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

Computational Mathematics and Mathematical Physics

Quantum Markovian Dynamics after the Bath Correlation Time

Alexander Evgen’evich Teretenkov

Published Jan 1, 2023• 4 citations
Publisher

Abstract

Abstract For a model of a multilevel system interacting with several baths at zero temperature, it is shown that its dynamics becomes Markovian after the bath correlation time. We take into account not only the contribution of the bath spectral density, which leads to a continuous correlation function, but also the ohmic contribution to the spectral density, which leads to a renormalization of both equations and initial conditions. An explicit Gorini–Kossakowski–Sudarshan–Lindblad equation describing the dynamics of the system after the bath correlation time is derived, and the form of initial conditions for this equation is obtained. They do not coincide with the exact initial conditions due to the renormalization associated with the ohmic contribution and due to the short initial non-Markovian time interval preceding the bath correlation time.

DOI
10.1134/s0965542523010128

Related papers

Quantum Markovian Dynamics after the Bath Correlation Time

Alexander Evgen’evich Teretenkov

For a model of a multilevel system interacting with several baths at zero temperature, it is shown that its dynamics becomes Markovian after the bath correlation time. We take into account not only the contribution of the bath spectral density, which leads to a continuous correlation function, but also the ohmic contribution to the spectral density, which leads to a renormalization of both equations and initial conditions. An explicit Gorini–Kossakowski–Sudarshan–Lindblad equation describing the dynamics of the system after the bath correlation time is derived, and the form of initial conditions for this equation is obtained. They do not coincide with the exact initial conditions due to the renormalization associated with the ohmic contribution and due to the short initial non-Markovian time interval preceding the bath correlation time.

Журнал вычислительной математики и математической физики•Jan 1, 2023•0 citations

Completely-positive non-signalling non-Markovian dynamics

Serhii Kryhin, Vivishek Sudhir

We define non-Markovian quantum dynamics as evolution in which the current state depends on all past states, and completely characterize its structure under the assumptions of complete positivity and non-signalling. The resulting continuous-time dynamics is an integro-differential equation that augments the Gorini-Kossakowski-Sudarshan-Lindblad equation with a memory integral, and is capable of describing the quantum state of systems exposed to noise with any integrable power spectral density with no further approximations. We then establish a formalism to evaluate multi-time correlations of measurement outcomes in this general setting, obviating the need for a regression theorem. As an application, we derive the emission spectrum of a driven two-level system coupled to a non-Markovian bath: the familiar Mollow triplet acquires a frequency-dependent linewidth that encodes the memory of the bath. Our work provides a rigorous yet transparent description of the quantum state of non-Markovian systems, opening the door for state estimation and state-based quantum control beyond the Markovian regime.

quant-pharXiv:2605.03197•May 6, 2026•0 citations

Derivation of Markovian Master Equation by Renormalization Group Method

Yasusada Nambu, Shingo Kukita

We present a derivation of the Markovian master equation by the renormalization group method. Starting from a naive perturbative solution of the von Neumann equation, the reduced density matrix with the coarse grained time steps is obtained using the assumption of short correlation time of the bath field. Then by applying the renormalization group method, we show that the dependence of the specific initial time on the perturbative solution can be removed and the Markovian semigroup master equation in the Gorini--Kossakowski--Lindblad--Sudarshan (GKLS) form is obtained in the weak coupling limit.

arXiv:1504.03070•Journal of the Physical Society of Japan•Oct 28, 2016•4 citations

Dynamics of Open Quantum Systems II, Markovian Approximation

Marco Merkli

A finite-dimensional quantum system is coupled to a bath of oscillators in thermal equilibrium at temperature T>0 . We show that for fixed, small values of the coupling constant λ , the true reduced dynamics of the system is approximated by the completely positive, trace preserving Markovian semigroup generated by the Davies-Lindblad generator. The difference between the true and the Markovian dynamics is O(|λ|1/4) for all times, meaning that the solution of the Gorini-Kossakowski-Sudarshan-Lindblad master equation is approximating the true dynamics to accuracy O(|λ|1/4) for all times. Our method is based on a recently obtained expansion of the full system-bath propagator. It applies to reservoirs with correlation functions decaying in time as 1/t4 or faster, which is a significant improvement relative to the previously required exponential decay.

arXiv:2105.00023•Quantum•Jan 3, 2022•24 citations

Memory Tensor for Non-Markovian Dynamics with Random Hamiltonian

Alexander Evgen’evich Teretenkov

In the theory of open quantum systems, the Markovian approximation is very widespread. Usually, it assumes the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation for density matrix dynamics and quantum regression formulae for multi-time correlation functions. Nevertheless, now, quantum non-Markovianity is being actively studied, especially the non-Markovianity of multi-time correlations. In this work, we consider dynamics with a random Hamiltonian, which can lead to GKSL dynamics of the density matrix for some special cases, but correlation functions generally do not satisfy the quantum regression formulae. Despite the fact that random Hamiltonians have been actively studied, dynamics with such Hamiltonians has been little discussed from the viewpoint of multi-time correlations. For specific models with a random Hamiltonian, we provide the formulae for multi-time correlations which occur instead of the usual regression formulae. Moreover, we introduce and calculate the memory tensor, which characterizes multi-time correlations against the Markovian ones. We think that, despite being applied to specific models, the methods developed in this work can be used in a much broader setup.

Mathematics•Sep 8, 2023•4 citations

Dynamics of Quantum Correlations in a Qubit-Oscillator System Interacting via a Dissipative Bath

R. Badveli, V. Jagadish, S. Akshaya +2 more

The entanglement dynamics in a bipartite system consisting of a qubit and a harmonic oscillator interacting only through their coupling with the same bath is studied. The considered model assumes that the qubit is coupled to the bath via the Jaynes-Cummings interaction, whilst the position of the oscillator is coupled to the position of the bath via a dipole interaction. We give a microscopic derivation of the Gorini–Kossakowski–Sudarshan–Lindblad equation for the considered model. Based on the Kossakowski matrix, we show that non-classical correlations including entanglement can be generated by the considered dynamics. We then analytically identify specific initial states for which entanglement is generated. This result is also supported by our numerical simulations.

arXiv:2002.04173•Open Systems & Information Dynamics•Mar 1, 2020•6 citations
Abstract For a model of a multilevel system interacting with several baths at zero temperature, it is shown that its dynamics becomes Markovian after the bath correlation time. We take into account not only the contribution of the bath spectral density, which leads to a continuous correlation function, but also the ohmic contribution to the spectral density, which leads to a renormalization of both equations and initial conditions. An explicit Gorini–Kossakowski–Sudarshan–Lindblad equation describing the dynamics of the system after the bath correlation time is derived, and the form of initial conditions for this equation is obtained. They do not coincide with the exact initial conditions due to the renormalization associated with the ohmic contribution and due to the short initial non-Markovian time interval preceding the bath correlation time.
For a model of a multilevel system interacting with several baths at zero temperature, it is shown that its dynamics becomes Markovian after the bath correlation time. We take into account not only the contribution of the bath spectral density, which leads to a continuous correlation function, but also the ohmic contribution to the spectral density, which leads to a renormalization of both equations and initial conditions. An explicit Gorini–Kossakowski–Sudarshan–Lindblad equation describing the dynamics of the system after the bath correlation time is derived, and the form of initial conditions for this equation is obtained. They do not coincide with the exact initial conditions due to the renormalization associated with the ohmic contribution and due to the short initial non-Markovian time interval preceding the bath correlation time.
We define non-Markovian quantum dynamics as evolution in which the current state depends on all past states, and completely characterize its structure under the assumptions of complete positivity and non-signalling. The resulting continuous-time dynamics is an integro-differential equation that augments the Gorini-Kossakowski-Sudarshan-Lindblad equation with a memory integral, and is capable of describing the quantum state of systems exposed to noise with any integrable power spectral density with no further approximations. We then establish a formalism to evaluate multi-time correlations of measurement outcomes in this general setting, obviating the need for a regression theorem. As an application, we derive the emission spectrum of a driven two-level system coupled to a non-Markovian bath: the familiar Mollow triplet acquires a frequency-dependent linewidth that encodes the memory of the bath. Our work provides a rigorous yet transparent description of the quantum state of non-Markovian systems, opening the door for state estimation and state-based quantum control beyond the Markovian regime.
We present a derivation of the Markovian master equation by the renormalization group method. Starting from a naive perturbative solution of the von Neumann equation, the reduced density matrix with the coarse grained time steps is obtained using the assumption of short correlation time of the bath field. Then by applying the renormalization group method, we show that the dependence of the specific initial time on the perturbative solution can be removed and the Markovian semigroup master equation in the Gorini--Kossakowski--Lindblad--Sudarshan (GKLS) form is obtained in the weak coupling limit.
A finite-dimensional quantum system is coupled to a bath of oscillators in thermal equilibrium at temperature T>0 . We show that for fixed, small values of the coupling constant λ , the true reduced dynamics of the system is approximated by the completely positive, trace preserving Markovian semigroup generated by the Davies-Lindblad generator. The difference between the true and the Markovian dynamics is O(|λ|1/4) for all times, meaning that the solution of the Gorini-Kossakowski-Sudarshan-Lindblad master equation is approximating the true dynamics to accuracy O(|λ|1/4) for all times. Our method is based on a recently obtained expansion of the full system-bath propagator. It applies to reservoirs with correlation functions decaying in time as 1/t4 or faster, which is a significant improvement relative to the previously required exponential decay.
In the theory of open quantum systems, the Markovian approximation is very widespread. Usually, it assumes the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation for density matrix dynamics and quantum regression formulae for multi-time correlation functions. Nevertheless, now, quantum non-Markovianity is being actively studied, especially the non-Markovianity of multi-time correlations. In this work, we consider dynamics with a random Hamiltonian, which can lead to GKSL dynamics of the density matrix for some special cases, but correlation functions generally do not satisfy the quantum regression formulae. Despite the fact that random Hamiltonians have been actively studied, dynamics with such Hamiltonians has been little discussed from the viewpoint of multi-time correlations. For specific models with a random Hamiltonian, we provide the formulae for multi-time correlations which occur instead of the usual regression formulae. Moreover, we introduce and calculate the memory tensor, which characterizes multi-time correlations against the Markovian ones. We think that, despite being applied to specific models, the methods developed in this work can be used in a much broader setup.
The entanglement dynamics in a bipartite system consisting of a qubit and a harmonic oscillator interacting only through their coupling with the same bath is studied. The considered model assumes that the qubit is coupled to the bath via the Jaynes-Cummings interaction, whilst the position of the oscillator is coupled to the position of the bath via a dipole interaction. We give a microscopic derivation of the Gorini–Kossakowski–Sudarshan–Lindblad equation for the considered model. Based on the Kossakowski matrix, we show that non-classical correlations including entanglement can be generated by the considered dynamics. We then analytically identify specific initial states for which entanglement is generated. This result is also supported by our numerical simulations.