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

arXiv:0707.1795 · Physical Review E

Stochastic analysis and simulation of spin star systems

Heinz‐Peter Breuer, Francesco Petruccione

Published Jul 6, 2007• 10 citations
arXivHTMLPDF

Abstract

We discuss two methods of an exact stochastic representation of the non-Markovian quantum dynamics of open systems. The first method employs a pair of stochastic product vectors in the total system's state space, while the second method uses a pair of state vectors in the open system's state space and a random operator acting on the state space of the environment. Both techniques lead to an exact solution of the von Neumann equation for the density matrix of the total system. Employing a spin star model describing a central spin coupled to the bath of surrounding spins, we perform Monte Carlo simulations for both variants of the stochastic dynamics. In addition, we derive an analytical expression for the expectation values of the stochastic dynamics to obtain the exact solution for the density matrix of the central spin.

DOI
10.1103/physreve.76.016701

Related papers

Exact quantum jump approach to open systems in bosonic and spin baths

Heinz‐Peter Breuer

A general method is developed which enables the exact treatment of the non-Markovian quantum dynamics of open systems through a Monte Carlo simulation technique. The method is based on a stochastic formulation of the von Neumann equation of the composite system and employs a pair of product states following a Markovian random jump process. The performance of the method is illustrated by means of stochastic simulations of the dynamics of open systems interacting with a Bosonic reservoir at zero temperature and with a spin bath in the strong coupling regime.

arXiv:quant-ph/0308052•Physical Review A•Feb 24, 2004•63 citations

Non-Markovian dynamics in a spin star system: Exact solution and approximation techniques

Heinz‐Peter Breuer, Daniel Klaus Burgarth, Francesco Petruccione

The reduced dynamics of a central spin coupled to a bath of $N$ spin-$\frac{1}{2}$ particles arranged in a spin star configuration is investigated. The exact time evolution of the reduced density operator is derived, and an analytical solution is obtained in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$ of an infinite number of bath spins, where the model shows complete relaxation and partial decoherence. It is demonstrated that the dynamics of the central spin cannot be treated within the Born-Markov approximation. The Nakajima-Zwanzig and the time-convolutionless projection operator technique are applied to the spin star system. The performance of the corresponding perturbation expansions of the non-Markovian equations of motion is examined through a comparison with the exact solution.

arXiv:quant-ph/0401051•Physical Review B•Jul 30, 2004•256 citations

Stochastic simulation of dissipation and non-Markovian effects in open quantum systems

Denis Lacroix A

The exact dynamics of a system coupled to an environment can be described by an integro-differential stochastic equation for the reduced density. The influence of the environment is incorporated through a mean field which is both stochastic and nonlocal in time and where the standard two-time correlation functions of the environment appear naturally. Since no approximation is made, the presented theory incorporates exactly dissipative and non-Markovian effects. Applications to the spin-boson model coupled to a heat bath with various coupling regimes and temperature show that the presented stochastic theory can be a valuable tool to simulate exactly the dynamics of open quantum systems. Links with the stochastic Schrödinger equation method and possible extensions to "imaginary time" propagation are discussed.

arXiv:0802.1981•Physical Review E•Apr 29, 2008•22 citations

Non-Markovian quantum state diffusion for spin environments

Valentin Link, Kimmo Luoma, Walter T. Strunz

Abstract We introduce an exact open system method to describe the dynamics of quantum systems that are strongly coupled to specific types of environments comprising of spins, such as central spin systems. Our theory is similar to the established non-Markovian quantum state diffusion theory, but for a spin bath instead of a Gaussian bath. The method allows us to represent the time-evolved reduced state of the system as an ensemble average of stochastically evolving pure states. We present a comprehensive theory for arbitrary linear spin environments at both zero and finite temperatures. Furthermore, we introduce a hierarchical expansion method that enables the numerical computation of the time evolution of the stochastic pure states, facilitating a numerical solution of the open system problem in relevant strong coupling regimes.

arXiv:2203.02417•New Journal of Physics•Aug 14, 2023•11 citations

Fermionic stochastic Schrödinger equation and master equation: An open-system model

Xinyu Zhao, Wufu Shi, Lian-Ao Wu +1 more

This paper considers the extension of the non-Markovian stochastic approach for quantum open systems strongly coupled to a fermionic bath to the models in which the system operators commute with the fermionic bath. This technique can also be a useful tool for studying open quantum systems coupled to a spin-chain environment, which can be further transformed into an effective fermionic bath. We derive an exact stochastic Schr\"odinger equation (SSE), called the fermionic quantum state diffusion (QSD) equation, from the first principle by using the fermionic coherent state representation. The reduced density operator for the open system can be recovered from the stochastic average of the solutions to the QSD equation over the Grassmann-type noise. By employing the exact fermionic QSD equation, we can derive the corresponding exact master equation. The power of our approach is illustrated by the applications of our stochastic approach to several models of interest including the one-qubit dissipative model, the coupled two-qubit dissipative model, the quantum Brownian motion model, and the $N$-fermion model coupled to a fermionic bath. Different effects caused by the fermionic and bosonic baths on the dynamics of open systems are also discussed.

arXiv:1203.2220•Physical Review A•Sep 19, 2012•52 citations

Non-Markovian stochastic Schrödinger equation: Matrix-product-state approach to the hierarchy of pure states

Xing Bao Gao, Jiajun Ren, Alexander Eisfeld +1 more

We derive a stochastic hierarchy of matrix product states (HOMPS) for non-Markovian dynamics in an open quantum system at finite temperature, which is numerically exact and efficient. HOMPS is obtained from the recently developed stochastic hierarchy of pure states (HOPS) by expressing HOPS in terms of formal creation and annihilation operators. The resulting stochastic first-order differential equation is then formulated in terms of matrix product states and matrix product operators. In this way the exponential complexity of HOPS can be reduced to scale polynomial with the number of particles. The validity and efficiency of HOMPS is demonstrated for the spin-boson model and long chains where each site is coupled to a structured, strongly non-Markovian environment.

arXiv:2109.06393•Physical Review A•Mar 31, 2022•35 citations
We discuss two methods of an exact stochastic representation of the non-Markovian quantum dynamics of open systems. The first method employs a pair of stochastic product vectors in the total system's state space, while the second method uses a pair of state vectors in the open system's state space and a random operator acting on the state space of the environment. Both techniques lead to an exact solution of the von Neumann equation for the density matrix of the total system. Employing a spin star model describing a central spin coupled to the bath of surrounding spins, we perform Monte Carlo simulations for both variants of the stochastic dynamics. In addition, we derive an analytical expression for the expectation values of the stochastic dynamics to obtain the exact solution for the density matrix of the central spin.
A general method is developed which enables the exact treatment of the non-Markovian quantum dynamics of open systems through a Monte Carlo simulation technique. The method is based on a stochastic formulation of the von Neumann equation of the composite system and employs a pair of product states following a Markovian random jump process. The performance of the method is illustrated by means of stochastic simulations of the dynamics of open systems interacting with a Bosonic reservoir at zero temperature and with a spin bath in the strong coupling regime.
The reduced dynamics of a central spin coupled to a bath of NNN spin-12\frac{1}{2}21​ particles arranged in a spin star configuration is investigated. The exact time evolution of the reduced density operator is derived, and an analytical solution is obtained in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$ of an infinite number of bath spins, where the model shows complete relaxation and partial decoherence. It is demonstrated that the dynamics of the central spin cannot be treated within the Born-Markov approximation. The Nakajima-Zwanzig and the time-convolutionless projection operator technique are applied to the spin star system. The performance of the corresponding perturbation expansions of the non-Markovian equations of motion is examined through a comparison with the exact solution.
The exact dynamics of a system coupled to an environment can be described by an integro-differential stochastic equation for the reduced density. The influence of the environment is incorporated through a mean field which is both stochastic and nonlocal in time and where the standard two-time correlation functions of the environment appear naturally. Since no approximation is made, the presented theory incorporates exactly dissipative and non-Markovian effects. Applications to the spin-boson model coupled to a heat bath with various coupling regimes and temperature show that the presented stochastic theory can be a valuable tool to simulate exactly the dynamics of open quantum systems. Links with the stochastic Schrödinger equation method and possible extensions to "imaginary time" propagation are discussed.
Abstract We introduce an exact open system method to describe the dynamics of quantum systems that are strongly coupled to specific types of environments comprising of spins, such as central spin systems. Our theory is similar to the established non-Markovian quantum state diffusion theory, but for a spin bath instead of a Gaussian bath. The method allows us to represent the time-evolved reduced state of the system as an ensemble average of stochastically evolving pure states. We present a comprehensive theory for arbitrary linear spin environments at both zero and finite temperatures. Furthermore, we introduce a hierarchical expansion method that enables the numerical computation of the time evolution of the stochastic pure states, facilitating a numerical solution of the open system problem in relevant strong coupling regimes.
This paper considers the extension of the non-Markovian stochastic approach for quantum open systems strongly coupled to a fermionic bath to the models in which the system operators commute with the fermionic bath. This technique can also be a useful tool for studying open quantum systems coupled to a spin-chain environment, which can be further transformed into an effective fermionic bath. We derive an exact stochastic Schr\"odinger equation (SSE), called the fermionic quantum state diffusion (QSD) equation, from the first principle by using the fermionic coherent state representation. The reduced density operator for the open system can be recovered from the stochastic average of the solutions to the QSD equation over the Grassmann-type noise. By employing the exact fermionic QSD equation, we can derive the corresponding exact master equation. The power of our approach is illustrated by the applications of our stochastic approach to several models of interest including the one-qubit dissipative model, the coupled two-qubit dissipative model, the quantum Brownian motion model, and the NNN-fermion model coupled to a fermionic bath. Different effects caused by the fermionic and bosonic baths on the dynamics of open systems are also discussed.
We derive a stochastic hierarchy of matrix product states (HOMPS) for non-Markovian dynamics in an open quantum system at finite temperature, which is numerically exact and efficient. HOMPS is obtained from the recently developed stochastic hierarchy of pure states (HOPS) by expressing HOPS in terms of formal creation and annihilation operators. The resulting stochastic first-order differential equation is then formulated in terms of matrix product states and matrix product operators. In this way the exponential complexity of HOPS can be reduced to scale polynomial with the number of particles. The validity and efficiency of HOMPS is demonstrated for the spin-boson model and long chains where each site is coupled to a structured, strongly non-Markovian environment.