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

Physical Review A

Rate equation for locally time-averaged dynamics in weakly damped systems

James P. Stone

Published Nov 1, 1984• 4 citations
Publisher

Abstract

We consider the density matrix for weakly damped quantum oscillators. It is proved that the locally time-averaged density matrix derived from the full density matrix satisfies a closed set of equations, which reduce under specified conditions to population rate equations. Rate coefficients in these equations include contributions from collision-induced transitions and internal collisionless transitions explicitly arranged as sequential processes. We compare solutions of the new rate equations with the full density matrix for a four-level driven system. It is shown that despite some notable success in matching locally time-averaged quantities by extending the Wilcox-Lamb formalism to the weak-damping limit, the new equations are more naturally suited for this limit. In contrast with the Wilcox-Lamb formalism, we show that the new equations give a representation of the local time average whose accuracy is limited only by the degree of time-scale separation between collisions and internal dynamics.

DOI
10.1103/physreva.30.2517

Related papers

Analytic approach to quantum-mechanical time averages

James P. Stone

We show that the time-averaged density matrix which involves the solution of any Liouville equation with a specified initial condition is also the steady-state solution of an appropriately constructed Bloch equation taken in the weak-damping limit. This leads to new explicit expressions for the time-averaged density matrix which, unlike expressions in general use, give a completely analytic relation between the time average and the initial condition and contain no numerically determined eigenvector components. We apply this approach to driven multilevel systems to find new expressions for the time-averaged level populations in a three-level system. A strong initial-condition dependence is exhibited even for an equally spaced resonantly driven system. Analytic forms specifically showing a Stark-shifted multiphoton resonance and the associated line shape are also derived.

Physical Review A•Aug 1, 1982•12 citations

Solutions of the damped oscillator Fokker-Planck equation

W. H. Louisell, John H. Marburger

The quantum theory of damping is presented and illustrated by means of a driven damped harmonic oscillator. The theory is formulated in the coherent state representation which illustrates very vividly the nearly classical nature of the problem. In this representation the reduced system density operator equation becomes a Fokker-Planck equation. Green's function solutions are found for the damped oscillator in closed form and as an eigenfunction expansion. In addition, a quantum regression theorem due to Lax is derived in the coherent state representation. The theorem allows two-time averages to be computed from one-time averages.

IEEE Journal of Quantum Electronics•Aug 1, 1967•54 citations

Density matrix for the damped harmonic oscillator within the Lindblad theory

Аурелиан Исар, Aurel Sǎndulescu, W. Scheid

The time evolution of the density matrix of the damped harmonic oscillator is studied within the Lindblad theory for open quantum systems. The density matrix is represented via a generating function, which is obtained by solving a time-dependent linear partial differential equation derived from the master equation of the damped harmonic oscillator. Illustrative examples for specific initial conditions of the density matrix are provided. It is also shown that various master equations for the damped quantum oscillator, for damped collective modes in deep inelastic collisions of heavy ions and in different models of quantum optics are particular cases of the Lindblad equation and that only some of these equations satisfy the quantum mechanical constraints on the diffusion coefficients.

Journal of Mathematical Physics•Sep 1, 1993•44 citations

Quantum-mechanical derivation of the Bloch equations: Beyond the weak-coupling limit

Brian B. Laird, J. Budimir, James L. Skinner

Two nondegenerate quantum levels coupled off-diagonally and linearly to a bath of quantum-mechanical harmonic oscillators are considered. In the weak-coupling limit one finds that the equations of motion for the reduced density-matrix elements separate naturally into two uncoupled pairs of linear equations for the diagonal and off-diagonal elements, which are known as the Bloch equations. The equations for the populations form the simplest two-component master equation, and the rate constant for the relaxation of nonequilibrium population distributions is 1/T1, defined as the sum of the ‘‘up’’ and ‘‘down’’ rate constants in the master equation. Detailed balance is satisfied for this master equation in that the ratio of these rate constants is equal to the ratio of the equilibrium populations. The relaxation rate constant for the off-diagonal density-matrix elements is known as 1/T2. One finds that this satisfies the well-known relation 1/T2=1/2T1. In this paper the weak-coupling limit is transcended by deriving the Bloch equations to fourth order in the coupling. The equations have the same form as in the weak-coupling limit, but the rate constants are calculated to fourth order. For the population-relaxation rate constants this results in an extension to fourth order of Fermi’s golden rule. We find that these higher-order rate constants do indeed satisfy detailed balance. Comparing the dephasing and population-relaxation rate constants, we find that in fourth order 1/T2≠1/2T1.

The Journal of Chemical Physics•Mar 15, 1991•160 citations

Damped quantum harmonic oscillator

Аурелиан Исар, A. Sãndulescu

In the framework of the Lindblad theory for open quantum systems the damping of the harmonic oscillator is studied. A generalization of the fundamental constraints on quantum mechanical diffusion coefficients which appear in the master equation for the damped quantum oscillator is presented; the Schrödinger and Heisenberg representations of the Lindblad equation are given explicitly. On the basis of these representations it is shown that various master equations for the damped quantum oscillator used in the literature are particular cases of the Lindblad equation and that the majority of these equations are not satisfying the constraints on quantum mechanical diffusion coefficients. Analytical expressions for the first two moments of coordinate and momentum are also obtained by using the characteristic function of the Lindblad master equation. The master equation is transformed into Fokker-Planck equations for quasiprobability distributions. A comparative study is made for the Glauber $P$ representation, the antinormal ordering $Q$ representation and the Wigner $W$ representation. It is proven that the variances for the damped harmonic oscillator found with these representations are the same. By solving the Fokker-Planck equations in the steady state, it is shown that the quasiprobability distributions are two-dimensional Gaussians with widths determined by the diffusion coefficients. The density matrix is represented via a generating function, which is obtained by solving a time-dependent linear partial differential equation derived from the master equation. Illustrative examples for specific initial conditions of the density matrix are provided.

arXiv:quant-ph/0602149•Feb 17, 2006•5 citations

The quantum mechanical density operator and its time evolution

Abraham Nitzan

Abstract This chapter introduces the density operator and the Liouville equation and outlines how reduced dynamical description of open quantum systems naturally leads to the appearance of relaxation processes. (1) The density operator and the quantum Liouville equation (the density matrix for a pure system, statistical mixtures, representations, coherences, thermodynamic equilibrium). (2) An example: The time evolution of a two-level system in the density matrix formalism. (3) Reduced descriptions. (4) Time evolution equations for reduced density operators: The quantum master equation (projection operators, The Nakajima–Zwanzig equation, Derivation of the quantum master equation using the thermal projector, The quantum master equation in the interaction representation, The Markovian limit—the Redfield equation, The secular approximation and the Lindblad equation). (5) The Lindblad equation in the ‘Heisenberg representation’. (6) Relaxation of a two-level system and a harmonic oscillator revisited. (7) The optically driven two-level system in a thermal environment—the Bloch equations. (8) The oscillating dipole model of a two-level molecule. (9) Analogy of a coupled two-level system to a spin ½ system in a magnetic field.

•Jun 14, 2024•0 citations
We consider the density matrix for weakly damped quantum oscillators. It is proved that the locally time-averaged density matrix derived from the full density matrix satisfies a closed set of equations, which reduce under specified conditions to population rate equations. Rate coefficients in these equations include contributions from collision-induced transitions and internal collisionless transitions explicitly arranged as sequential processes. We compare solutions of the new rate equations with the full density matrix for a four-level driven system. It is shown that despite some notable success in matching locally time-averaged quantities by extending the Wilcox-Lamb formalism to the weak-damping limit, the new equations are more naturally suited for this limit. In contrast with the Wilcox-Lamb formalism, we show that the new equations give a representation of the local time average whose accuracy is limited only by the degree of time-scale separation between collisions and internal dynamics.
We show that the time-averaged density matrix which involves the solution of any Liouville equation with a specified initial condition is also the steady-state solution of an appropriately constructed Bloch equation taken in the weak-damping limit. This leads to new explicit expressions for the time-averaged density matrix which, unlike expressions in general use, give a completely analytic relation between the time average and the initial condition and contain no numerically determined eigenvector components. We apply this approach to driven multilevel systems to find new expressions for the time-averaged level populations in a three-level system. A strong initial-condition dependence is exhibited even for an equally spaced resonantly driven system. Analytic forms specifically showing a Stark-shifted multiphoton resonance and the associated line shape are also derived.
The quantum theory of damping is presented and illustrated by means of a driven damped harmonic oscillator. The theory is formulated in the coherent state representation which illustrates very vividly the nearly classical nature of the problem. In this representation the reduced system density operator equation becomes a Fokker-Planck equation. Green's function solutions are found for the damped oscillator in closed form and as an eigenfunction expansion. In addition, a quantum regression theorem due to Lax is derived in the coherent state representation. The theorem allows two-time averages to be computed from one-time averages.
The time evolution of the density matrix of the damped harmonic oscillator is studied within the Lindblad theory for open quantum systems. The density matrix is represented via a generating function, which is obtained by solving a time-dependent linear partial differential equation derived from the master equation of the damped harmonic oscillator. Illustrative examples for specific initial conditions of the density matrix are provided. It is also shown that various master equations for the damped quantum oscillator, for damped collective modes in deep inelastic collisions of heavy ions and in different models of quantum optics are particular cases of the Lindblad equation and that only some of these equations satisfy the quantum mechanical constraints on the diffusion coefficients.
Two nondegenerate quantum levels coupled off-diagonally and linearly to a bath of quantum-mechanical harmonic oscillators are considered. In the weak-coupling limit one finds that the equations of motion for the reduced density-matrix elements separate naturally into two uncoupled pairs of linear equations for the diagonal and off-diagonal elements, which are known as the Bloch equations. The equations for the populations form the simplest two-component master equation, and the rate constant for the relaxation of nonequilibrium population distributions is 1/T1, defined as the sum of the ‘‘up’’ and ‘‘down’’ rate constants in the master equation. Detailed balance is satisfied for this master equation in that the ratio of these rate constants is equal to the ratio of the equilibrium populations. The relaxation rate constant for the off-diagonal density-matrix elements is known as 1/T2. One finds that this satisfies the well-known relation 1/T2=1/2T1. In this paper the weak-coupling limit is transcended by deriving the Bloch equations to fourth order in the coupling. The equations have the same form as in the weak-coupling limit, but the rate constants are calculated to fourth order. For the population-relaxation rate constants this results in an extension to fourth order of Fermi’s golden rule. We find that these higher-order rate constants do indeed satisfy detailed balance. Comparing the dephasing and population-relaxation rate constants, we find that in fourth order 1/T2≠1/2T1.
In the framework of the Lindblad theory for open quantum systems the damping of the harmonic oscillator is studied. A generalization of the fundamental constraints on quantum mechanical diffusion coefficients which appear in the master equation for the damped quantum oscillator is presented; the Schrödinger and Heisenberg representations of the Lindblad equation are given explicitly. On the basis of these representations it is shown that various master equations for the damped quantum oscillator used in the literature are particular cases of the Lindblad equation and that the majority of these equations are not satisfying the constraints on quantum mechanical diffusion coefficients. Analytical expressions for the first two moments of coordinate and momentum are also obtained by using the characteristic function of the Lindblad master equation. The master equation is transformed into Fokker-Planck equations for quasiprobability distributions. A comparative study is made for the Glauber PPP representation, the antinormal ordering QQQ representation and the Wigner WWW representation. It is proven that the variances for the damped harmonic oscillator found with these representations are the same. By solving the Fokker-Planck equations in the steady state, it is shown that the quasiprobability distributions are two-dimensional Gaussians with widths determined by the diffusion coefficients. The density matrix is represented via a generating function, which is obtained by solving a time-dependent linear partial differential equation derived from the master equation. Illustrative examples for specific initial conditions of the density matrix are provided.
Abstract This chapter introduces the density operator and the Liouville equation and outlines how reduced dynamical description of open quantum systems naturally leads to the appearance of relaxation processes. (1) The density operator and the quantum Liouville equation (the density matrix for a pure system, statistical mixtures, representations, coherences, thermodynamic equilibrium). (2) An example: The time evolution of a two-level system in the density matrix formalism. (3) Reduced descriptions. (4) Time evolution equations for reduced density operators: The quantum master equation (projection operators, The Nakajima–Zwanzig equation, Derivation of the quantum master equation using the thermal projector, The quantum master equation in the interaction representation, The Markovian limit—the Redfield equation, The secular approximation and the Lindblad equation). (5) The Lindblad equation in the ‘Heisenberg representation’. (6) Relaxation of a two-level system and a harmonic oscillator revisited. (7) The optically driven two-level system in a thermal environment—the Bloch equations. (8) The oscillating dipole model of a two-level molecule. (9) Analogy of a coupled two-level system to a spin ½ system in a magnetic field.