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

Annals of Physics

Exact master equation for an open Jaynes–Cummings system

Akane Watanabe, Hiromichi Nakazato

Published Apr 27, 2022• 3 citations
Publisher

Abstract

A master equation for the Jaynes-Cummings (JC) system interacting with a bosonic reservoir through a conserved quantity is derived analytically without any approximation.The derivation is valid for any initial JC state that has no correlation with the initial reservoir at any temperature and yields a Gorini-Kossakowski-Lindblad-Sudarshan-type master equation with time-dependent coefficients.Several physical quantities of the open JC system are examined and the validity or limit of some of the standard approximations used to derive the master equation is discussed on the basis of the exact results obtained here.

DOI
10.1016/j.aop.2022.168890

Related papers

Exact time-evolution of the dispersive Jaynes–Cummings model: the effect of initial correlation and master equation approach

Masashi Ban

Time-evolution of the dispersive Jaynes–Cummings model interacting with a bosonic reservoir is considered in the presence of an initial correlation between the cavity field and reservoir. When there is at most one excitation in the whole system, the exact time-evolution can be obtained. The effects of the initial correlation on the decoherence of the two-level system and cavity field are investigated. Furthermore, the validity of the time-convolutionless quantum master equation in the second-order approximation with respect to the system-reservoir interaction is examined by means of the exact solution.

Journal of Modern Optics•Mar 14, 2011•5 citations

Analytical solution of the open dispersive Jaynes-Cummings model and perturbative analytical solution of the open quantum Rabi model

Antti Vaaranta, Marco Cattaneo, Paolo Muratore-Ginanneschi

The Jaynes-Cummings and quantum Rabi models are fundamental to cavity and circuit quantum electrodynamics, as they describe the simplest form of light-matter interaction, where a single qubit is coupled to a single bosonic mode. A scenario that is commonly encountered in the experimental practice arises when the bosonic mode interacts with an external dissipative thermal bath, making the qubit-boson system open. In this work, we present a new analytical solution of the Lindblad master equations for the open dispersive Jaynes-Cummings model and a perturbative analytical solution of the open quantum Rabi model in the limit of weak qubit-boson coupling $g$, using the holomorphic formalism in Bargmann space. Specifically, we derive the most general solution of the local Lindblad master equation for the open dispersive Jaynes-Cummings model coupled to a thermal bath, with the only assumptions that the initial state of the qubit-boson system is separable. Additionally, we obtain a perturbative analytical solution for the open quantum Rabi model up to second order in $g$. Notably, our findings include a new formula for the qubit's steady state at zeroth order, showing that the stationary populations depend on both qubit and boson frequencies in the quantum Rabi model, but not in the Jaynes-Cummings model, regardless of the value of $g$. Our results are of general interest to the study of open quantum systems in the context of light-matter interaction.

arXiv:2411.18310•Physical Review A•May 20, 2025•4 citations

Unified Gorini-Kossakowski-Lindblad-Sudarshan quantum master equation beyond the secular approximation

Anton Sergeevich Trushechkin

Derivation of a quantum master equation for a system weakly coupled to a bath which takes into account nonsecular effects, but nevertheless has the mathematically correct Gorini-Kossakowski-Lindblad-Sudarshan form (in particular, it preserves positivity of the density operator) and also satisfies the standard thermodynamic properties is a known long-standing problem in theory of open quantum systems. The nonsecular terms are important when some energy levels of the system or their differences (Bohr frequencies) are nearly degenerate. We provide a fully rigorous derivation of such equation based on a formalization of the weak-coupling limit for the general case.

arXiv:2103.12042•Physical Review A•Jun 29, 2021•104 citations

Out-of-equilibrium open quantum systems: A comparison of approximate quantum master equation approaches with exact results

Archak Purkayastha, Abhishek Dhar, Manas Kulkarni

We present the Born-Markov approximated Redfield quantum master equation (RQME) description for an open system of noninteracting particles (bosons or fermions) on an arbitrary lattice of $N$ sites in any dimension and weakly connected to multiple reservoirs at different temperatures and chemical potentials. The RQME can be reduced to the Lindblad equation, of various forms, by making further approximations. By studying the $N=2$ case, we show that RQME gives results which agree with exact analytical results for steady-state properties and with exact numerics for time-dependent properties over a wide range of parameters. In comparison, the Lindblad equations have a limited domain of validity in nonequilibrium. We conclude that it is indeed justified to use microscopically derived full RQME to go beyond the limitations of Lindblad equations in out-of-equilibrium systems. We also derive closed-form analytical results for out-of-equilibrium time dynamics of two-point correlation functions. These results explicitly show the approach to steady state and thermalization. These results are experimentally relevant for cold atoms, cavity QED, and far-from-equilibrium quantum dot experiments.

arXiv:1511.03778•Physical Review A•Jun 13, 2016•121 citations

Lectures on Open Quantum Systems

Marco Merkli, Ángel Neira

These notes are a short introduction to the mathematical theory of open quantum systems. They are meant to serve as an entry point into a broad research area which has applications across the quantum sciences dealing with systems subjected to external noise. The guiding idea is to let the key structures of the theory emerge from a concrete model. By working through the dissipative Jaynes-Cummings model the reader will dis- cover explicitly how irreversible dynamics arises from a unitary system-reservoir evolution. The notions of the continuous mode limit, correlation functions, spectral density appear in a natural manner and lead to the evolution equation of the open system in form of a master equation. This sets the stage for the more general analysis of completely positive, trace preserving (CPTP) maps and the study of quantum dynamical semigroups. We motivate and prove the Kraus representation theorem, the dilation theorem and the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) theorem. Working through the exercises (for which full solutions are supplied) will reinforce the ideas introduced in the main text.

quant-pharXiv:2603.11925•Mar 13, 2026•0 citations

Exact non-Markovian master equation for the spin-boson and Jaynes-Cummings models

Luca Ferialdi

We provide the exact non-Markovian master equation for a two-level system interacting with a thermal bosonic bath, and we write the solution of such a master equation in terms of the Bloch vector. We show that previous approximated results are particular limits of our exact master equation. We generalize these results to more complex systems involving an arbitrary number of two-level systems coupled to different thermal baths, providing the exact master equations also for these systems. As an example of this general case we derive the master equation for the Jaynes-Cummings model.

arXiv:1609.00645•Physical Review A•Feb 6, 2017•33 citations
A master equation for the Jaynes-Cummings (JC) system interacting with a bosonic reservoir through a conserved quantity is derived analytically without any approximation.The derivation is valid for any initial JC state that has no correlation with the initial reservoir at any temperature and yields a Gorini-Kossakowski-Lindblad-Sudarshan-type master equation with time-dependent coefficients.Several physical quantities of the open JC system are examined and the validity or limit of some of the standard approximations used to derive the master equation is discussed on the basis of the exact results obtained here.
Time-evolution of the dispersive Jaynes–Cummings model interacting with a bosonic reservoir is considered in the presence of an initial correlation between the cavity field and reservoir. When there is at most one excitation in the whole system, the exact time-evolution can be obtained. The effects of the initial correlation on the decoherence of the two-level system and cavity field are investigated. Furthermore, the validity of the time-convolutionless quantum master equation in the second-order approximation with respect to the system-reservoir interaction is examined by means of the exact solution.
The Jaynes-Cummings and quantum Rabi models are fundamental to cavity and circuit quantum electrodynamics, as they describe the simplest form of light-matter interaction, where a single qubit is coupled to a single bosonic mode. A scenario that is commonly encountered in the experimental practice arises when the bosonic mode interacts with an external dissipative thermal bath, making the qubit-boson system open. In this work, we present a new analytical solution of the Lindblad master equations for the open dispersive Jaynes-Cummings model and a perturbative analytical solution of the open quantum Rabi model in the limit of weak qubit-boson coupling ggg, using the holomorphic formalism in Bargmann space. Specifically, we derive the most general solution of the local Lindblad master equation for the open dispersive Jaynes-Cummings model coupled to a thermal bath, with the only assumptions that the initial state of the qubit-boson system is separable. Additionally, we obtain a perturbative analytical solution for the open quantum Rabi model up to second order in ggg. Notably, our findings include a new formula for the qubit's steady state at zeroth order, showing that the stationary populations depend on both qubit and boson frequencies in the quantum Rabi model, but not in the Jaynes-Cummings model, regardless of the value of ggg. Our results are of general interest to the study of open quantum systems in the context of light-matter interaction.
Derivation of a quantum master equation for a system weakly coupled to a bath which takes into account nonsecular effects, but nevertheless has the mathematically correct Gorini-Kossakowski-Lindblad-Sudarshan form (in particular, it preserves positivity of the density operator) and also satisfies the standard thermodynamic properties is a known long-standing problem in theory of open quantum systems. The nonsecular terms are important when some energy levels of the system or their differences (Bohr frequencies) are nearly degenerate. We provide a fully rigorous derivation of such equation based on a formalization of the weak-coupling limit for the general case.
We present the Born-Markov approximated Redfield quantum master equation (RQME) description for an open system of noninteracting particles (bosons or fermions) on an arbitrary lattice of NNN sites in any dimension and weakly connected to multiple reservoirs at different temperatures and chemical potentials. The RQME can be reduced to the Lindblad equation, of various forms, by making further approximations. By studying the N=2N=2N=2 case, we show that RQME gives results which agree with exact analytical results for steady-state properties and with exact numerics for time-dependent properties over a wide range of parameters. In comparison, the Lindblad equations have a limited domain of validity in nonequilibrium. We conclude that it is indeed justified to use microscopically derived full RQME to go beyond the limitations of Lindblad equations in out-of-equilibrium systems. We also derive closed-form analytical results for out-of-equilibrium time dynamics of two-point correlation functions. These results explicitly show the approach to steady state and thermalization. These results are experimentally relevant for cold atoms, cavity QED, and far-from-equilibrium quantum dot experiments.
These notes are a short introduction to the mathematical theory of open quantum systems. They are meant to serve as an entry point into a broad research area which has applications across the quantum sciences dealing with systems subjected to external noise. The guiding idea is to let the key structures of the theory emerge from a concrete model. By working through the dissipative Jaynes-Cummings model the reader will dis- cover explicitly how irreversible dynamics arises from a unitary system-reservoir evolution. The notions of the continuous mode limit, correlation functions, spectral density appear in a natural manner and lead to the evolution equation of the open system in form of a master equation. This sets the stage for the more general analysis of completely positive, trace preserving (CPTP) maps and the study of quantum dynamical semigroups. We motivate and prove the Kraus representation theorem, the dilation theorem and the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) theorem. Working through the exercises (for which full solutions are supplied) will reinforce the ideas introduced in the main text.
We provide the exact non-Markovian master equation for a two-level system interacting with a thermal bosonic bath, and we write the solution of such a master equation in terms of the Bloch vector. We show that previous approximated results are particular limits of our exact master equation. We generalize these results to more complex systems involving an arbitrary number of two-level systems coupled to different thermal baths, providing the exact master equations also for these systems. As an example of this general case we derive the master equation for the Jaynes-Cummings model.