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

St Petersburg Mathematical Journal

On some properties of the time evolution of quadratic open quantum systems

D. Robert

Published Apr 12, 2019• 0 citations
Publisher

Abstract

The paper is aimed at revisiting some mathematical properties that appear in open quantum systems with quadratic Hamiltonians. In particular, a discussion is included of some properties related to the notions of purity, separability, and entanglement, introduced by physicists in quantum information theory. The results are applied, among other things, to coupled harmonic oscillators appearing in the quantum Brownian motion. Moreover, the master equation is revisited, which was obtained for the time evolution of an open system coupled to an environment modeled by any quadratic Hamiltonians.

DOI
10.1090/spmj/1556

Related papers

Time Evolution of States for Open Quantum Systems. The quadratic case

Robert, Didier

Our main goal in this paper is to extend to any system of coupled quadratic Hamiltonians some properties known for systems of quantum harmonic oscillators related with the Brownian Quantum Motion model. In a first part we get a rather general formula for the purity (or the linear entropy) in a short time approximation. In a second part we establish a master equation (or a Fokker-Planck type equation) for the time evolution of the reduced matrix density for bilinearly coupled quadratic Hamiltonians. The Hamiltonians and the bilinear coupling can be time dependent. Moreover we give an explicit formula for the solution of this master equation so that the time evolution of the reduced density at time $t$ is connected with the reduced density at initial time $t_0$ for $t_0 \leq t

arXiv:1208.1598•Aug 8, 2012•0 citations

Open Quantum Systems and Coherent States

Didier Robert, Monique Combescure

In this chapter we study some properties of systems of coupled quadratic quantum Hamiltonians. The well known example is coupled harmonic oscillators appearing in the Quantum Brownian Motion. An open system is a physical (S) interacting with the environment (E) such that the total system \((S\cup E)\) is isolated from the rest of the universe. So we have two independent Hamiltonians \(\hat{H}_S\), \(\hat{H}_E\) and an interaction \(\hat{V}\) between (S) and (E). The main problem discussed for open systems is to understand the dynamical action of the interaction \(\hat{V}\) with the environment on the time dependent evolution of (S) which is now irreversible because a part of its energy escapes to the environment. An important physical effect of the interaction of the environment on a quantum system is decoherence which give an explanation for the mechanism driving the quantum-to-classical transition. We restrict ourself to few mathematical problems related with this very active field of research in physics [155, 200, 231, 270]. In a first part we study the purity of a state of (S) (or the linear entropy) in a short time approximation. In a second part we discuss the separability of bipartite systems. In a third part we establish a master equation (or a Fokker-Planck type equation) for the time evolution of the reduced matrix density for coupled quadratic Hamiltonians by a bilinear inter-action. We give an explicit formula for the solution of this master equation so that the time evolution of the reduced density at time t is written as a convolution integral for the reduced density at initial time \(t_0=0\), with a Gaussian kernel, for \(0 \le t < t_c\) where \(t_c\in ]0, \infty ]\) is a critical time such that reversibility is lost for \(t > t_c\) if \(t_c<+\infty \). Some manifestations of decoherence, for \(t>0\) small enough, are proved on our simple model which is similar to a representation of the Schrödinger cat.

Theoretical and mathematical physics•Jan 1, 2021•0 citations

Demystifying decoherence and the master equation of quantum Brownian motion

John King Gamble, John F. Lindner

Quantum decoherence is a proposed mechanism for the emergence of classical physics from the quantum mechanics. It has been developed extensively in recent years, but is sufficiently technically complicated to discourage widespread understanding. In this paper we provide a gentle introduction to quantum decoherence. We introduce state operators and their density matrix representations to describe composite systems, such as an experiment and its environment. We illustrate how the loss of information about a subsystem can cause a quantum system to appear classical. We first analyze a discrete example of phase randomization, then a Bell state, and finally a continuous system. In the final case we provide an accessible derivation of a major early result of decoherence theory, the master equation of quantum Brownian motion. We conclude by applying the master equation to the decoherence of a simple harmonic oscillator, with results reminiscent of our earlier discrete examples.

American Journal of Physics•Feb 12, 2009•14 citations

Open environments for quantum open systems

Michael R. Gallis

The majority of quantum open system models in the literature are simplistic in the sense that they only explicitly account for that part of the environment that directly interacts with the system of interest. A quantum open system with an open environment is examined using the projection operator method in the weak coupling limit. The openness of environment is modelled by nonunitary evolution of the Lindblad form. Under certain conditions, the resulting master equation for the system is insensitive to the initial state of the environment and to initial entanglements between the system and environment for time scales greater than the environment relaxation timescales. For the particular case of an environment consisting of a harmonic oscillator bath, the resulting master equations are demonstrated to have the algebraic form for completely positive evolution. The open environment model is illustrated for the particular case of a system linearly coupled to an oscillator bath.

arXiv:quant-ph/0210054•Oct 8, 2002•0 citations

Entanglement Generation and Evolution in Open Quantum Systems

Аурелиан Исар

In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we study the continuous variable entanglement for a system consisting of two independent harmonic oscillators interacting with a general environment. We solve the Kossakowski-Lindblad master equation for the time evolution of the considered system and describe the entanglement in terms of the covariance matrix for an arbitrary Gaussian input state. Using Peres–Simon necessary and sufficient criterion for separability of two-mode Gaussian states, we show that for certain values of diffusion and dissipation coefficients describing the environment, the state keeps for all times its initial type: separable or entangled. In other cases, entanglement generation, entanglement sudden death or a periodic collapse and revival of entanglement take place. We analyze also the time evolution of the logarithmic negativity, which characterizes the degree of entanglement of the quantum state.

arXiv:0912.5436•Open Systems & Information Dynamics•Aug 13, 2009•34 citations

Adjoint master equation for quantum Brownian motion

Matteo Carlesso, Angelo Bassi

Quantum Brownian motion is a fundamental model for a proper understanding of open quantum systems in different contexts such as chemistry, condensed-matter physics, biophysics, and optomechanics. In this paper we propose a different approach to describe this model. We provide an exact and analytic equation for the time evolution of the operators and we show that the corresponding equation for the states is equivalent to well-known results in the literature. The dynamics is expressed in terms of the spectral density, regardless of the strength of the coupling between the system and the bath. Our derivation allows to compute the time evolution of physically relevant quantities in a much easier way than previous formulations. An example is explicitly studied.

arXiv:1602.05116•Physical Review A•May 23, 2017•31 citations
The paper is aimed at revisiting some mathematical properties that appear in open quantum systems with quadratic Hamiltonians. In particular, a discussion is included of some properties related to the notions of purity, separability, and entanglement, introduced by physicists in quantum information theory. The results are applied, among other things, to coupled harmonic oscillators appearing in the quantum Brownian motion. Moreover, the master equation is revisited, which was obtained for the time evolution of an open system coupled to an environment modeled by any quadratic Hamiltonians.
Our main goal in this paper is to extend to any system of coupled quadratic Hamiltonians some properties known for systems of quantum harmonic oscillators related with the Brownian Quantum Motion model. In a first part we get a rather general formula for the purity (or the linear entropy) in a short time approximation. In a second part we establish a master equation (or a Fokker-Planck type equation) for the time evolution of the reduced matrix density for bilinearly coupled quadratic Hamiltonians. The Hamiltonians and the bilinear coupling can be time dependent. Moreover we give an explicit formula for the solution of this master equation so that the time evolution of the reduced density at time ttt is connected with the reduced density at initial time t0t_0t0​ for $t_0 \leq t
In this chapter we study some properties of systems of coupled quadratic quantum Hamiltonians. The well known example is coupled harmonic oscillators appearing in the Quantum Brownian Motion. An open system is a physical (S) interacting with the environment (E) such that the total system (S∪E)(S\cup E)(S∪E) is isolated from the rest of the universe. So we have two independent Hamiltonians H^S\hat{H}_SH^S​, H^E\hat{H}_EH^E​ and an interaction V^\hat{V}V^ between (S) and (E). The main problem discussed for open systems is to understand the dynamical action of the interaction V^\hat{V}V^ with the environment on the time dependent evolution of (S) which is now irreversible because a part of its energy escapes to the environment. An important physical effect of the interaction of the environment on a quantum system is decoherence which give an explanation for the mechanism driving the quantum-to-classical transition. We restrict ourself to few mathematical problems related with this very active field of research in physics [155, 200, 231, 270]. In a first part we study the purity of a state of (S) (or the linear entropy) in a short time approximation. In a second part we discuss the separability of bipartite systems. In a third part we establish a master equation (or a Fokker-Planck type equation) for the time evolution of the reduced matrix density for coupled quadratic Hamiltonians by a bilinear inter-action. We give an explicit formula for the solution of this master equation so that the time evolution of the reduced density at time t is written as a convolution integral for the reduced density at initial time t0=0t_0=0t0​=0, with a Gaussian kernel, for 0≤t<tc0 \le t < t_c0≤t<tc​ where tc∈]0,∞]t_c\in ]0, \infty ]tc​∈]0,∞] is a critical time such that reversibility is lost for t>tct > t_ct>tc​ if tc<+∞t_c<+\infty tc​<+∞. Some manifestations of decoherence, for t>0t>0t>0 small enough, are proved on our simple model which is similar to a representation of the Schrödinger cat.
Quantum decoherence is a proposed mechanism for the emergence of classical physics from the quantum mechanics. It has been developed extensively in recent years, but is sufficiently technically complicated to discourage widespread understanding. In this paper we provide a gentle introduction to quantum decoherence. We introduce state operators and their density matrix representations to describe composite systems, such as an experiment and its environment. We illustrate how the loss of information about a subsystem can cause a quantum system to appear classical. We first analyze a discrete example of phase randomization, then a Bell state, and finally a continuous system. In the final case we provide an accessible derivation of a major early result of decoherence theory, the master equation of quantum Brownian motion. We conclude by applying the master equation to the decoherence of a simple harmonic oscillator, with results reminiscent of our earlier discrete examples.
The majority of quantum open system models in the literature are simplistic in the sense that they only explicitly account for that part of the environment that directly interacts with the system of interest. A quantum open system with an open environment is examined using the projection operator method in the weak coupling limit. The openness of environment is modelled by nonunitary evolution of the Lindblad form. Under certain conditions, the resulting master equation for the system is insensitive to the initial state of the environment and to initial entanglements between the system and environment for time scales greater than the environment relaxation timescales. For the particular case of an environment consisting of a harmonic oscillator bath, the resulting master equations are demonstrated to have the algebraic form for completely positive evolution. The open environment model is illustrated for the particular case of a system linearly coupled to an oscillator bath.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we study the continuous variable entanglement for a system consisting of two independent harmonic oscillators interacting with a general environment. We solve the Kossakowski-Lindblad master equation for the time evolution of the considered system and describe the entanglement in terms of the covariance matrix for an arbitrary Gaussian input state. Using Peres–Simon necessary and sufficient criterion for separability of two-mode Gaussian states, we show that for certain values of diffusion and dissipation coefficients describing the environment, the state keeps for all times its initial type: separable or entangled. In other cases, entanglement generation, entanglement sudden death or a periodic collapse and revival of entanglement take place. We analyze also the time evolution of the logarithmic negativity, which characterizes the degree of entanglement of the quantum state.
Quantum Brownian motion is a fundamental model for a proper understanding of open quantum systems in different contexts such as chemistry, condensed-matter physics, biophysics, and optomechanics. In this paper we propose a different approach to describe this model. We provide an exact and analytic equation for the time evolution of the operators and we show that the corresponding equation for the states is equivalent to well-known results in the literature. The dynamics is expressed in terms of the spectral density, regardless of the strength of the coupling between the system and the bath. Our derivation allows to compute the time evolution of physically relevant quantities in a much easier way than previous formulations. An example is explicitly studied.