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

The Journal of Chemical Physics

Variational treatment of a harmonic oscillator coupled to a dissipative heat bath

P. E. Parris, R. Silbey

Published Jun 1, 1987• 10 citations
Publisher

Abstract

We consider the problem of a single quantum oscillator coupled linearly to a heat bath of independent harmonic modes. An exact solution is presented for the system-oscillator observables of interest. The exact results are then used to evaluate the utility of a variational approach to the problem that has proven useful recently in elucidating the dynamics of dissipatively coupled systems. We find that the variational approach does provide a good description for most, but not all, observables of interest. Both the exact and the variational treatment demonstrate the important role played by the low-frequency bath modes in determining qualitative features of the dynamical behavior.

DOI
10.1063/1.452426

Related papers

Exact results for a damped quantum-mechanical harmonic oscillator

Peter S. Riseborough, Peter Hänggi A, Ulrich Weiß

Exact calculations for a quantum oscillator linearly coupled to a thermal reservoir are presented. Explicit results (variances and correlations) for the response and the spontaneous fluctuations are evaluated for two different models of the heat bath; a reservoir with a spectrum which is similar to that of acoustic phonons and another which is analogous to optic phonons. The low-temperature behavior is shown to depend sensitively on the spectral density of the thermal reservoir. The mass and the frequency of the oscillator become renormalized through the coupling to the bath. These renormalizations are intrinsically frequency dependent. Moreover, the commonly used form of Ehrenfest's equation for the dynamics of the linearly damped oscillator is shown to only be a reasonable approximation within certain frequency regimes.

Physical Review A•Jan 1, 1985•131 citations

Dissipative dynamics of a two-state system coupled to a heat bath

Lit-Deh Chang, Sudip Chakravarty

The dynamics of a two-state system coupled to the dissipative degrees of freedom of the environment appears in many different contexts. In this paper we have given a general formulation of the problem as well as a detailed comparison of two a priori very different heat baths: (a) a collection of harmonic oscillators coupled to the system, and (b) a collection of fermions coupled to the system. Consideration of such heat baths is motivated by the assertion that the weakly excited states of the environment can be modeled by a collection of elementary excitations following Landau. Our conclusion is that the case of Ohmic dissipation of harmonic-oscillator heat bath is very similar to the fermionic bath, although a crucially important dimensionless parameter \ensuremath{\alpha} assumes a different range of values in the two different cases.

Physical review. B, Condensed matter•Jan 1, 1985•115 citations

Dissipative quantum oscillator with two competing heat baths

Heiner Kohler, Fernando Sols

We study the dissipative dynamics of a harmonic oscillator which couples linearly through its position and its momentum to two independent heat baths at the same temperature. We argue that this model describes a large spin in a ferromagnet. We find that some effects of the two heat baths partially cancel each other. This leads to unexpected features such as underdamped oscillations and long relaxation times in the strong coupling regime. Such a partial frustration of dissipation can be ascribed to the canonically conjugate character of position and momentum. We compare this model to the scenario where a single heat bath couples linearly to both the position and the momentum of the central oscillator. In that case less surprising behaviour occurs for strong coupling. The dynamical evolution of the quantum purity for a single and a double wave packet is also investigated.

arXiv:cond-mat/0608484•New Journal of Physics•Aug 25, 2006•21 citations

Probing the dynamics of an open quantum system via a single qubit

Pablo Carlos López Vázquez, T. Gorin

Abstract We investigate strategies to monitor the dynamics and determine the characteristics of a dissipative harmonic oscillator coupled to a finite temperature heat bath. For this purpose we use a single qubit coupled via dephasing to the oscillator. The whole system is described by a quantum master equation, which can be solved analytically. The solution is surprisingly simple and allows one to read off all relevant parameters from the behavior of the qubit coherence as a function of time. Analyzing the qubit coherence for a short time, turns out to be sufficient to predict the complete dynamics of the oscillator, including its asymptotic state at thermal equilibrium.

Journal of Physics A Mathematical and Theoretical•Jun 22, 2020•5 citations

Decoherence of Coupled Quantum Oscillators: Going Beyond the Caldeira-Leggett Approximation

Björn Bartels

We calculate the reduced density matrix for a system of coupled harmonic oscillators in a bosonic heat bath using the Born-Markov approximation and show that the expectation values of position and momentum evolve like classical quantities. We consider the cases where every oscillator is coupled to its own reservoir and where all oscillators are coupled to the same heat bath, pointing out the differences between these two models. In particular, for systems coupled to a common reservoir, we present an uncommon dissipation mechanism, which disappears, when the dynamics of the system are gouverned by the low frequency modes of the environment (Caldeira-Leggett limit). We also address the interesting phenomenon of reservoir-induced interactions.

arXiv:1106.1371•Jun 7, 2011•0 citations

The quantum harmonic oscillator in a dissipative bath of anyon pairs

Nils-Henrik Meyer, Axel Pelster, Michael Thorwart

Abstract We generalize the formalism of open quantum systems to introduce anyon baths. In particular, we work out a dissipative anyon bath composed of independent pairs of one-dimensional Grundberg–Hansson harmonically bound anyons, which are characterized by one statistical parameter. Using a mapping of these anyons to a bosonic bath with rescaled oscillator frequencies, we show that the original bilinear system-bath coupling assumes a particular non-polynomial form. To determine the relaxation properties, we use the imaginary-time path integral formalism together with a generalization of Wick’s theorem in the form of a smearing formula. The latter allows to approximately calculate the anyon bath spectral density, which acquires a nontrivial temperature dependence. The corresponding relaxation dynamics of the dissipative harmonic oscillator in an anyon bath is found. Well defined limits are revealed for both low and high temperatures. Anyonic features turn out to be most pronounced in the regime of intermediate temperatures.

arXiv:2604.22342•New Journal of Physics•Jun 24, 2026•0 citations
We consider the problem of a single quantum oscillator coupled linearly to a heat bath of independent harmonic modes. An exact solution is presented for the system-oscillator observables of interest. The exact results are then used to evaluate the utility of a variational approach to the problem that has proven useful recently in elucidating the dynamics of dissipatively coupled systems. We find that the variational approach does provide a good description for most, but not all, observables of interest. Both the exact and the variational treatment demonstrate the important role played by the low-frequency bath modes in determining qualitative features of the dynamical behavior.
Exact calculations for a quantum oscillator linearly coupled to a thermal reservoir are presented. Explicit results (variances and correlations) for the response and the spontaneous fluctuations are evaluated for two different models of the heat bath; a reservoir with a spectrum which is similar to that of acoustic phonons and another which is analogous to optic phonons. The low-temperature behavior is shown to depend sensitively on the spectral density of the thermal reservoir. The mass and the frequency of the oscillator become renormalized through the coupling to the bath. These renormalizations are intrinsically frequency dependent. Moreover, the commonly used form of Ehrenfest's equation for the dynamics of the linearly damped oscillator is shown to only be a reasonable approximation within certain frequency regimes.
The dynamics of a two-state system coupled to the dissipative degrees of freedom of the environment appears in many different contexts. In this paper we have given a general formulation of the problem as well as a detailed comparison of two a priori very different heat baths: (a) a collection of harmonic oscillators coupled to the system, and (b) a collection of fermions coupled to the system. Consideration of such heat baths is motivated by the assertion that the weakly excited states of the environment can be modeled by a collection of elementary excitations following Landau. Our conclusion is that the case of Ohmic dissipation of harmonic-oscillator heat bath is very similar to the fermionic bath, although a crucially important dimensionless parameter \ensuremath{\alpha} assumes a different range of values in the two different cases.
We study the dissipative dynamics of a harmonic oscillator which couples linearly through its position and its momentum to two independent heat baths at the same temperature. We argue that this model describes a large spin in a ferromagnet. We find that some effects of the two heat baths partially cancel each other. This leads to unexpected features such as underdamped oscillations and long relaxation times in the strong coupling regime. Such a partial frustration of dissipation can be ascribed to the canonically conjugate character of position and momentum. We compare this model to the scenario where a single heat bath couples linearly to both the position and the momentum of the central oscillator. In that case less surprising behaviour occurs for strong coupling. The dynamical evolution of the quantum purity for a single and a double wave packet is also investigated.
Abstract We investigate strategies to monitor the dynamics and determine the characteristics of a dissipative harmonic oscillator coupled to a finite temperature heat bath. For this purpose we use a single qubit coupled via dephasing to the oscillator. The whole system is described by a quantum master equation, which can be solved analytically. The solution is surprisingly simple and allows one to read off all relevant parameters from the behavior of the qubit coherence as a function of time. Analyzing the qubit coherence for a short time, turns out to be sufficient to predict the complete dynamics of the oscillator, including its asymptotic state at thermal equilibrium.
We calculate the reduced density matrix for a system of coupled harmonic oscillators in a bosonic heat bath using the Born-Markov approximation and show that the expectation values of position and momentum evolve like classical quantities. We consider the cases where every oscillator is coupled to its own reservoir and where all oscillators are coupled to the same heat bath, pointing out the differences between these two models. In particular, for systems coupled to a common reservoir, we present an uncommon dissipation mechanism, which disappears, when the dynamics of the system are gouverned by the low frequency modes of the environment (Caldeira-Leggett limit). We also address the interesting phenomenon of reservoir-induced interactions.
Abstract We generalize the formalism of open quantum systems to introduce anyon baths. In particular, we work out a dissipative anyon bath composed of independent pairs of one-dimensional Grundberg–Hansson harmonically bound anyons, which are characterized by one statistical parameter. Using a mapping of these anyons to a bosonic bath with rescaled oscillator frequencies, we show that the original bilinear system-bath coupling assumes a particular non-polynomial form. To determine the relaxation properties, we use the imaginary-time path integral formalism together with a generalization of Wick’s theorem in the form of a smearing formula. The latter allows to approximately calculate the anyon bath spectral density, which acquires a nontrivial temperature dependence. The corresponding relaxation dynamics of the dissipative harmonic oscillator in an anyon bath is found. Well defined limits are revealed for both low and high temperatures. Anyonic features turn out to be most pronounced in the regime of intermediate temperatures.