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

Illustrative Overview of Reaction Coordinate Mapping and Derivation of Quantum Master Equation with RCPT Techniques

Abhirup Chatterjee

Published Nov 18, 2025• 0 citations
Publisher

Abstract

In this tutorial we want to present a thorough overview about the well known technique known as the Reaction Coordinate Mapping which is one of the most used theoretical tool in the context of Open quantum system and Quantum thermodynamics of quantum systems which is not weakly coupled to the reservoir. Here we discuss different aspects of the Reaction Coordinate mapping along with detailed mathematical derivations for the Bosonic and Fermionic mappings which can be achieved through the Symplectic and unitary transformations. In the later part of this article we discuss the derivation of the Quantum master equation for the extended systems (System+Reaction Coordinate) coupled to the residual reservoir in the weak coupling limit described by the usual Redfield Quantum Master equation. We observe that though the dynamics of the Extended system is Markovian in nature because of the weak coupling between the extended system and the Residual reservoir but the Reaction coordinate transformation enables us to analyses the dynamics of the system which is essentially non-Markovian.

DOI
10.20944/preprints202511.1317.v1

Related papers

Modeling the Non Markovian Dynamics of Open Quantum System Strongly Coupled to Reservoirs Through Quantum Master Equation by Method of Reaction Coordinate Mapping and the Mesoscopic Lead Construction

Abhirup Chatterjee

In this article we want to present a thorough overview about the well known technique known as the Reaction Coordinate Mapping which is one of the most used theoretical tool in the context of Open quantum system and Quantum thermodynamics of quantum systems which is not weakly coupled to the reservoir. Here we discuss different aspects of the Reaction Coordinate mapping along with detailed mathematical derivations for the Bosonic and Fermionic mappings which can be achieved through the Symplectic and unitary transformations. In the later part of this article we discuss the derivation of the Quantum master equation for the extended systems (System+Reaction Coordinate) coupled to the residual reservoir in the weak coupling limit described by the usual Redfield Quantum Master equation. We observe that though the dynamics of the Extended system is Markovian in nature because of the weak coupling between the extended system and the Residual reservoir but the Reaction coordinate transformation enables us to analyses the dynamics of the system which is essentially non-Markovian.

•Nov 21, 2025•0 citations

Modeling the Non Markovian Dynamics of Open Quantum System Strongly Coupled to Reservoirs Through Quantum Master Equation by Method of Reaction Coordinate Mapping and the Mesoscopic Lead Construction

Abhirup Chatterjee

In this article we want to present a thorough overview about the well known technique known as the Reaction Coordinate Mapping which is one of the most used theoretical tool in the context of Open quantum system and Quantum thermodynamics of quantum systems which is not weakly coupled to the reservoir. Here we discuss different aspects of the Reaction Coordinate mapping along with detailed mathematical derivations for the Bosonic and Fermionic mappings which can be achieved through the Symplectic and unitary transformations. In the later part of this article we discuss the derivation of the Quantum master equation for the extended systems (System+Reaction Coordinate) coupled to the residual reservoir in the weak coupling limit described by the usual Redfield Quantum Master equation. We observe that though the dynamics of the Extended system is Markovian in nature because of the weak coupling between the extended system and the Residual reservoir but the Reaction coordinate transformation enables us to analyses the dynamics of the system which is essentially non-Markovian. The idea of the mesoscopic lead construction and brute force chain mapping techniques has been introduced which can be used to bypass the strong system bath coupling situations.

•Nov 24, 2025•0 citations

Capturing non-Markovian dynamics with the reaction coordinate method

Nicholas Anto-Sztrikacs, Dvira Segal

The reaction coordinate (RC) technique is emerging as a significant tool in the study of quantum dissipative dynamics and quantum thermodynamics. With the objective to further establish this tool, here we explore to what extent the method can capture non-Markovian dynamics of open quantum systems. As a case study, we focus on the pure decoherence model of a spin coupled to a harmonic reservoir. We compare the spin dynamics and measures for non-Markovianity from the exact analytical solution to simulations based on the RC method at the level of a second-order quantum master equation. We find that the RC method can quantitatively capture non-Markovian effects at strong system-bath coupling and for structured baths. This is rationalized by the fact that the collective RC bath mode, which is made part of the system, maintains system-bath correlations. Lastly, we apply our RC method and study the spin-boson model in the non-Markovian regime.

arXiv:2110.02455•Physical Review A•Nov 23, 2021•34 citations

RC-HEOM Hybrid Method for Non-Perturbative Open System Dynamics

Po-Rong Lai, Jhen-Dong Lin, Yi-Te Huang +3 more

The Hierarchical equations of motion (HEOM) method is an important non-perturbative technique, allowing numerically exact treatment of open quantum systems with strong coupling and non-Markovian memory. However, its encoding of bath memory into auxiliary density operators often limits direct access to detailed bath information. In contrast, the reaction-coordinate (RC) mapping allows direct and transparent access to the dominant collective bath mode, but its perturbative and often Markovian treatment of the residual bath restricts its reliability. In this work, we introduce RC-HEOM, a hybrid method that unifies the strengths of both approaches by combining RC mapping with a fully non-perturbative HEOM description of the residual bath. RC-HEOM simultaneously retains exact non-Markovian memory and access to the RC mode, which enables analysis of system-RC information. Applying this method to the Anderson impurity models, we directly track the emergence of the Kondo singlet from the growth of the Kondo resonance and uncover a nontrivial RC-mediated coherence revival. These results demonstrate that RC-HEOM is a promising method for characterizing open quantum systems in regimes that are difficult to access with conventional master-equation methods.

quant-pharXiv:2603.22833•Mar 25, 2026•0 citations

Non-Markovian exceptional points and information backflow: a Heisenberg reaction-coordinate approach

Daniel Holst, Gianmichele Blasi, Shishir Khandelwal

The primary pathway to the investigation of exceptional points (EPs) in open quantum systems has been the use of Markovian master equations. However, recently it has been realised that they can arise in settings that show non-Markovian behaviour. The main approach in this direction has been the use of pseudomodes, which allow the influence of a structured environment to be replaced by new damped auxiliary modes. In this work, we use a Heisenberg-equation approach which relies on the reaction-coordinate formalism to obtain exact analytical dynamics. First, we show how our method considerably simplifies the analysis compared to the use of master equations. Second, we utilise it to discuss non-Markovian revivals in systems with second and higher-order EPs, showing that the notion that EPs demarcate the onset of revivals is not generally true. We then utilise our results to discuss the dynamical effects of higher-order non-Markovian EPs, deriving bounds on the number of revivals and investigating enhanced asymptotic relaxation near higher-order EPs. Finally, we investigate non-Markovian EPs with engineered reservoirs and show how they can be utilised to control the rate of irreversible information loss.

quant-pharXiv:2610.26620•Oct 20, 2026•0 citations

Non-Markovian quantum heat statistics with the reaction coordinate mapping

Mike Shubrook, Jake Iles-Smith, Ahsan Nazir

Abstract The definition of heat in quantum mechanics is ambiguous. Complications arise in particular when the coupling between a quantum system and a thermal environment is non-negligible, as the boundary between the two becomes blurred, making the distinction between system and environment difficult to draw. The reaction coordinate (RC) mapping can be used in such regimes to redraw the boundary between the system and environment. In this paper we combine the RC technique with a two-point measurement protocol to compare two different definitions of heat: energetic changes with respect to the full environment Hamiltonian (prior to the mapping), and energetic changes with respect to the residual environment Hamiltonian (after the mapping). We find that the latter definition displays behaviour more expected of a heat bath in the highly non-Markovian regime considered. Video summaries of this paper are available as supplementary material on the article webpage .

arXiv:2408.08829•Quantum Science and Technology•Mar 28, 2025•6 citations
In this tutorial we want to present a thorough overview about the well known technique known as the Reaction Coordinate Mapping which is one of the most used theoretical tool in the context of Open quantum system and Quantum thermodynamics of quantum systems which is not weakly coupled to the reservoir. Here we discuss different aspects of the Reaction Coordinate mapping along with detailed mathematical derivations for the Bosonic and Fermionic mappings which can be achieved through the Symplectic and unitary transformations. In the later part of this article we discuss the derivation of the Quantum master equation for the extended systems (System+Reaction Coordinate) coupled to the residual reservoir in the weak coupling limit described by the usual Redfield Quantum Master equation. We observe that though the dynamics of the Extended system is Markovian in nature because of the weak coupling between the extended system and the Residual reservoir but the Reaction coordinate transformation enables us to analyses the dynamics of the system which is essentially non-Markovian.
In this article we want to present a thorough overview about the well known technique known as the Reaction Coordinate Mapping which is one of the most used theoretical tool in the context of Open quantum system and Quantum thermodynamics of quantum systems which is not weakly coupled to the reservoir. Here we discuss different aspects of the Reaction Coordinate mapping along with detailed mathematical derivations for the Bosonic and Fermionic mappings which can be achieved through the Symplectic and unitary transformations. In the later part of this article we discuss the derivation of the Quantum master equation for the extended systems (System+Reaction Coordinate) coupled to the residual reservoir in the weak coupling limit described by the usual Redfield Quantum Master equation. We observe that though the dynamics of the Extended system is Markovian in nature because of the weak coupling between the extended system and the Residual reservoir but the Reaction coordinate transformation enables us to analyses the dynamics of the system which is essentially non-Markovian.
In this article we want to present a thorough overview about the well known technique known as the Reaction Coordinate Mapping which is one of the most used theoretical tool in the context of Open quantum system and Quantum thermodynamics of quantum systems which is not weakly coupled to the reservoir. Here we discuss different aspects of the Reaction Coordinate mapping along with detailed mathematical derivations for the Bosonic and Fermionic mappings which can be achieved through the Symplectic and unitary transformations. In the later part of this article we discuss the derivation of the Quantum master equation for the extended systems (System+Reaction Coordinate) coupled to the residual reservoir in the weak coupling limit described by the usual Redfield Quantum Master equation. We observe that though the dynamics of the Extended system is Markovian in nature because of the weak coupling between the extended system and the Residual reservoir but the Reaction coordinate transformation enables us to analyses the dynamics of the system which is essentially non-Markovian. The idea of the mesoscopic lead construction and brute force chain mapping techniques has been introduced which can be used to bypass the strong system bath coupling situations.
The reaction coordinate (RC) technique is emerging as a significant tool in the study of quantum dissipative dynamics and quantum thermodynamics. With the objective to further establish this tool, here we explore to what extent the method can capture non-Markovian dynamics of open quantum systems. As a case study, we focus on the pure decoherence model of a spin coupled to a harmonic reservoir. We compare the spin dynamics and measures for non-Markovianity from the exact analytical solution to simulations based on the RC method at the level of a second-order quantum master equation. We find that the RC method can quantitatively capture non-Markovian effects at strong system-bath coupling and for structured baths. This is rationalized by the fact that the collective RC bath mode, which is made part of the system, maintains system-bath correlations. Lastly, we apply our RC method and study the spin-boson model in the non-Markovian regime.
The Hierarchical equations of motion (HEOM) method is an important non-perturbative technique, allowing numerically exact treatment of open quantum systems with strong coupling and non-Markovian memory. However, its encoding of bath memory into auxiliary density operators often limits direct access to detailed bath information. In contrast, the reaction-coordinate (RC) mapping allows direct and transparent access to the dominant collective bath mode, but its perturbative and often Markovian treatment of the residual bath restricts its reliability. In this work, we introduce RC-HEOM, a hybrid method that unifies the strengths of both approaches by combining RC mapping with a fully non-perturbative HEOM description of the residual bath. RC-HEOM simultaneously retains exact non-Markovian memory and access to the RC mode, which enables analysis of system-RC information. Applying this method to the Anderson impurity models, we directly track the emergence of the Kondo singlet from the growth of the Kondo resonance and uncover a nontrivial RC-mediated coherence revival. These results demonstrate that RC-HEOM is a promising method for characterizing open quantum systems in regimes that are difficult to access with conventional master-equation methods.
The primary pathway to the investigation of exceptional points (EPs) in open quantum systems has been the use of Markovian master equations. However, recently it has been realised that they can arise in settings that show non-Markovian behaviour. The main approach in this direction has been the use of pseudomodes, which allow the influence of a structured environment to be replaced by new damped auxiliary modes. In this work, we use a Heisenberg-equation approach which relies on the reaction-coordinate formalism to obtain exact analytical dynamics. First, we show how our method considerably simplifies the analysis compared to the use of master equations. Second, we utilise it to discuss non-Markovian revivals in systems with second and higher-order EPs, showing that the notion that EPs demarcate the onset of revivals is not generally true. We then utilise our results to discuss the dynamical effects of higher-order non-Markovian EPs, deriving bounds on the number of revivals and investigating enhanced asymptotic relaxation near higher-order EPs. Finally, we investigate non-Markovian EPs with engineered reservoirs and show how they can be utilised to control the rate of irreversible information loss.
Abstract The definition of heat in quantum mechanics is ambiguous. Complications arise in particular when the coupling between a quantum system and a thermal environment is non-negligible, as the boundary between the two becomes blurred, making the distinction between system and environment difficult to draw. The reaction coordinate (RC) mapping can be used in such regimes to redraw the boundary between the system and environment. In this paper we combine the RC technique with a two-point measurement protocol to compare two different definitions of heat: energetic changes with respect to the full environment Hamiltonian (prior to the mapping), and energetic changes with respect to the residual environment Hamiltonian (after the mapping). We find that the latter definition displays behaviour more expected of a heat bath in the highly non-Markovian regime considered. Video summaries of this paper are available as supplementary material on the article webpage .