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

International Journal of Pure and Apllied Mathematics

ON A QUANTUM SYSTEM EXPOSED TO A HEAT BATH

Eugen Grycko, W. Kirsch, Tobias Mühlenbruch

Published Jun 28, 2015• 0 citations
Publisher

Abstract

An appropriate generator matrix induces in a natural way a Markov semigroup which can be applied to the construction of a continuous time Markov process whose asymptotic distribution is independent of the initial state.We propose a generator associated with a prescribed (arbitrary) distribution which turns out to be reversible for the corresponding Markov process.As an application, we show that the Markov process offers a physically plausible description of a quantum system exposed to a heat bath.The system approaches its Gibbs state corresponding to the temperature of the heat bath if its initial state commutes with the Hamiltonian.

DOI
10.12732/ijpam.v102i3.11

Related papers

Thermodynamic Formalism for Continuous-Time Quantum Markov Semigroups: the Detailed Balance Condition, Entropy, Pressure and Equilibrium Quantum Processes

Jader E. Brasil, Josué Knorst, Artur Oscar Lopes

Let [Formula: see text] denote the set of [Formula: see text] by [Formula: see text] complex matrices. Consider continuous time quantum semigroups [Formula: see text], [Formula: see text], where [Formula: see text] is the infinitesimal generator. If we assume that [Formula: see text], we will call [Formula: see text], [Formula: see text] a quantum Markov semigroup. Given a stationary density matrix [Formula: see text], for the quantum Markov semigroup [Formula: see text], [Formula: see text], we can define a continuous time stationary quantum Markov process, denoted by [Formula: see text], [Formula: see text] Given an a priori Laplacian operator [Formula: see text], we will present a natural concept of entropy for a class of density matrices on [Formula: see text]. Given a Hermitian operator [Formula: see text] (which plays the role of a Hamiltonian), we will study a version of the variational principle of pressure for [Formula: see text]. A density matrix [Formula: see text] maximizing pressure will be called an equilibrium density matrix. From [Formula: see text] we will derive a new infinitesimal generator [Formula: see text]. Finally, the continuous time quantum Markov process defined by the semigroup [Formula: see text], [Formula: see text], and an initial stationary density matrix, will be called the continuous time equilibrium quantum Markov process for the Hamiltonian [Formula: see text]. It corresponds to the quantum thermodynamical equilibrium for the action of the Hamiltonian [Formula: see text].

arXiv:2201.05094•Open Systems & Information Dynamics•Dec 1, 2023•1 citation

Quantum trajectories in random environment: the statistical model for a heat bath

Ion Nechita, Clément Pellegrini

In this paper, we derive the stochastic master equations corresponding to the statistical model of a heat bath. These stochastic differential equations are obtained as continuous time limits of discrete models of quantum repeated measurements. Physically, they describe the evolution of a small system in contact with a heat bath undergoing continuous measurement. The equations obtained in the present work are qualitatively different from the ones derived in [6], where the Gibbs model of heat bath has been studied. It is shown that the statistical model of a heat bath has a clear physical interpretation in terms of emissions and absorptions of photons. Our approach yields models of random environment and unravelings of stochastic master equations. The equations are rigorously obtained as solutions of martingale problems using the convergence of Markov generators.

arXiv:0908.2754•Confluentes Mathematici•May 30, 2009•5 citations

Quantum Trajectories in Random Environment: the Statistical Model for a Heat Bath

Ion Nechita, Clément Pellegrini

In this article, we derive the stochastic master equations corresponding to the sta-tistical model of a heat bath. These stochastic differential equations are obtained as continuous time limits of discrete models of quantum repeated measurements. Physically, they describe the evolution of a small system in contact with a heat bath undergoing continuous measurement. The equations obtained in the present work are qualitatively different from the ones derived in [6], where the Gibbs model of heat bath has been studied. It is shown that the statistical model of a heat bath provides clear physical interpretation in terms of emissions and absorptions of photons. Our approach yields models of random environment and unravelings of stochastic master equations. The equations are rigorously obtained as solutions of martingale problems using the convergence of Markov generators.

•Jan 1, 2009•4 citations

Markovian repeated interaction quantum systems

Jean-François Bougron, Alain Joye, Claude‐Alain Pillet

We study a class of dynamical semigroups [Formula: see text] that emerge, by a Feynman–Kac type formalism, from a random quantum dynamical system [Formula: see text] driven by a Markov chain [Formula: see text]. We show that the almost sure large time behavior of the system can be extracted from the large [Formula: see text] asymptotics of the semigroup, which is in turn directly related to the spectral properties of the generator [Formula: see text]. As a physical application, we consider the case where the [Formula: see text]’s are the reduced dynamical maps describing the repeated interactions of a system [Formula: see text] with thermal probes [Formula: see text]. We study the full statistics of the entropy in this system and derive a fluctuation theorem for the heat exchanges and the associated linear response formulas.

arXiv:2202.05321•Reviews in Mathematical Physics•Jun 4, 2022•7 citations

STRONGLY CONTINUOUS SEMIGROUP ON DECOHERENCE-FREE SUBALGEBRA (DFS) OF QUANTUM MARKOV SEMIGROUP (QMS)

Ezekiel Abiodun Oluwafemi, Micheal O. Ogundiran, U.S. Idiong +1 more

This study examines the strongly continuous semigroup on the decoherence-free subalgebra (DFS) of the QMS. Initially, we derived the infinitesimal generator (IG) for the DFS. A theorem characterizing a strongly continuous semigroup on the decoherence-free subalgebra was established. Strongly continuous semigroups offer a natural framework for modeling the time evolution of quantum systems influenced by a quantum Markov semigroup. When paired with the concept of decoherence-free subalgebras, they enable the modeling of systems that are immune to decoherence due to the specific nature of their interaction with the environment. The dynamics on these subalgebras are governed by the restricted semigroup, which can be analyzed through operator theory and the tools of quantum Markov processes. This approach is particularly relevant for applications in quantum error correction, quantum computing, and quantum control, where isolating parts of a system from decoherence effects is crucial.

FUDMA Journal of Sciences•Feb 15, 2026•0 citations

A characterization of quantum Markov semigroups of weak coupling limit type

Franco Fagnola, Roberto B. Quezada

We characterize generators of quantum Markov semigroups leaving invariant a maximal abelian purely atomic algebra and certain operator subspaces associated with it in a natural way. From this result, we also establish a characterization of generators of quantum Markov semigroups of weak coupling limit type associated with a nondegenerate Hamiltonian.

Infinite Dimensional Analysis Quantum Probability and Related Topics•Jun 1, 2019•8 citations
An appropriate generator matrix induces in a natural way a Markov semigroup which can be applied to the construction of a continuous time Markov process whose asymptotic distribution is independent of the initial state.We propose a generator associated with a prescribed (arbitrary) distribution which turns out to be reversible for the corresponding Markov process.As an application, we show that the Markov process offers a physically plausible description of a quantum system exposed to a heat bath.The system approaches its Gibbs state corresponding to the temperature of the heat bath if its initial state commutes with the Hamiltonian.
Let [Formula: see text] denote the set of [Formula: see text] by [Formula: see text] complex matrices. Consider continuous time quantum semigroups [Formula: see text], [Formula: see text], where [Formula: see text] is the infinitesimal generator. If we assume that [Formula: see text], we will call [Formula: see text], [Formula: see text] a quantum Markov semigroup. Given a stationary density matrix [Formula: see text], for the quantum Markov semigroup [Formula: see text], [Formula: see text], we can define a continuous time stationary quantum Markov process, denoted by [Formula: see text], [Formula: see text] Given an a priori Laplacian operator [Formula: see text], we will present a natural concept of entropy for a class of density matrices on [Formula: see text]. Given a Hermitian operator [Formula: see text] (which plays the role of a Hamiltonian), we will study a version of the variational principle of pressure for [Formula: see text]. A density matrix [Formula: see text] maximizing pressure will be called an equilibrium density matrix. From [Formula: see text] we will derive a new infinitesimal generator [Formula: see text]. Finally, the continuous time quantum Markov process defined by the semigroup [Formula: see text], [Formula: see text], and an initial stationary density matrix, will be called the continuous time equilibrium quantum Markov process for the Hamiltonian [Formula: see text]. It corresponds to the quantum thermodynamical equilibrium for the action of the Hamiltonian [Formula: see text].
In this paper, we derive the stochastic master equations corresponding to the statistical model of a heat bath. These stochastic differential equations are obtained as continuous time limits of discrete models of quantum repeated measurements. Physically, they describe the evolution of a small system in contact with a heat bath undergoing continuous measurement. The equations obtained in the present work are qualitatively different from the ones derived in [6], where the Gibbs model of heat bath has been studied. It is shown that the statistical model of a heat bath has a clear physical interpretation in terms of emissions and absorptions of photons. Our approach yields models of random environment and unravelings of stochastic master equations. The equations are rigorously obtained as solutions of martingale problems using the convergence of Markov generators.
In this article, we derive the stochastic master equations corresponding to the sta-tistical model of a heat bath. These stochastic differential equations are obtained as continuous time limits of discrete models of quantum repeated measurements. Physically, they describe the evolution of a small system in contact with a heat bath undergoing continuous measurement. The equations obtained in the present work are qualitatively different from the ones derived in [6], where the Gibbs model of heat bath has been studied. It is shown that the statistical model of a heat bath provides clear physical interpretation in terms of emissions and absorptions of photons. Our approach yields models of random environment and unravelings of stochastic master equations. The equations are rigorously obtained as solutions of martingale problems using the convergence of Markov generators.
We study a class of dynamical semigroups [Formula: see text] that emerge, by a Feynman–Kac type formalism, from a random quantum dynamical system [Formula: see text] driven by a Markov chain [Formula: see text]. We show that the almost sure large time behavior of the system can be extracted from the large [Formula: see text] asymptotics of the semigroup, which is in turn directly related to the spectral properties of the generator [Formula: see text]. As a physical application, we consider the case where the [Formula: see text]’s are the reduced dynamical maps describing the repeated interactions of a system [Formula: see text] with thermal probes [Formula: see text]. We study the full statistics of the entropy in this system and derive a fluctuation theorem for the heat exchanges and the associated linear response formulas.
This study examines the strongly continuous semigroup on the decoherence-free subalgebra (DFS) of the QMS. Initially, we derived the infinitesimal generator (IG) for the DFS. A theorem characterizing a strongly continuous semigroup on the decoherence-free subalgebra was established. Strongly continuous semigroups offer a natural framework for modeling the time evolution of quantum systems influenced by a quantum Markov semigroup. When paired with the concept of decoherence-free subalgebras, they enable the modeling of systems that are immune to decoherence due to the specific nature of their interaction with the environment. The dynamics on these subalgebras are governed by the restricted semigroup, which can be analyzed through operator theory and the tools of quantum Markov processes. This approach is particularly relevant for applications in quantum error correction, quantum computing, and quantum control, where isolating parts of a system from decoherence effects is crucial.