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

Communications in Theoretical Physics

Exact Solution for Non-Markovian Master Equation Using Hyper-operator Approach

Hong‐Chen Fu, Zhirui Gong

Published Sep 1, 2019• 3 citations
Publisher

Abstract

Abstract Non-Markovian master equation of Harmonic oscillator and two-level systems are investigated using the hyper-operator approach. Exact solution of time evolution operator of Harmonic oscillator is obtained exactly. For two-level system the time evolution operator is exactly found and coefficients satisfy ordinary differential equation.

DOI
10.1088/0253-6102/71/9/1089

Related papers

Density-matrix operatorial solution of the non-Markovian master equation for quantum Brownian motion

F. Intravaia, Sabrina Maniscalco, ANTONINO MESSINA

An original method to exactly solve the non-Markovian master equation describing the interaction of a single harmonic oscillator with a quantum environment in the weak-coupling limit is reported. By using a superoperatorial approach, we succeed in deriving the operatorial solution for the density matrix of the system. Our method is independent of the physical properties of the environment. We show the usefulness of our solution deriving explicit expressions for the dissipative time evolution of some observables of physical interest for the system, such as, for example, its mean energy.

arXiv:quant-ph/0211196•Physical Review A•Apr 22, 2003•76 citations

Solution of the post-Markovian master equation for a damping harmonic oscillator

X. L. Huang, Jing Nie, Jiao Chen +1 more

We solve the post-Markovian master equation for a damping harmonic oscillator by virtue of the damping basis approach. With our solution, we study some features for the harmonic oscillator in non-Markovian channel. The results show that the non-Markovian effect governed by the post-Markovian master equation keeps the semi-positivity and slows the energy decay to environment. Moreover, different from the Markovian process, at zero temperature, a coherent state under time evolution does not remain a coherent state.

Physica Scripta•Jul 25, 2008•3 citations

Non-Markovian master equation for a damped oscillator with time-varying parameters

K. W. Chang, C. K. Law

We derive an exact non-Markovian master equation that generalizes the previous work [Hu, Paz and Zhang, Phys. Rev. D 45, 2843 (1992)] to damped harmonic oscillators with time-varying parameters. This is achieved by exploiting the linearity of the system and operator solution in Heisenberg picture. Our equation governs the non-Markovian quantum dynamics when the system is modulated by external devices. As an application, we apply our equation to parity kick decoupling problems. The time-dependent dissipative coefficients in the master equation are shown to be modified drastically when the system is driven by $\ensuremath{\pi}$ pulses. For coherence protection to be effective, our numerical results indicate that kicking period should be shorter than memory time of the bath. The effects of using soft pulses in an ohmic bath are also discussed.

arXiv:1003.5975•Physical Review A•May 10, 2010•29 citations

Effective Hamiltonian approach to the master equation

X X Yi, S X Yu

A method of exactly solving the master equation is presented in this paper. The explicit form of the solution is determined by the time evolution of a composite system including an auxiliary system and the open system in question. The effective Hamiltonian governing the time evolution of the composed system is derived from the master equation. Two examples, the dissipative two-level system and the damped harmonic oscillator, are presented to illustrate the solution procedure.

arXiv:quant-ph/0008090•Journal of Optics B Quantum and Semiclassical Optics•Nov 13, 2001•24 citations

Non-Markovian quantum trajectories versus master equations: Finite-temperature heat bath

Ting Yu

The interrelationship between the non-Markovian stochastic Schr\"odinger equations and the corresponding non-Markovian master equations is investigated in the finite-temperature regimes. We show that the general finite-temperature non-Markovian trajectories can be used to derive the corresponding non-Markovian master equations. A simple, yet important solvable example is the well-known damped harmonic oscillator model in which a harmonic oscillator is coupled to a finite-temperature reservoir in the rotating-wave approximation. The exact convolutionless master equation for the damped harmonic oscillator is obtained by averaging the quantum trajectories, relying upon no assumption of coupling strength or time scale. The master equation derived in this way automatically preserves the positivity, Hermiticity, and unity.

arXiv:quant-ph/0402086•Physical Review A•Jun 18, 2004•138 citations

Exact Solution for a Non-Markovian Dissipative Quantum Dynamics

Luca Ferialdi, Angelo Bassi

We provide the exact analytic solution of the stochastic Schrödinger equation describing a harmonic oscillator interacting with a non-Markovian and dissipative environment. This result represents an arrival point in the study of non-Markovian dynamics via stochastic differential equations. It is also one of the few exactly solvable models for infinite-dimensional systems. We compute the Green's function; in the case of a free particle and with an exponentially correlated noise, we discuss the evolution of Gaussian wave functions.

arXiv:1204.4348•Physical Review Letters•Apr 26, 2012•36 citations
Abstract Non-Markovian master equation of Harmonic oscillator and two-level systems are investigated using the hyper-operator approach. Exact solution of time evolution operator of Harmonic oscillator is obtained exactly. For two-level system the time evolution operator is exactly found and coefficients satisfy ordinary differential equation.
An original method to exactly solve the non-Markovian master equation describing the interaction of a single harmonic oscillator with a quantum environment in the weak-coupling limit is reported. By using a superoperatorial approach, we succeed in deriving the operatorial solution for the density matrix of the system. Our method is independent of the physical properties of the environment. We show the usefulness of our solution deriving explicit expressions for the dissipative time evolution of some observables of physical interest for the system, such as, for example, its mean energy.
We solve the post-Markovian master equation for a damping harmonic oscillator by virtue of the damping basis approach. With our solution, we study some features for the harmonic oscillator in non-Markovian channel. The results show that the non-Markovian effect governed by the post-Markovian master equation keeps the semi-positivity and slows the energy decay to environment. Moreover, different from the Markovian process, at zero temperature, a coherent state under time evolution does not remain a coherent state.
We derive an exact non-Markovian master equation that generalizes the previous work [Hu, Paz and Zhang, Phys. Rev. D 45, 2843 (1992)] to damped harmonic oscillators with time-varying parameters. This is achieved by exploiting the linearity of the system and operator solution in Heisenberg picture. Our equation governs the non-Markovian quantum dynamics when the system is modulated by external devices. As an application, we apply our equation to parity kick decoupling problems. The time-dependent dissipative coefficients in the master equation are shown to be modified drastically when the system is driven by $\ensuremath{\pi}$ pulses. For coherence protection to be effective, our numerical results indicate that kicking period should be shorter than memory time of the bath. The effects of using soft pulses in an ohmic bath are also discussed.
The interrelationship between the non-Markovian stochastic Schr\"odinger equations and the corresponding non-Markovian master equations is investigated in the finite-temperature regimes. We show that the general finite-temperature non-Markovian trajectories can be used to derive the corresponding non-Markovian master equations. A simple, yet important solvable example is the well-known damped harmonic oscillator model in which a harmonic oscillator is coupled to a finite-temperature reservoir in the rotating-wave approximation. The exact convolutionless master equation for the damped harmonic oscillator is obtained by averaging the quantum trajectories, relying upon no assumption of coupling strength or time scale. The master equation derived in this way automatically preserves the positivity, Hermiticity, and unity.
We provide the exact analytic solution of the stochastic Schrödinger equation describing a harmonic oscillator interacting with a non-Markovian and dissipative environment. This result represents an arrival point in the study of non-Markovian dynamics via stochastic differential equations. It is also one of the few exactly solvable models for infinite-dimensional systems. We compute the Green's function; in the case of a free particle and with an exponentially correlated noise, we discuss the evolution of Gaussian wave functions.