accardi, Alexander Nikolaevich Pechen, Igor Vasil'evich Volovich

A rigorous derivation of quantum Langevin equation from microscopic dynamics in the low density limit is given. We consider a quantum model of a microscopic system (test particle) coupled with a reservoir (gas of light Bose particles) via interaction of scattering type. We formulate a mathematical procedure (the so-called stochastic golden rule) which allows us to determine the quantum Langevin equation in the limit of large time and small density of particles of the reservoir. The quantum Langevin equation describes not only dynamics of the system but also the reservoir. We show that the generator of the corresponding master equation has the Lindblad form of most general generators of completely positive semigroups.

arXiv:math-ph/0206032•Infinite Dimensional Analysis Quantum Probability and Related Topics•Sep 1, 2003•18 citations

Quantum stochastic equation for a test particle interacting with a dilute Bose gas

Alexander Nikolaevich Pechen

We use the stochastic limit method to study long time quantum dynamics of a test particle interacting with a dilute Bose gas. The case of arbitrary form factors and an arbitrary, not necessarily equilibrium, quasifree low density state of the Bose gas is considered. Starting from microscopic dynamics we derive in the low density limit a quantum white noise equation for the evolution operator. This equation is equivalent to a quantum stochastic equation driven by a quantum Poisson process with intensity S−1, where S is a one-particle S matrix. The novelty of our approach is that the equations are derived directly in terms of correlators, without use of a Fock–anti-Fock (or Gel’fand–Naimark–Segal) representation. Advantages of our approach are the simplicity of derivation of the limiting equation and that the algebra of the master fields and the Ito table do not depend on the initial state of the Bose gas. The notion of a causal state is introduced. We construct master fields (white noise and number operators) describing the dynamics in the low density limit and prove the convergence of chronological (causal) correlators of the field operators to correlators of the master fields in the causal state.

arXiv:math-ph/0303020•Journal of Mathematical Physics•Dec 19, 2003•18 citations

Quantum decoherence and the Glauber dynamics from the Stochastic limit

Luigi Accardi, Sergei Vladimirovich Kozyrev

The effects of decoherence for quantum system coupled with a bosonic field are investigated. An application of the stochastic golden rule shows that in the stochastic limit the dynamics of such a system is described by a quantum stochastic differential equation. The corresponding master equation describes convergence of a system to equilibrium. In particular it predicts exponential damping for off-diagonal matrix elements of the system density matrix, moreover these elements for a generic system will decay at least as \exp(-tN{kT\over\hbar}), where N is a number of particles in the system. As an application of the described technique a derivation from first principles (i.e. starting from a Hamiltonian description) of a quantum extension of the Glauber dynamics for systems of spins is given.

arXiv:quant-ph/0005029•May 6, 2000•0 citations

Derivation of exact master equation with stochastic description: Models in quantum optics

Haifeng Li, Jiushu Shao

The methodology of stochastic description for dissipation, a generic scheme to decouple the interaction between two subsystems, is applied to the study of dissipative dynamics in quantum optics. It is shown that the influence of the coupled thermal or vacuum field on the quantum mode can be exactly represented by the induced stochastic fields. The quantum mode thereby satisfies a stochastic differential equation and dissipation effect due to the coupling with the environment is obtained through statistical averaging. Within the framework of stochastic description, it is demonstrated how to derive the master equation for a single optical mode interacting with the bosonic bath. A numerical algorithm for solving the master equation in which the coefficients are determined by a set of integral equations is discussed and a comparison with the known results is displayed. The derivation of the master equation for the spontaneous decay of two-state atoms in the vacuum is also presented.

arXiv:1205.4616•May 21, 2012•0 citations

The stochastic limit of quantum spin systems

L. Accardi

The stochastic limit for the system of spins interacting with a boson field is investigated. In the finite volume an application of the stochastic golden rule shows that in the limit the dynamics of a quantum system is described by a quantum white noise equation that after taking of normal order is equivalent to quantum stochastic differential equation (QSDE). For the quantum Langevin equation the dynamics is well defined and is a quantum flow on the infinite lattice system.

arXiv:quant-ph/9911077•AIP conference proceedings•Jan 1, 2000•2 citations

The stochastic limit of quantum spin systems

Accardi, L, Kozyrev, S V

The stochastic limit for the system of spins interacting with a boson field is investigated. In the finite volume an application of the stochastic golden rule shows that in the limit the dynamics of a quantum system is described by a quantum white noise equation that after taking of normal order is equivalent to quantum stochastic differential equation (QSDE). For the quantum Langevin equation the dynamics is well defined and is a quantum flow on the infinite lattice system.

•Nov 18, 1999•4 citations
A new derivation of the quantum stochastic differential equation for the evolution operator in the low density limit is presented. We use the distribution approach and derive a new algebra for quadratic master fields in the low density limit by using the energy representation. We formulate the stochastic golden rule in the low density limit case for a system coupling with a Bose field via quadratic interaction. In particular, the vacuum expectation value of the evolution operator is computed and its exponential decay is shown.
A rigorous derivation of quantum Langevin equation from microscopic dynamics in the low density limit is given. We consider a quantum model of a microscopic system (test particle) coupled with a reservoir (gas of light Bose particles) via interaction of scattering type. We formulate a mathematical procedure (the so-called stochastic golden rule) which allows us to determine the quantum Langevin equation in the limit of large time and small density of particles of the reservoir. The quantum Langevin equation describes not only dynamics of the system but also the reservoir. We show that the generator of the corresponding master equation has the Lindblad form of most general generators of completely positive semigroups.
We use the stochastic limit method to study long time quantum dynamics of a test particle interacting with a dilute Bose gas. The case of arbitrary form factors and an arbitrary, not necessarily equilibrium, quasifree low density state of the Bose gas is considered. Starting from microscopic dynamics we derive in the low density limit a quantum white noise equation for the evolution operator. This equation is equivalent to a quantum stochastic equation driven by a quantum Poisson process with intensity S−1, where S is a one-particle S matrix. The novelty of our approach is that the equations are derived directly in terms of correlators, without use of a Fock–anti-Fock (or Gel’fand–Naimark–Segal) representation. Advantages of our approach are the simplicity of derivation of the limiting equation and that the algebra of the master fields and the Ito table do not depend on the initial state of the Bose gas. The notion of a causal state is introduced. We construct master fields (white noise and number operators) describing the dynamics in the low density limit and prove the convergence of chronological (causal) correlators of the field operators to correlators of the master fields in the causal state.
The effects of decoherence for quantum system coupled with a bosonic field are investigated. An application of the stochastic golden rule shows that in the stochastic limit the dynamics of such a system is described by a quantum stochastic differential equation. The corresponding master equation describes convergence of a system to equilibrium. In particular it predicts exponential damping for off-diagonal matrix elements of the system density matrix, moreover these elements for a generic system will decay at least as \exp(-tN{kT\over\hbar}), where N is a number of particles in the system. As an application of the described technique a derivation from first principles (i.e. starting from a Hamiltonian description) of a quantum extension of the Glauber dynamics for systems of spins is given.
The methodology of stochastic description for dissipation, a generic scheme to decouple the interaction between two subsystems, is applied to the study of dissipative dynamics in quantum optics. It is shown that the influence of the coupled thermal or vacuum field on the quantum mode can be exactly represented by the induced stochastic fields. The quantum mode thereby satisfies a stochastic differential equation and dissipation effect due to the coupling with the environment is obtained through statistical averaging. Within the framework of stochastic description, it is demonstrated how to derive the master equation for a single optical mode interacting with the bosonic bath. A numerical algorithm for solving the master equation in which the coefficients are determined by a set of integral equations is discussed and a comparison with the known results is displayed. The derivation of the master equation for the spontaneous decay of two-state atoms in the vacuum is also presented.