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

Journal of Experimental and Theoretical Physics

Quantum theory of open systems based on stochastic differential equations of generalized Langevin (non-Wiener) type

Ashat M. Basharov

Published Sep 1, 2012• 29 citations
Publisher

Abstract

It is shown that the effective Hamiltonian representation, as it is formulated in author’s papers, serves as a basis for distinguishing, in a broadband environment of an open quantum system, independent noise sources that determine, in terms of the stationary quantum Wiener and Poisson processes in the Markov approximation, the effective Hamiltonian and the equation for the evolution operator of the open system and its environment. General stochastic differential equations of generalized Langevin (non-Wiener) type for the evolution operator and the kinetic equation for the density matrix of an open system are obtained, which allow one to analyze the dynamics of a wide class of localized open systems in the Markov approximation. The main distinctive features of the dynamics of open quantum systems described in this way are the stabilization of excited states with respect to collective processes and an additional frequency shift of the spectrum of the open system. As an illustration of the general approach developed, the photon dynamics in a single-mode cavity without losses on the mirrors is considered, which contains identical intracavity atoms coupled to the external vacuum electromagnetic field. For some atomic densities, the photons of the cavity mode are “locked” inside the cavity, thus exhibiting a new phenomenon of radiation trapping and non-Wiener dynamics.

DOI
10.1134/s1063776112070035

Related papers

A theory of open systems based on stochastic differential equations

Ashat M. Basharov

For a model of an open quantum system—a concentrated ensemble consisting of similar atoms and interacting with a one-dimensional quantum vacuum environment with a zero photon density—quantum stochastic differential equations of a non-Wiener type of the general form have been obtained; based on the equations, kinetic equations describing a wide class of physical systems are derived. The distinctive feature of such systems is effects of suppression of collective spontaneous emission and stabilization of the excited state. For the open classical system exposed to the action of noise in the form of a Levy process of the general non-Gaussian kind, kinetic equations of the Fokker-Planck type with fractional derivatives have been obtained based on classical non-Wiener stochastic differential equations. This emphasizes the common base of the developed theory for different types of open systems, which is expressed in using the mathematical formalism of stochastic differential equations of the general non-Wiener type.

Optics and Spectroscopy•Apr 1, 2014•9 citations

Quantum theory of open systems in the representation of effective Hamiltonian and stochastic differential equations

Ashat M. Basharov

An original general approach to the description of open quantum systems in the Markovian approximation was formulated for the example of the dynamics of high-Q cavity mode photons in the presence of nonresonant cavity atoms bound with an external vacuum electromagnetic field.

Bulletin of the Russian Academy of Sciences Physics•Mar 1, 2012•0 citations

Langevin and generalized Langevin types of spontaneous emission of quantum particles

Ashat M. Basharov

In the effective Hamiltonian representation, we have obtained a quantum stochastic differential equation of a generalized Langevin type for the evolution operator of an atomic ensemble in a microcavity in an external broadband quantized field and in a nonresonant field of the microcavity. We show that, depending on the number of particles in the atomic ensemble, its dynamics demonstrates both the Langevin and the generalized Langevin types of the two-photon spontaneous decay. In this case, one photon is emitted into the cavity mode, whereas the other photon is emitted into the external broadband electromagnetic field. The Langevin type is determined by a considerable Stark interaction of the atomic ensemble with the broadband photon-free quantized field. We show that, here, the Stark interaction is represented by a quantized Poisson process and, depending on its magnitude (at certain numbers of atoms in the ensemble), the two-photon collective spontaneous emission of microcavity atoms can be completely suppressed. In this case, the two-photon spontaneous emission of the singly excited atomic ensemble is described by the two-level model, while the atom-photon cluster of the microcavity under the described conditions is an artificial two-level quantum particle with a strong Stark interaction.

Optics and Spectroscopy•Mar 1, 2012•0 citations

Stochastic Differential Equations in the Theory of Open Quantum Systems

A. M. Basharov

Abstract Quantum stochastic differential equations are shown to naturally arise because of the mathematical background of the correctness of the Schrödinger equation, which describes the interaction of an open quantum system with broadband quantized fields in the Markov approximation. The main classical and quantum stochastic processes appear in the notation in the Markov approximation of the operator of the electric broadband quantized field strength and the effective Hamiltonian of the interaction of a quantum system with it obtained by the methods of algebraic resonance perturbation theory. All basic stochastic quantum processes, including the counting and two-quantum Lévy processes, are constructed based on quantum white noise, which differs significantly from the classical case. At the same time, the apparatus of stochastic differential equations demonstrates consistency and parallelism.

Bulletin of the Russian Academy of Sciences Physics•Jul 1, 2026•0 citations

Generalized Langevin Dynamics in a Broadband Quantized Field

Andrew I. Trubilko

Abstract A generalized Langevin dynamics of a two-level system (atom, qubit) in a broadband quantized electromagnetic field is considered. The analysis is carried out using a kinetic equation of an open system, obtained based on a quantum stochastic differential equation. The latter is associated with the terms of the effective Hamiltonian of the problem up to the second order in the constant of interaction of the atom with the field.

Bulletin of the Russian Academy of Sciences Physics•Jul 1, 2026•0 citations

The Theory of Open Quantum Systems

Heinz‐Peter Breuer, Francesco Petruccione

Abstract This book treats the central physical concepts and mathematical techniques used to investigate the dynamics of open quantum systems. To provide a self-contained presentation, the text begins with a survey of classical probability theory and with an introduction to the foundations of quantum mechanics, with particular emphasis on its statistical interpretation and on the formulation of generalized measurement theory through quantum operations and effects. The fundamentals of density matrix theory, quantum Markov processes, and completely positive dynamical semigroups are developed. The most important master equations used in quantum optics and condensed matter theory are derived and applied to the study of many examples. Special attention is paid to the Markovian and non-Markovian theory of environment induced decoherence, its role in the dynamical description of the measurement process, and to the experimental observation of decohering electromagnetic field states. The book includes the modern formulation of open quantum systems in terms of stochastic processes in Hilbert space. Stochastic wave function methods and Monte Carlo algorithms are designed and applied to important examples from quantum optics and atomic physics. The fundamentals of the treatment of non-Markovian quantum processes in open systems are developed on the basis of various mathematical techniques, such as projection superoperator methods and influence functional techniques. In addition, the book expounds the relativistic theory of quantum measurements and the density matrix theory of relativistic quantum electrodynamics.

•Jan 25, 2007•5,936 citations
It is shown that the effective Hamiltonian representation, as it is formulated in author’s papers, serves as a basis for distinguishing, in a broadband environment of an open quantum system, independent noise sources that determine, in terms of the stationary quantum Wiener and Poisson processes in the Markov approximation, the effective Hamiltonian and the equation for the evolution operator of the open system and its environment. General stochastic differential equations of generalized Langevin (non-Wiener) type for the evolution operator and the kinetic equation for the density matrix of an open system are obtained, which allow one to analyze the dynamics of a wide class of localized open systems in the Markov approximation. The main distinctive features of the dynamics of open quantum systems described in this way are the stabilization of excited states with respect to collective processes and an additional frequency shift of the spectrum of the open system. As an illustration of the general approach developed, the photon dynamics in a single-mode cavity without losses on the mirrors is considered, which contains identical intracavity atoms coupled to the external vacuum electromagnetic field. For some atomic densities, the photons of the cavity mode are “locked” inside the cavity, thus exhibiting a new phenomenon of radiation trapping and non-Wiener dynamics.
For a model of an open quantum system—a concentrated ensemble consisting of similar atoms and interacting with a one-dimensional quantum vacuum environment with a zero photon density—quantum stochastic differential equations of a non-Wiener type of the general form have been obtained; based on the equations, kinetic equations describing a wide class of physical systems are derived. The distinctive feature of such systems is effects of suppression of collective spontaneous emission and stabilization of the excited state. For the open classical system exposed to the action of noise in the form of a Levy process of the general non-Gaussian kind, kinetic equations of the Fokker-Planck type with fractional derivatives have been obtained based on classical non-Wiener stochastic differential equations. This emphasizes the common base of the developed theory for different types of open systems, which is expressed in using the mathematical formalism of stochastic differential equations of the general non-Wiener type.
In the effective Hamiltonian representation, we have obtained a quantum stochastic differential equation of a generalized Langevin type for the evolution operator of an atomic ensemble in a microcavity in an external broadband quantized field and in a nonresonant field of the microcavity. We show that, depending on the number of particles in the atomic ensemble, its dynamics demonstrates both the Langevin and the generalized Langevin types of the two-photon spontaneous decay. In this case, one photon is emitted into the cavity mode, whereas the other photon is emitted into the external broadband electromagnetic field. The Langevin type is determined by a considerable Stark interaction of the atomic ensemble with the broadband photon-free quantized field. We show that, here, the Stark interaction is represented by a quantized Poisson process and, depending on its magnitude (at certain numbers of atoms in the ensemble), the two-photon collective spontaneous emission of microcavity atoms can be completely suppressed. In this case, the two-photon spontaneous emission of the singly excited atomic ensemble is described by the two-level model, while the atom-photon cluster of the microcavity under the described conditions is an artificial two-level quantum particle with a strong Stark interaction.
Abstract Quantum stochastic differential equations are shown to naturally arise because of the mathematical background of the correctness of the Schrödinger equation, which describes the interaction of an open quantum system with broadband quantized fields in the Markov approximation. The main classical and quantum stochastic processes appear in the notation in the Markov approximation of the operator of the electric broadband quantized field strength and the effective Hamiltonian of the interaction of a quantum system with it obtained by the methods of algebraic resonance perturbation theory. All basic stochastic quantum processes, including the counting and two-quantum Lévy processes, are constructed based on quantum white noise, which differs significantly from the classical case. At the same time, the apparatus of stochastic differential equations demonstrates consistency and parallelism.
Abstract This book treats the central physical concepts and mathematical techniques used to investigate the dynamics of open quantum systems. To provide a self-contained presentation, the text begins with a survey of classical probability theory and with an introduction to the foundations of quantum mechanics, with particular emphasis on its statistical interpretation and on the formulation of generalized measurement theory through quantum operations and effects. The fundamentals of density matrix theory, quantum Markov processes, and completely positive dynamical semigroups are developed. The most important master equations used in quantum optics and condensed matter theory are derived and applied to the study of many examples. Special attention is paid to the Markovian and non-Markovian theory of environment induced decoherence, its role in the dynamical description of the measurement process, and to the experimental observation of decohering electromagnetic field states. The book includes the modern formulation of open quantum systems in terms of stochastic processes in Hilbert space. Stochastic wave function methods and Monte Carlo algorithms are designed and applied to important examples from quantum optics and atomic physics. The fundamentals of the treatment of non-Markovian quantum processes in open systems are developed on the basis of various mathematical techniques, such as projection superoperator methods and influence functional techniques. In addition, the book expounds the relativistic theory of quantum measurements and the density matrix theory of relativistic quantum electrodynamics.