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

Dynamics and relaxation in interacting systems: Semigroup methods

Ronnie Kosloff, Mark A. Ratner, William B. Davis

Published May 1, 1997• 75 citations
Publisher

Abstract

The dynamical study of chemical systems whose evolution is governed by quantum mechanics can now be computed fairly effectively for small systems in which the evolution is entirely Hamiltonian. When such Hamiltonian systems interact with their environment, however, relaxation and dephasing terms are introduced into the evolution. To include the effect of these terms, several methods are in current use. This paper is devoted to an exposition, analysis, and several simple applications of the semigroup technique for dealing with these non-Hamiltonian evolution terms. We discuss the nature of the semigroup terms, how they arise and how they are applied, and some of their advantages and disadvantages compared to other methods including dissipation. Specific applications to three simple two-site problems are given.

DOI
10.1063/1.473726

Related papers

Semigroup Representations, Site Couplings, and Relaxation in Quantum Systems

William B. Davis, Michael R. Wasielewski, Ronnie Kosloff +1 more

The semigroup formalism permits a general description of relaxation processes in quantum mechanical systems coupled to Markovian baths. From its definition, the semigroup formalism should be expressed in terms of the dissipative and relaxation coupling of the eigenstates of the quantum system to the bath modes. Often, for convenience both in calculation and in physical interpretation, it is more straightforward to couple local states of the quantum mechanical system to the bath. Within a simple two-site system, we examine the generalities of these two different coupling schemes, noting situations in which they are the same and situations for which substantial differences occur.

The Journal of Physical Chemistry A•Sep 30, 1998•31 citations

Operator Algebras and Quantum Dynamics: Semigroup Methods in Infinite Dimensions

Anam Ghafoor, Ume Roman, Sawaiba Yousaf

Background: Operator algebras and quantum dynamical semigroups are the mathematical tools of choice for understanding open quantum systems, especially in infinite-dimensional spaces. Such tools are extremely important for the time development of quantum systems out of equilibium under the influence of environment, where conventional finite dimensional methods do not include all possibilities for interaction. Objective: The objects of the present research are the r-properties and the decompositions of generators of quantum dynamical semigroups on infinite-dimensional spaces including their Hamiltonian and dissipative part, semigroup continuity, stationary states, and positivity improvement. The aim is to set a fundament in order to think about long-time dynamics in open quantum systems. Methods: The theorem Hille–Yosida is used to get conditions for the semigroup strongly continuous generated by unbounded operators. The time-evolution operator of the system is approximated with both Gaussian quadrature and Monte Carlo integration during infinitesimal time slices, by using Inf-LCHS in conjunction. Generators were studied under decomposition of their Hamiltonian and dissipative parts, while commutant algebras were investigated for irreducibility and positivity enhancement. Results: Most systems studied for these properties presented a singular decomposition, strong and weak semigroup continuity, existence of stationary states and positivity improvement. The counterexamples were the systems with in a lack of verification for dissipative parts, and they lost smoothing property, uniqueness of decomposition, partial semigroup continuity or large errors in approximation when we perform numerically. Conclusions: The combination of the methods of operator algebra with state-of-the-art simulation techniques yields a powerful approach to study infinite-dimensional quantum dynamics. Although dissipative components are not yet completely characterized, the results improve our understanding of generator structures and long-term system behavior which will help future work in quantum statistical mechanics and quantum information theory.

Annual Methodological Archive Research Review•Dec 16, 2025•0 citations

Brownian motion

A. O. Caldeira

In the two preceding chapters of this book we have analyzed many interesting physical phenomena in magnetic and superconducting systems which could adequately be described by phenomenological dynamical equations in terms of collective classical variables. One unavoidable consequence of this approach is that, as we are always dealing with variables that describe only part of the whole system, the interaction with the remaining degrees of freedom shows up through the presence of non-conservative terms which describe the relaxation of those variables to equilibrium. Those phenomenological equations are able to describe a very rich diversity of physical phenomena, in particular, those which can be studied in the context of quantum mechanics. Since these are genuine dynamical equations, there is no reason why they should be restricted to classical physics. However, as we do not yet know how to treat dissipative effects in quantum mechanics, we have deliberately neglected those terms when trying to describe quantum mechanical effects of our collective variables. In this chapter we will describe the general approach to dealing with dissipation in quantum mechanics. However, before we embark on this enterprise we should spend some time learning a little bit about the classical behavior of dissipative systems. In this way we can develop some intuition on how systems evolve during a dissipative process and, hopefully, this will be useful later on when we deal with quantum mechanical systems. The immediate problem we have to face concerns the choice of dissipative system to be studied.

Cambridge University Press eBooks•Mar 20, 2014•0 citations

Studies Of Electronic, Magnetic And Entanglement Properties Of Correlated Models In Low-Dimensional Systems

Shaon Sahoo

This thesis consists of six chapters. The first chapter gives an introduction to the field of low-dimensional magnetic and electronic systems and relevant numerical techniques. The recent developments in molecular magnets are highlighted. The numerical techniques are reviewed along with their advantages and disadvantages from the present perspective. Study of entanglement of a system can give a great insight into the system. At the last part of this chapter a general overview is given regarding entanglement, its measures and its significance in studying many-body systems. Chapter 2 deals with the technique that has been developed by us for the full symmetry adaptation of non-relativistic Hamiltonians. It is advantageous both computationally and physically/chemically to exploit both spin and spatial symmetries of a system. It has been a long-standing problem to target a state which has definite total spin and also belongs to a definite irreducible representation of a point group, particularly for non-Abelian point groups. A very general technique is discussed in this chapter which is a hybrid method based on valence-bond basis and the basis of the z-component of the total spin. This technique is not only applicable to a system with arbitrary site spins and belonging to any point group symmetry, it is also quite easy to implement computationally. To demonstrate the power of the method, it is applied to the molecular magnetic system, Cu6Fe8, with cubic symmetry. In chapter 3, the extension of the previous hybrid technique to electronic systems is discussed. The power of the method is illustrated by applying it to a model icosahedral half-filled electronic system. This model spans a huge Hilbert space (dimension 1,778,966) and is in the largest non-Abelian point group. All the eigenstates of the model are obtained using our technique. Chapter 4 deals with the thermodynamic properties of an important class of single-chain magnets (SCMs). This class of SCMs has alternate isotropic spin-1/2 units and anisotropic high spin units with the anisotropy axes being non-collinear. Here anisotropy is assumed to be large and negative, as a result, anisotropic units behave like canted spins at low temperatures; but even then simple Ising-type model does not capture the essential physics of the system due to quantum mechanical nature of the isotropic units. A transfer matrix (TM) method is developed to study statistical behavior of this class of SCMs. For the first time, it is also discussed in detail that how weak inter-chain interactions can be treated by a TM method. The finite size effect is also discussed which becomes important for low temperature dynamics. This technique is applied to a real helical chain magnet, which has been studied experimentally. In the fifth chapter a bipartite entanglement entropy of finite systems is studied using exact diagonalization techniques to examine how the entanglement changes in the presence of long-range interactions. The PariserParrPople model with long-range interactions is used for this…

•Sep 1, 2012•0 citations

Quantum dynamics with dissipation: A treatment of dephasing in the stochastic Schrödinger equation

Dmitrii E. Makarov, Horia I. Metiu

The behavior of a quantum system coupled to a reservoir is usually described by an equation for the reduced density matrix. If the dephasing and the change of population in these equations can be described phenomenologically, then it is possible to develop a stochastic Schrödinger equation to describe quantum dynamics with dissipation. In this article, we propose a method for treating dephasing, within the stochastic Schrödinger equation, for many-level systems.

The Journal of Chemical Physics•Dec 8, 1999•37 citations

Strategies to simulate dephasing-assisted quantum transport on digital quantum computers

Federico Gallina, Matteo Bruschi, Barbara Fresch

Abstract Simulating charge and energy transfer in extended molecular networks requires an effective model to include the environment because it significantly affects the quantum dynamics. A prototypical effect known as environment-assisted quantum transport (ENAQT) consists in the enhancement of the transfer efficiency by the interaction with an environment. A simple description of this phenomenon is obtained by a quantum master equation describing a quantum walk over the molecular network in the presence of inter-site decoherence. We consider the problem of simulating the dynamics underlying ENAQT in a digital quantum computer. Two different quantum algorithms are introduced, the first one based on stochastic Hamiltonians and the second one based on a collision scheme. We test both algorithms by simulating ENAQT in a small molecular network on a quantum computer emulator and provide a comparative analysis of the two approaches. Both algorithms can be implemented in a memory efficient encoding with the number of required qubits scaling logarithmically with the size of the simulated system while the number of gates increases quadratically. We discuss the algorithmic quantum trajectories generated by the two simulation strategies showing that they realize distinct unravellings of the site-dephasing master equation. In our approach, the non-unitary dynamics of the open system is obtained through effective representations of the environment, paving the way to digital quantum simulations of quantum transport influenced by structured environments.

arXiv:2111.02897•New Journal of Physics•Feb 1, 2022•14 citations
The dynamical study of chemical systems whose evolution is governed by quantum mechanics can now be computed fairly effectively for small systems in which the evolution is entirely Hamiltonian. When such Hamiltonian systems interact with their environment, however, relaxation and dephasing terms are introduced into the evolution. To include the effect of these terms, several methods are in current use. This paper is devoted to an exposition, analysis, and several simple applications of the semigroup technique for dealing with these non-Hamiltonian evolution terms. We discuss the nature of the semigroup terms, how they arise and how they are applied, and some of their advantages and disadvantages compared to other methods including dissipation. Specific applications to three simple two-site problems are given.
The semigroup formalism permits a general description of relaxation processes in quantum mechanical systems coupled to Markovian baths. From its definition, the semigroup formalism should be expressed in terms of the dissipative and relaxation coupling of the eigenstates of the quantum system to the bath modes. Often, for convenience both in calculation and in physical interpretation, it is more straightforward to couple local states of the quantum mechanical system to the bath. Within a simple two-site system, we examine the generalities of these two different coupling schemes, noting situations in which they are the same and situations for which substantial differences occur.
Background: Operator algebras and quantum dynamical semigroups are the mathematical tools of choice for understanding open quantum systems, especially in infinite-dimensional spaces. Such tools are extremely important for the time development of quantum systems out of equilibium under the influence of environment, where conventional finite dimensional methods do not include all possibilities for interaction. Objective: The objects of the present research are the r-properties and the decompositions of generators of quantum dynamical semigroups on infinite-dimensional spaces including their Hamiltonian and dissipative part, semigroup continuity, stationary states, and positivity improvement. The aim is to set a fundament in order to think about long-time dynamics in open quantum systems. Methods: The theorem Hille–Yosida is used to get conditions for the semigroup strongly continuous generated by unbounded operators. The time-evolution operator of the system is approximated with both Gaussian quadrature and Monte Carlo integration during infinitesimal time slices, by using Inf-LCHS in conjunction. Generators were studied under decomposition of their Hamiltonian and dissipative parts, while commutant algebras were investigated for irreducibility and positivity enhancement. Results: Most systems studied for these properties presented a singular decomposition, strong and weak semigroup continuity, existence of stationary states and positivity improvement. The counterexamples were the systems with in a lack of verification for dissipative parts, and they lost smoothing property, uniqueness of decomposition, partial semigroup continuity or large errors in approximation when we perform numerically. Conclusions: The combination of the methods of operator algebra with state-of-the-art simulation techniques yields a powerful approach to study infinite-dimensional quantum dynamics. Although dissipative components are not yet completely characterized, the results improve our understanding of generator structures and long-term system behavior which will help future work in quantum statistical mechanics and quantum information theory.
In the two preceding chapters of this book we have analyzed many interesting physical phenomena in magnetic and superconducting systems which could adequately be described by phenomenological dynamical equations in terms of collective classical variables. One unavoidable consequence of this approach is that, as we are always dealing with variables that describe only part of the whole system, the interaction with the remaining degrees of freedom shows up through the presence of non-conservative terms which describe the relaxation of those variables to equilibrium. Those phenomenological equations are able to describe a very rich diversity of physical phenomena, in particular, those which can be studied in the context of quantum mechanics. Since these are genuine dynamical equations, there is no reason why they should be restricted to classical physics. However, as we do not yet know how to treat dissipative effects in quantum mechanics, we have deliberately neglected those terms when trying to describe quantum mechanical effects of our collective variables. In this chapter we will describe the general approach to dealing with dissipation in quantum mechanics. However, before we embark on this enterprise we should spend some time learning a little bit about the classical behavior of dissipative systems. In this way we can develop some intuition on how systems evolve during a dissipative process and, hopefully, this will be useful later on when we deal with quantum mechanical systems. The immediate problem we have to face concerns the choice of dissipative system to be studied.
This thesis consists of six chapters. The first chapter gives an introduction to the field of low-dimensional magnetic and electronic systems and relevant numerical techniques. The recent developments in molecular magnets are highlighted. The numerical techniques are reviewed along with their advantages and disadvantages from the present perspective. Study of entanglement of a system can give a great insight into the system. At the last part of this chapter a general overview is given regarding entanglement, its measures and its significance in studying many-body systems. Chapter 2 deals with the technique that has been developed by us for the full symmetry adaptation of non-relativistic Hamiltonians. It is advantageous both computationally and physically/chemically to exploit both spin and spatial symmetries of a system. It has been a long-standing problem to target a state which has definite total spin and also belongs to a definite irreducible representation of a point group, particularly for non-Abelian point groups. A very general technique is discussed in this chapter which is a hybrid method based on valence-bond basis and the basis of the z-component of the total spin. This technique is not only applicable to a system with arbitrary site spins and belonging to any point group symmetry, it is also quite easy to implement computationally. To demonstrate the power of the method, it is applied to the molecular magnetic system, Cu6Fe8, with cubic symmetry. In chapter 3, the extension of the previous hybrid technique to electronic systems is discussed. The power of the method is illustrated by applying it to a model icosahedral half-filled electronic system. This model spans a huge Hilbert space (dimension 1,778,966) and is in the largest non-Abelian point group. All the eigenstates of the model are obtained using our technique. Chapter 4 deals with the thermodynamic properties of an important class of single-chain magnets (SCMs). This class of SCMs has alternate isotropic spin-1/2 units and anisotropic high spin units with the anisotropy axes being non-collinear. Here anisotropy is assumed to be large and negative, as a result, anisotropic units behave like canted spins at low temperatures; but even then simple Ising-type model does not capture the essential physics of the system due to quantum mechanical nature of the isotropic units. A transfer matrix (TM) method is developed to study statistical behavior of this class of SCMs. For the first time, it is also discussed in detail that how weak inter-chain interactions can be treated by a TM method. The finite size effect is also discussed which becomes important for low temperature dynamics. This technique is applied to a real helical chain magnet, which has been studied experimentally. In the fifth chapter a bipartite entanglement entropy of finite systems is studied using exact diagonalization techniques to examine how the entanglement changes in the presence of long-range interactions. The PariserParrPople model with long-range interactions is used for this…
Abstract Simulating charge and energy transfer in extended molecular networks requires an effective model to include the environment because it significantly affects the quantum dynamics. A prototypical effect known as environment-assisted quantum transport (ENAQT) consists in the enhancement of the transfer efficiency by the interaction with an environment. A simple description of this phenomenon is obtained by a quantum master equation describing a quantum walk over the molecular network in the presence of inter-site decoherence. We consider the problem of simulating the dynamics underlying ENAQT in a digital quantum computer. Two different quantum algorithms are introduced, the first one based on stochastic Hamiltonians and the second one based on a collision scheme. We test both algorithms by simulating ENAQT in a small molecular network on a quantum computer emulator and provide a comparative analysis of the two approaches. Both algorithms can be implemented in a memory efficient encoding with the number of required qubits scaling logarithmically with the size of the simulated system while the number of gates increases quadratically. We discuss the algorithmic quantum trajectories generated by the two simulation strategies showing that they realize distinct unravellings of the site-dephasing master equation. In our approach, the non-unitary dynamics of the open system is obtained through effective representations of the environment, paving the way to digital quantum simulations of quantum transport influenced by structured environments.