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

Effects of a thermal reservoir on variational functions in open systems

Matthias Jakob, Stig Stenholm

Published Jul 9, 2004• 16 citations
Publisher

Abstract

We connect the theory of a Lyapunov functional for an open system with the thermophysical concept of the relative entropy or equivalently with the entropy production rate. In order to do this we consider the environment of the open system as a thermal reservoir, which introduces the temperature as a thermophysical quantity. The general theory of the Lyapunov functional of an open system constructs a metric operator which contains an element of ambiguity in its scaling. This ambiguity can be removed by assuming a thermostatistic environment which fixes the scaling of the metric operator in a unique manner. The Lyapunov functional thus acquires a thermophysical meaning and can be connected with a relative entropy. In this case the metric operator is related to the inverse of the stationary density operator of the system. We illustrate the theory on a two-level atomic system and a degenerate three-level atom which are exposed to a thermal electromagnetic environment.

DOI
10.1103/physreva.70.012104

Related papers

Out-of-equilibrium quantum thermodynamics in the Bloch sphere: Temperature and internal entropy production

Andrés Vallejo, A. Romanelli, R. Donangelo

An explicit expression for the temperature of an open two-level quantum system is obtained as a function of local properties under the hypothesis of weak interaction with the environment. This temperature is defined for both equilibrium and out-of-equilibrium states and coincides with the environment temperature if the system reaches thermal equilibrium with a heat reservoir. Additionally, we show that within this theoretical framework the total entropy production can be partitioned into two contributions: one due to heat transfer and another, associated to internal irreversibilities, related to the loss of internal coherence by the qubit. The positiveness of the heat capacity is established, as well as its consistency with the well-known results at thermal equilibrium. We apply these concepts to two different systems and show that they behave in analogous ways as their classical counterparts.

arXiv:2004.04845•Physical review. E•Apr 27, 2020•7 citations

Variational functions in degenerate open quantum systems

Matthias Jakob, Stig Stenholm

We have derived a Lyapunov functional for a degenerate open atomic system. This functional develops monotonically towards its stationary state. The open system is described by a Lindblad-type master equation. For the construction of the variational functional it is necessary that the Lindblad operator can be diagonalized. Since the generator of motion is non-Hermitian, diagonalization is, in general, only possible if the eigenvalues are nondegenerate. In this paper, we propose that in a physical system the biorthogonal eigenbasis of the Lindblad operator remains complete even when degeneracy is present. Thus diagonalization of the Lindblad operator, and consequently the construction of the variational functional, is still possible. We discuss the reasons and illustrate the theory of the variational functional for a driven $\ensuremath{\Lambda}$-type three-level atom with degenerate ground state. The degeneracy has interesting effects on the variational functional in the steady state with respect to its interpretation as an entropic quantity. In case of the driven three-level atom, the dark state turns out to be an isentropic state.

Physical Review A•Apr 20, 2004•18 citations

Equilibrium states of open quantum systems in the strong coupling regime

Yiğit Subaşı, Christen H. Fleming, Jacob M. Taylor +1 more

In this work we investigate the late-time steady states of open quantum systems coupled to a thermal reservoir in the strong coupling regime. In general such systems do not necessarily relax to a Boltzmann distribution if the coupling to the thermal reservoir is nonvanishing or equivalently if the relaxation time scales are finite. Using a variety of nonequilibrium formalisms valid for non-Markovian processes, we show that starting from a product state of the closed system = system+environment, with the environment in its thermal state, the open system which results from coarse graining the environment will evolve towards an equilibrium state at late times. This state can be expressed as the reduced state of the closed system thermal state at the temperature of the environment. For a linear (harmonic) system and environment, which is exactly solvable, we are able to show in a rigorous way that all multitime correlations of the open system evolve towards those of the closed system thermal state. Multitime correlations are especially relevant in the non-Markovian regime, since they cannot be generated by the dynamics of the single-time correlations. For more general systems, which cannot be exactly solved, we are able to provide a general proof that all single-time correlations of the open system evolve to those of the closed system thermal state, to first order in the relaxation rates. For the special case of a zero-temperature reservoir, we are able to explicitly construct the reduced closed system thermal state in terms of the environmental correlations.

arXiv:1206.2707•Physical Review E•Dec 26, 2012•87 citations

Nonequilibrium Thermodynamics and Steady State Density Matrix for Quantum Open Systems

H. Ness

We consider the generic model of a finite-size quantum electron system connected to two (temperature and particle) reservoirs. The quantum open system is driven out of equilibrium by the presence of both potential temperature and chemical differences between the two reservoirs. The nonequilibrium (NE) thermodynamical properties of such a quantum open system are studied for the steady state regime. In such a regime, the corresponding NE density matrix is built on the so-called generalised Gibbs ensembles. From different expressions of the NE density matrix, we can identify the terms related to the entropy production in the system. We show, for a simple model, that the entropy production rate is always a positive quantity. Alternative expressions for the entropy production are also obtained from the Gibbs–von Neumann conventional formula and discussed in detail. Our results corroborate and expand earlier works found in the literature.

arXiv:1703.10531•Entropy•Apr 2, 2017•17 citations

Perturbation theory for open quantum systems at the steady state

Edgar A. Gómez, Jorge David Castaño‐Yepes, Saravana Prakash Thirumuruganandham

We propose a theoretical method for studying open quantum systems at the steady state within the framework of the perturbation theory. This approach allow us to compute the density operator of the system perturbatively and, in principle, to obtain an expression for its density matrix elements explicitly. To illustrate our methodology, we provide a case study of a nonlinear two-level system coupled to a thermal reservoir and special attention is given to the relationship between nonlinearity and nonclassicality. In particular, this relation is quantified by considering the quantum measures as the nonclassical depth, nonclassical distance and the quantum fidelity. We find that the nonclassicality is originated at least by second order contribution of nonlinear terms in quantum dynamics.

Results in Physics•Jun 22, 2018•8 citations

Non-equilibrium quantum thermodynamics with time-evolving matrix product operators

Maria Popovic, Mark T. Mitchison, Aidan Strathearn +3 more

We present a numerically exact method to compute the full counting statistics of heat transfer in non-Markovian open quantum systems, which is based on the time-evolving matrix product operator (TEMPO) algorithm. This approach is applied to the paradigmatic spin-boson model in order to calculate the mean and fluctuations of the heat transferred to the environment during thermal equilibration. We show that system-reservoir correlations make a significant contribution to the heat statistics at low temperature and present a variational theory that quantitatively explains our numerical results. We also demonstrate a fluctuation-dissipation relation connecting the mean and variance of the heat distribution at high temperature. Our results reveal that system-bath interactions make a significant contribution to heat dissipation even when the dynamics of the open system is effectively Markovian. The method presented here provides a flexible and general tool to predict the fluctuations of heat transfer in open quantum systems in non-perturbative regimes.

•Aug 14, 2020•1 citation
We connect the theory of a Lyapunov functional for an open system with the thermophysical concept of the relative entropy or equivalently with the entropy production rate. In order to do this we consider the environment of the open system as a thermal reservoir, which introduces the temperature as a thermophysical quantity. The general theory of the Lyapunov functional of an open system constructs a metric operator which contains an element of ambiguity in its scaling. This ambiguity can be removed by assuming a thermostatistic environment which fixes the scaling of the metric operator in a unique manner. The Lyapunov functional thus acquires a thermophysical meaning and can be connected with a relative entropy. In this case the metric operator is related to the inverse of the stationary density operator of the system. We illustrate the theory on a two-level atomic system and a degenerate three-level atom which are exposed to a thermal electromagnetic environment.
An explicit expression for the temperature of an open two-level quantum system is obtained as a function of local properties under the hypothesis of weak interaction with the environment. This temperature is defined for both equilibrium and out-of-equilibrium states and coincides with the environment temperature if the system reaches thermal equilibrium with a heat reservoir. Additionally, we show that within this theoretical framework the total entropy production can be partitioned into two contributions: one due to heat transfer and another, associated to internal irreversibilities, related to the loss of internal coherence by the qubit. The positiveness of the heat capacity is established, as well as its consistency with the well-known results at thermal equilibrium. We apply these concepts to two different systems and show that they behave in analogous ways as their classical counterparts.
We have derived a Lyapunov functional for a degenerate open atomic system. This functional develops monotonically towards its stationary state. The open system is described by a Lindblad-type master equation. For the construction of the variational functional it is necessary that the Lindblad operator can be diagonalized. Since the generator of motion is non-Hermitian, diagonalization is, in general, only possible if the eigenvalues are nondegenerate. In this paper, we propose that in a physical system the biorthogonal eigenbasis of the Lindblad operator remains complete even when degeneracy is present. Thus diagonalization of the Lindblad operator, and consequently the construction of the variational functional, is still possible. We discuss the reasons and illustrate the theory of the variational functional for a driven $\ensuremath{\Lambda}$-type three-level atom with degenerate ground state. The degeneracy has interesting effects on the variational functional in the steady state with respect to its interpretation as an entropic quantity. In case of the driven three-level atom, the dark state turns out to be an isentropic state.
In this work we investigate the late-time steady states of open quantum systems coupled to a thermal reservoir in the strong coupling regime. In general such systems do not necessarily relax to a Boltzmann distribution if the coupling to the thermal reservoir is nonvanishing or equivalently if the relaxation time scales are finite. Using a variety of nonequilibrium formalisms valid for non-Markovian processes, we show that starting from a product state of the closed system = system+environment, with the environment in its thermal state, the open system which results from coarse graining the environment will evolve towards an equilibrium state at late times. This state can be expressed as the reduced state of the closed system thermal state at the temperature of the environment. For a linear (harmonic) system and environment, which is exactly solvable, we are able to show in a rigorous way that all multitime correlations of the open system evolve towards those of the closed system thermal state. Multitime correlations are especially relevant in the non-Markovian regime, since they cannot be generated by the dynamics of the single-time correlations. For more general systems, which cannot be exactly solved, we are able to provide a general proof that all single-time correlations of the open system evolve to those of the closed system thermal state, to first order in the relaxation rates. For the special case of a zero-temperature reservoir, we are able to explicitly construct the reduced closed system thermal state in terms of the environmental correlations.
We consider the generic model of a finite-size quantum electron system connected to two (temperature and particle) reservoirs. The quantum open system is driven out of equilibrium by the presence of both potential temperature and chemical differences between the two reservoirs. The nonequilibrium (NE) thermodynamical properties of such a quantum open system are studied for the steady state regime. In such a regime, the corresponding NE density matrix is built on the so-called generalised Gibbs ensembles. From different expressions of the NE density matrix, we can identify the terms related to the entropy production in the system. We show, for a simple model, that the entropy production rate is always a positive quantity. Alternative expressions for the entropy production are also obtained from the Gibbs–von Neumann conventional formula and discussed in detail. Our results corroborate and expand earlier works found in the literature.
We propose a theoretical method for studying open quantum systems at the steady state within the framework of the perturbation theory. This approach allow us to compute the density operator of the system perturbatively and, in principle, to obtain an expression for its density matrix elements explicitly. To illustrate our methodology, we provide a case study of a nonlinear two-level system coupled to a thermal reservoir and special attention is given to the relationship between nonlinearity and nonclassicality. In particular, this relation is quantified by considering the quantum measures as the nonclassical depth, nonclassical distance and the quantum fidelity. We find that the nonclassicality is originated at least by second order contribution of nonlinear terms in quantum dynamics.
We present a numerically exact method to compute the full counting statistics of heat transfer in non-Markovian open quantum systems, which is based on the time-evolving matrix product operator (TEMPO) algorithm. This approach is applied to the paradigmatic spin-boson model in order to calculate the mean and fluctuations of the heat transferred to the environment during thermal equilibration. We show that system-reservoir correlations make a significant contribution to the heat statistics at low temperature and present a variational theory that quantitatively explains our numerical results. We also demonstrate a fluctuation-dissipation relation connecting the mean and variance of the heat distribution at high temperature. Our results reveal that system-bath interactions make a significant contribution to heat dissipation even when the dynamics of the open system is effectively Markovian. The method presented here provides a flexible and general tool to predict the fluctuations of heat transfer in open quantum systems in non-perturbative regimes.