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

Reconsideration of Mori's Theory of Generalized Brownian Motion and Its Application

Masao Hirokawa

Published Apr 20, 1994• 0 citations
OpenAlex

Abstract

For every system with a passably general Hamiltonian of a quantum harmonic oscillator coupled to infinitely many scalar bosons, we estimate Heisenberg pictures of the annihilation and creation operators, describing the quantum harmonic oscillator, from only data on the autocorrelation function of the operators such that the pictures restore their autocorrelation functions to original ones, by reconsidering Mori's theory of generalized Brownian motion. And we then investigate the Liouville space and Liouville operator associated with the estimation.

Related papers

Generalized Langevin equations of Mori and quantum stochastic processes

Hiroshi Nakazawa

The structure of the projection-operator theory of Mori for quantum Brownian motion is reconsidered, emphasizing the nature of underlying quantum processes. Quantum-mechanical assumptions of the theory are extracted to sets of axioms. The Hilbert space scrV formed with relevant observables for quantum generalized Langevin equations (QGLE's) includes, in general, elements that cannot be operators on state vectors of the original system. The definition of Liouville operator L, as a mapping from one operator to another, thus requires its reconstruction in scrV. The work shows that the mentioned axioms not only ensure a unified reconstruction of L (hence of QGLE's) in systems in stationary states that may be nonequilibrium as well as equilibrium, but also extend intrinsically the domain of applicability of QGLE's. Illustrations of the setting for these extended QGLE's are presented by concrete examples of Hilbert spaces formed with quantum-mechanical Gaussian and non-Gaussian random variables, showing specifically the significance of a response-function-type inner product that assigns \ensuremath{\delta} correlation to the quantized Gaussian white noise which is an operator-valued distributional process with Kubo-Martin-Schwinger property.

Physical Review A•May 1, 1989•2 citations

Brownian Motion of a Quantum Oscillator

G. S. Agarwal

The theory of Brownian motion of a quantum oscillator is developed. The Brownian motion is described by a model Hamiltonian which is taken to be the one describing the interaction between this oscillator and a reservoir. Use is made of the master equation recently derived by the author, to obtain the equation of motion for the various reduced phase-space distribution functions that are obtained by mapping the density operator onto $c$-number functions. The equations of motion for the reduced phase-space distribution functions are found to be of the Fokker-Planck type. On transforming the Fokker-Planck equation to real variables, it is found to have the same form as the Fokker-Planck equation obtained by Wang and Uhlenbeck to describe the Brownian motion of a classical oscillator. The Fokker-Planck equation is solved for the conditional probability (Green's function) which is found to be in the form of a two-dimensional Gaussian distribution. This solution is then used to obtain various time-dependent quantum statistical properties of the oscillator. Next, the entropy for a quantum oscillator undergoing Brownian motion is calculated and we show that this system approaches equilibrium as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally we show that in the weak-coupling limit the Fokker-Planck equation reduces to the one obtained by making the usual rotating-wave approximation.

Physical Review A•Aug 1, 1971•208 citations

Quantum Brownian Motion in the Heisenberg picture: derivation of the adjoint master equation and applications

Matteo Carlesso, Angelo Bassi

Open quantum system theory is fundamental for a proper description of quantum systems in different contexts such as chemistry, condensed matter physics, bio-physics and opto-mechamics. Quantum brownian motion which is analytically described by the Hu-Paz-Zhang (HPZ) model represents one of the most important examples of an open quantum system. In this paper we propose a novel derivation of its master equation, based on the Heisenberg picture. We provide an exact and analytic equation both for the operators as well as for the states. The result, equivalent to the one derived originally by HPZ, is expressed in terms of the spectral density, regardless the strength of the coupling between the system and the environment and allows to compute the time evolution of physically relevant quantities in a much easier way, since it is directly expressed in terms of the spectral density. An example is explicitly studied.

•Feb 16, 2016•3 citations

On the Theory of Brownian Motion and Spin Relaxation

Takeyasu Kotera, Morikazu Toda

The Brownian motion of a harmonic oscillator is treated quantum-mechanically and the same method is applied to the relaxation of a spin. The method is quite general and applicable to a system displaced from the equilibrium state by an arbitrary amount initially. The oscillator or the spin is interacting with a thermostat composed of a great number of harmonic oscillators. The equation of motion is solved in operator form taking average as to the thermostat to the second order of the interaction parameter. The Langevin equation, the Kramers-Chandra-sekhar equation for a harmonic oscillator and the Bloch equation for a static field are derived.

Journal of the Physical Society of Japan•Nov 1, 1959•7 citations

Reconstruction approach to quantum dynamics of bosonic systems

Akseli Mäkinen, Joni Ikonen, Matti Partanen +1 more

We propose an approach to analytically solve the quantum dynamics of bosonic systems. The method is based on reconstructing the quantum state of the system from the moments of its annihilation operators, dynamics of which is solved in the Heisenberg picture. The proposed dynamical reconstruction method is general in the sense that it does not require assumptions on the initial conditions of the system such as separability, or the structure of the system such as linearity. It is an alternative to the standard master-equation approaches, which are analytically demanding especially for large multipartite quantum systems. To demonstrate the proposed technique, we apply it to a system consisting of two coupled damped quantum harmonic oscillators.

arXiv:1904.11353•Physical Review A•Oct 11, 2019•1 citation

Quantum Brownian motion with inhomogeneous damping and diffusion

Pietro Massignan, Aniello Lampo, Jan Wehr +1 more

We analyze the microscopic model of quantum Brownian motion, describing a Brownian particle interacting with a bosonic bath through a coupling which is linear in the creation and annihilation operators of the bath, but may be a nonlinear function of the position of the particle. Physically, this corresponds to a configuration in which damping and diffusion are spatially inhomogeneous. We derive systematically the quantum master equation for the Brownian particle in the Born-Markov approximation and we discuss the appearance of additional terms, for various polynomials forms of the coupling. We discuss the cases of linear and quadratic coupling in great detail and we derive, using Wigner function techniques, the stationary solutions of the master equation for a Brownian particle in a harmonic trapping potential. We predict quite generally Gaussian stationary states, and we compute the aspect ratio and the spread of the distributions. In particular, we find that these solutions may be squeezed (superlocalized) with respect to the position of the Brownian particle. We analyze various restrictions to the validity of our theory posed by non-Markovian effects and by the Heisenberg principle. We further study the dynamical stability of the system, by applying a Gaussian approximation to the time-dependent Wigner function, and we compute the decoherence rates of coherent quantum superpositions in position space. Finally, we propose a possible experimental realization of the physics discussed here, by considering an impurity particle embedded in a degenerate quantum gas.

arXiv:1410.8448•Physical Review A•Mar 23, 2015•36 citations
For every system with a passably general Hamiltonian of a quantum harmonic oscillator coupled to infinitely many scalar bosons, we estimate Heisenberg pictures of the annihilation and creation operators, describing the quantum harmonic oscillator, from only data on the autocorrelation function of the operators such that the pictures restore their autocorrelation functions to original ones, by reconsidering Mori's theory of generalized Brownian motion. And we then investigate the Liouville space and Liouville operator associated with the estimation.
The structure of the projection-operator theory of Mori for quantum Brownian motion is reconsidered, emphasizing the nature of underlying quantum processes. Quantum-mechanical assumptions of the theory are extracted to sets of axioms. The Hilbert space scrV formed with relevant observables for quantum generalized Langevin equations (QGLE's) includes, in general, elements that cannot be operators on state vectors of the original system. The definition of Liouville operator L, as a mapping from one operator to another, thus requires its reconstruction in scrV. The work shows that the mentioned axioms not only ensure a unified reconstruction of L (hence of QGLE's) in systems in stationary states that may be nonequilibrium as well as equilibrium, but also extend intrinsically the domain of applicability of QGLE's. Illustrations of the setting for these extended QGLE's are presented by concrete examples of Hilbert spaces formed with quantum-mechanical Gaussian and non-Gaussian random variables, showing specifically the significance of a response-function-type inner product that assigns \ensuremath{\delta} correlation to the quantized Gaussian white noise which is an operator-valued distributional process with Kubo-Martin-Schwinger property.
The theory of Brownian motion of a quantum oscillator is developed. The Brownian motion is described by a model Hamiltonian which is taken to be the one describing the interaction between this oscillator and a reservoir. Use is made of the master equation recently derived by the author, to obtain the equation of motion for the various reduced phase-space distribution functions that are obtained by mapping the density operator onto ccc-number functions. The equations of motion for the reduced phase-space distribution functions are found to be of the Fokker-Planck type. On transforming the Fokker-Planck equation to real variables, it is found to have the same form as the Fokker-Planck equation obtained by Wang and Uhlenbeck to describe the Brownian motion of a classical oscillator. The Fokker-Planck equation is solved for the conditional probability (Green's function) which is found to be in the form of a two-dimensional Gaussian distribution. This solution is then used to obtain various time-dependent quantum statistical properties of the oscillator. Next, the entropy for a quantum oscillator undergoing Brownian motion is calculated and we show that this system approaches equilibrium as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally we show that in the weak-coupling limit the Fokker-Planck equation reduces to the one obtained by making the usual rotating-wave approximation.
Open quantum system theory is fundamental for a proper description of quantum systems in different contexts such as chemistry, condensed matter physics, bio-physics and opto-mechamics. Quantum brownian motion which is analytically described by the Hu-Paz-Zhang (HPZ) model represents one of the most important examples of an open quantum system. In this paper we propose a novel derivation of its master equation, based on the Heisenberg picture. We provide an exact and analytic equation both for the operators as well as for the states. The result, equivalent to the one derived originally by HPZ, is expressed in terms of the spectral density, regardless the strength of the coupling between the system and the environment and allows to compute the time evolution of physically relevant quantities in a much easier way, since it is directly expressed in terms of the spectral density. An example is explicitly studied.
The Brownian motion of a harmonic oscillator is treated quantum-mechanically and the same method is applied to the relaxation of a spin. The method is quite general and applicable to a system displaced from the equilibrium state by an arbitrary amount initially. The oscillator or the spin is interacting with a thermostat composed of a great number of harmonic oscillators. The equation of motion is solved in operator form taking average as to the thermostat to the second order of the interaction parameter. The Langevin equation, the Kramers-Chandra-sekhar equation for a harmonic oscillator and the Bloch equation for a static field are derived.
We propose an approach to analytically solve the quantum dynamics of bosonic systems. The method is based on reconstructing the quantum state of the system from the moments of its annihilation operators, dynamics of which is solved in the Heisenberg picture. The proposed dynamical reconstruction method is general in the sense that it does not require assumptions on the initial conditions of the system such as separability, or the structure of the system such as linearity. It is an alternative to the standard master-equation approaches, which are analytically demanding especially for large multipartite quantum systems. To demonstrate the proposed technique, we apply it to a system consisting of two coupled damped quantum harmonic oscillators.
We analyze the microscopic model of quantum Brownian motion, describing a Brownian particle interacting with a bosonic bath through a coupling which is linear in the creation and annihilation operators of the bath, but may be a nonlinear function of the position of the particle. Physically, this corresponds to a configuration in which damping and diffusion are spatially inhomogeneous. We derive systematically the quantum master equation for the Brownian particle in the Born-Markov approximation and we discuss the appearance of additional terms, for various polynomials forms of the coupling. We discuss the cases of linear and quadratic coupling in great detail and we derive, using Wigner function techniques, the stationary solutions of the master equation for a Brownian particle in a harmonic trapping potential. We predict quite generally Gaussian stationary states, and we compute the aspect ratio and the spread of the distributions. In particular, we find that these solutions may be squeezed (superlocalized) with respect to the position of the Brownian particle. We analyze various restrictions to the validity of our theory posed by non-Markovian effects and by the Heisenberg principle. We further study the dynamical stability of the system, by applying a Gaussian approximation to the time-dependent Wigner function, and we compute the decoherence rates of coherent quantum superpositions in position space. Finally, we propose a possible experimental realization of the physics discussed here, by considering an impurity particle embedded in a degenerate quantum gas.