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 Physics A Mathematical and General

Stochastic evolution of a harmonic oscillator in a photon reservoir

David Applebaum, Paul W. Robinson

Published May 11, 1987• 3 citations
Publisher

Abstract

The evolution of a harmonic oscillator in a photon reservoir is described by a quantum stochastic differential equation via the Wigner-Weisskopf approximation. The rotating wave approximation is shown to be equivalent to the requirement of quantum detailed balance and the irreversible evolution of the initial equilibrium state is explicitly computed.

DOI
10.1088/0305-4470/20/7/024

Related papers

Quantum dynamics of a harmonic oscillator in a non-equilibrium bath without the rotating wave approximation

Daeimohammad, Mohsen

The aim of this paper is to investigate dynamical properties of a harmonic oscillator coupled to a non-equilibrium bath without the rotating wave approximation. At first, we investigate the theory of this model and obtain the Heisenberg equations of motion for the oscillator and bath annihilation operators in the absence and in the presence of a rotating wave approximation. Then, we obtain the fluctuation and fluctuation-dissipation relations in the absence and in the presence of a rotating wave approximation for the non-equilibrium bath annihilation operator, respectively. After that, we study the decay of system modes using the Wigner-Weisskopf approximation. Finally, we calculate the time-dependent spectrum of a cavity field mode coupled to a non-equilibrium reservoir. We show that even under the condition in which the RWA is considered to be valid (weak interaction), there are significant effects of virtual-photon field on the diffusion coefficient, the energy density, line width and the shift frequency.

Iranian Journal of Physics Research•May 1, 2022•0 citations

The stochastic Wigner–Weisskopf atom

David Applebaum

Using techniques of quantum stochastic calculus, a dilation scheme is developed to describe the evolution of a two-level atom, which is initially in a (nontracial) Gibbs state, through a radiation field. The Wigner–Weisskopf approximation is assumed but not the rotating wave approximation, the latter being shown to be equivalent to the requirement that the dilation satisfies the quantum detailed balance condition with respect to the initial state.

Journal of Mathematical Physics•Aug 1, 1987•0 citations

Quantum theory of dissipation of a harmonic oscillator coupled to a non-equilibrium bath; Wigner-Weisskopf decay and physical spectra

Jyotipratim Ray Chaudhuri, Bimalendu Deb, Gautam Gangopadhyay +1 more

We extend the quantum theory of dissipation in the context of a system-reservoir model, where the reservoir in question is kept in a non-equilibrium condition. Based on a systematic separation of time scales involved in the dynamics, appropriate generalizations of the fluctuation-dissipation and Einstein's relations have been pointed out. We show that the Wigner-Weisskopf decay of the system mode results in a rate constant which, depending on the relaxation of a non-equilibrium bath, is dynamically modified. We also calculate the time-dependent spectra of a cavity mode with a suitable gain when the cavity is kept in contact with a non-equilibrium bath.

arXiv:physics/9902037•Journal of Physics B Atomic Molecular and Optical Physics•Sep 14, 1998•7 citations

Asymptotic regime of quantal stochastic and dissipative motion

Marcelo A. Despósito, Silvina M. Gatica, E. Susana Hernández

We analyze the irreversible behavior of a quantum harmonic oscillator immersed in an arbitrary thermal reservoir. For the fully coupled (FC) model, i.e., the coupling between subsystems linear in the coordinate of the oscillator, we construct a generalized master equation for the reduced density operator of the oscillator. We apply the rotating-wave approximation (RWA) and investigate its connection with the FC master equation. By means of the Wigner distribution, we obtain the corresponding Fokker-Planck equations in the semiclassical representation. We find that, when the RWA is not imposed, the correct classical limit is obtained (Kramer's equation). Then, we establish the conditions for the equivalence between the FC Fokker-Planck equation and that obtained in the RWA. Quantum maps for a kicked oscillator immersed in a heat bath are obtained in both cases. These maps are studied in the classical and semiclassical limits, and it is shown that they coincide for low kicking frequency compared to the damping rate.

Physical Review A•Sep 1, 1992•11 citations

Quantum-Mechanical Description of Two Coupled Harmonic Oscillators

Lee E. Estes, Thomas H. Keil, Lorenzo M. Narducci

Systems of interacting harmonic oscillators have recently received considerable attention as models for describing a variety of physical problems. We have investigated the validity of the rotating-wave approximation which constitutes the traditional approach to the solution of the dynamical problem by comparing it with the exact solution. A numerical comparison has been made and the limits of validity of the rotating-wave approximation has been established in terms of the strength of oscillator interaction.In particular, the time development of dynamical operators and certain transition probabilities have been compared. In the region where the rotating-wave approximation is valid, the time evolution of the quasiprobability distribution $P(\ensuremath{\alpha}, t)$ of one oscillator is given for several initial conditions. A counting scheme similar to the argument given by Feynman for the driven harmonic oscillator, is proposed for the interpretation of the time-dependent transition amplitudes between number states.

Physical Review•Nov 5, 1968•74 citations

Energy distribution of the quantum harmonic oscillator under random time-dependent perturbations

Josselin C. Garnier

This paper investigates the evolution of a quantum particle in a harmonic oscillator driven by time-dependent forces. The perturbations are small, but they act long enough so that we can solve the problem in the asymptotic framework corresponding to a perturbation amplitude that tends to zero and a perturbation duration that tends to infinity. We describe the effective evolution equation of the state vector, which reads as a stochastic partial differential equation. We exhibit a closed-form equation for the transition probabilities, which can be interpreted in terms of a jump process. Using standard probability tools, we are then able to compute explicitly the probabilities for observing the different energy eigenstates and give the exact statistical distribution of the energy of the particle.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Oct 1, 1999•8 citations
The evolution of a harmonic oscillator in a photon reservoir is described by a quantum stochastic differential equation via the Wigner-Weisskopf approximation. The rotating wave approximation is shown to be equivalent to the requirement of quantum detailed balance and the irreversible evolution of the initial equilibrium state is explicitly computed.
The aim of this paper is to investigate dynamical properties of a harmonic oscillator coupled to a non-equilibrium bath without the rotating wave approximation. At first, we investigate the theory of this model and obtain the Heisenberg equations of motion for the oscillator and bath annihilation operators in the absence and in the presence of a rotating wave approximation. Then, we obtain the fluctuation and fluctuation-dissipation relations in the absence and in the presence of a rotating wave approximation for the non-equilibrium bath annihilation operator, respectively. After that, we study the decay of system modes using the Wigner-Weisskopf approximation. Finally, we calculate the time-dependent spectrum of a cavity field mode coupled to a non-equilibrium reservoir. We show that even under the condition in which the RWA is considered to be valid (weak interaction), there are significant effects of virtual-photon field on the diffusion coefficient, the energy density, line width and the shift frequency.
We extend the quantum theory of dissipation in the context of a system-reservoir model, where the reservoir in question is kept in a non-equilibrium condition. Based on a systematic separation of time scales involved in the dynamics, appropriate generalizations of the fluctuation-dissipation and Einstein's relations have been pointed out. We show that the Wigner-Weisskopf decay of the system mode results in a rate constant which, depending on the relaxation of a non-equilibrium bath, is dynamically modified. We also calculate the time-dependent spectra of a cavity mode with a suitable gain when the cavity is kept in contact with a non-equilibrium bath.
We analyze the irreversible behavior of a quantum harmonic oscillator immersed in an arbitrary thermal reservoir. For the fully coupled (FC) model, i.e., the coupling between subsystems linear in the coordinate of the oscillator, we construct a generalized master equation for the reduced density operator of the oscillator. We apply the rotating-wave approximation (RWA) and investigate its connection with the FC master equation. By means of the Wigner distribution, we obtain the corresponding Fokker-Planck equations in the semiclassical representation. We find that, when the RWA is not imposed, the correct classical limit is obtained (Kramer's equation). Then, we establish the conditions for the equivalence between the FC Fokker-Planck equation and that obtained in the RWA. Quantum maps for a kicked oscillator immersed in a heat bath are obtained in both cases. These maps are studied in the classical and semiclassical limits, and it is shown that they coincide for low kicking frequency compared to the damping rate.
Systems of interacting harmonic oscillators have recently received considerable attention as models for describing a variety of physical problems. We have investigated the validity of the rotating-wave approximation which constitutes the traditional approach to the solution of the dynamical problem by comparing it with the exact solution. A numerical comparison has been made and the limits of validity of the rotating-wave approximation has been established in terms of the strength of oscillator interaction.In particular, the time development of dynamical operators and certain transition probabilities have been compared. In the region where the rotating-wave approximation is valid, the time evolution of the quasiprobability distribution $P(\ensuremath{\alpha}, t)$ of one oscillator is given for several initial conditions. A counting scheme similar to the argument given by Feynman for the driven harmonic oscillator, is proposed for the interpretation of the time-dependent transition amplitudes between number states.
This paper investigates the evolution of a quantum particle in a harmonic oscillator driven by time-dependent forces. The perturbations are small, but they act long enough so that we can solve the problem in the asymptotic framework corresponding to a perturbation amplitude that tends to zero and a perturbation duration that tends to infinity. We describe the effective evolution equation of the state vector, which reads as a stochastic partial differential equation. We exhibit a closed-form equation for the transition probabilities, which can be interpreted in terms of a jump process. Using standard probability tools, we are then able to compute explicitly the probabilities for observing the different energy eigenstates and give the exact statistical distribution of the energy of the particle.