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:1006.3647

STOCHASTIC SCHRÖDINGER EQUATIONS AND MEMORY

Alberto Barchielli, Paolo Di Tella, Clément Pellegrini, Francesco Petruccione

Published Jan 1, 2011• 6 citations
arXivHTMLPDF

Abstract

By starting from the stochastic Schrödinger equation and quantum trajectory theory, we introduce memory effects by considering stochastic adapted coefficients. As an example of a natural non-Markovian extension of the theory of white noise quantum trajectories we use an Ornstein-Uhlenbeck coloured noise as the output driving process. Under certain conditions a random Hamiltonian evolution is recovered. Moreover, we show that our non-Markovian stochastic Schrödinger equations unravel some master equations with memory kernels.

DOI
10.1142/9789814338745_0003

Related papers

Stochastic Schrödinger equations with coloured noise

Alberto Barchielli, Clément Pellegrini, Francesco Petruccione

A natural non-Markovian extension of the theory of white noise quantum trajectories is presented. In order to introduce memory effects in the formalism an Ornstein-Uhlenbeck coloured noise is considered as the output driving process. Under certain conditions a random Hamiltonian evolution is recovered. Moreover, non-Markovian stochastic Schrödinger equations which unravel non-Markovian master equations are derived. As a prototypical example, an harmonic oscillator is considered, able to emit light and with memory terms in the dynamics. The signature of the non-Markovian dynamics is seen in the spectrum of the emitted light.

arXiv:0911.2554•Europhysics Letters (EPL)•Jul 1, 2010•35 citations

Stochastic Schrödinger Equations for Markovian and non-Markovian Cases

I. Semina, Vitalii Semin, Francesco Petruccione +1 more

Firstly, the Markovian stochastic Schrödinger equations are presented, together with their connections with the theory of measurements in continuous time. Moreover, the stochastic evolution equations are translated into a simulation algorithm, which is illustrated by two concrete examples — the damped harmonic oscillator and a two-level atom with homodyne photodetection. We then consider how to introduce memory effects in the stochastic Schrödinger equation via coloured noise. Specifically, the approach by using the Ornstein-Uhlenbeck process is illustrated and a simulation for the non-Markovian process proposed. Finally, an analytical approximation technique is tested with the help of the stochastic simulation in a model of a dissipative qubit.

arXiv:1310.0644•Open Systems & Information Dynamics•Mar 12, 2014•17 citations

Quantum trajectories: Memory and continuous observation

Alberto Barchielli, Clément Pellegrini, Francesco Petruccione

Starting from a generalization of the quantum trajectory theory [based on the stochastic Schr\"odinger equation (SSE)], non-Markovian models of quantum dynamics are derived. In order to describe non-Markovian effects, the approach used in this article is based on the introduction of random coefficients in the usual linear SSE. A major interest is that this allows a consistent theory of quantum measurement in continuous time to be developed for these non-Markovian quantum trajectory models. In this context, the notions of ``instrument,'' ``a priori,'' and ``a posteriori'' states can be introduced. The key point is that by starting from a stochastic equation on the Hilbert space of the system, we are able to respect the complete positivity of the mean dynamics for the statistical operator and the requirements of the axioms of quantum measurement theory. The flexibility of the theory is next illustrated by a concrete physical model of a noisy oscillator where non-Markovian effects come from the random environment, colored noises, randomness in the stimulating light, and delay effects. The statistics of the emitted photons and the heterodyne and homodyne spectra are studied, and we show how these quantities are sensitive to the non-Markovian features of the system dynamics, so that, in principle, the observation and analysis of the fluorescent light could reveal the presence of non-Markovian effects and allow for a measure of the spectra of the noises affecting the system dynamics.

arXiv:1207.1610•Physical Review A•Dec 14, 2012•20 citations

Quantum trajectories and reduced dynamics in time-correlated environments

Pietro De Checchi, Federico Gallina, Barbara Fresch +1 more

The stochastic Schrödinger equation (SSE) provides a trajectory-level route to simulate the dynamics of open quantum systems with applications ranging from molecular processes to quantum technologies. We study a colored-noise extension of the SSE based on an Ornstein-Uhlenbeck (OU) noise drive and benchmark its ensemble-averaged dynamics against the standard white-noise SSE and against a fluctuating OU random Hamiltonian. When the environment exhibits a finite correlation time, averaging over pure-state trajectories yields master equations that are generally open-form and not of Lindblad type, yet remain positive by construction. By considering the differential of the OU process, we define an effective correlated noise, whose properties we analyze and use to formulate an SSE unraveling of its associated open-form quantum master equation. We show that the averaged dissipator separates into a Lindblad contribution stemming from the white-noise component and additional correlation terms arising from the fluctuations of the OU Hamiltonian. To obtain a practical closed description and physical intuition, we introduce a Redfield-inspired perturbative closure for these correlation terms, providing an effective master equation for the colored SSE. For a two-level system, the resulting dynamics exhibit long-lived coherences, nontrivial stationary (including oscillatory) states, and multi-timescale relaxation, rationalized through the components of a time-dependent Redfield tensor.

arXiv:2507.17864•The Journal of Chemical Physics•Apr 22, 2026•0 citations

Non-Markovian master equations with quantum trajectory unravellings

James D. Cresser, Sarah Croke

Non-Markovian master equations are constructed from underlying classical stochastic processes. These equations therefore have quantum trajectory unravellings, including unravellings that have an immediate measurement interpretation, usually only found for Markovian (Lindblad) master equations.

The Rochester Conferences on Coherence and Quantum Optics and the Quantum Information and Measurement meeting•Jan 1, 2007•0 citations

Convolutionless Non-Markovian master equations and quantum trajectories: Brownian motion

Walter T. Strunz, Ting Yu

Stochastic Schr\"odinger equations for quantum trajectories offer an alternative and sometimes superior approach to the study of open quantum system dynamics. Here we show that recently established convolutionless non-Markovian stochastic Schr\"odinger equations may serve as a powerful tool for the derivation of convolutionless master equations for non-Markovian open quantum systems. The most interesting example is quantum Brownian motion (QBM) of a harmonic oscillator coupled to a heat bath of oscillators, one of the most employed exactly soluble models of open system dynamics. We show explicitly how to establish the direct connection between the exact convolutionless master equation of QBM and the corresponding convolutionless exact stochastic Schr\"odinger equation.

arXiv:quant-ph/0312103•Physical Review A•May 24, 2004•165 citations
By starting from the stochastic Schrödinger equation and quantum trajectory theory, we introduce memory effects by considering stochastic adapted coefficients. As an example of a natural non-Markovian extension of the theory of white noise quantum trajectories we use an Ornstein-Uhlenbeck coloured noise as the output driving process. Under certain conditions a random Hamiltonian evolution is recovered. Moreover, we show that our non-Markovian stochastic Schrödinger equations unravel some master equations with memory kernels.
A natural non-Markovian extension of the theory of white noise quantum trajectories is presented. In order to introduce memory effects in the formalism an Ornstein-Uhlenbeck coloured noise is considered as the output driving process. Under certain conditions a random Hamiltonian evolution is recovered. Moreover, non-Markovian stochastic Schrödinger equations which unravel non-Markovian master equations are derived. As a prototypical example, an harmonic oscillator is considered, able to emit light and with memory terms in the dynamics. The signature of the non-Markovian dynamics is seen in the spectrum of the emitted light.
Firstly, the Markovian stochastic Schrödinger equations are presented, together with their connections with the theory of measurements in continuous time. Moreover, the stochastic evolution equations are translated into a simulation algorithm, which is illustrated by two concrete examples — the damped harmonic oscillator and a two-level atom with homodyne photodetection. We then consider how to introduce memory effects in the stochastic Schrödinger equation via coloured noise. Specifically, the approach by using the Ornstein-Uhlenbeck process is illustrated and a simulation for the non-Markovian process proposed. Finally, an analytical approximation technique is tested with the help of the stochastic simulation in a model of a dissipative qubit.
Starting from a generalization of the quantum trajectory theory [based on the stochastic Schr\"odinger equation (SSE)], non-Markovian models of quantum dynamics are derived. In order to describe non-Markovian effects, the approach used in this article is based on the introduction of random coefficients in the usual linear SSE. A major interest is that this allows a consistent theory of quantum measurement in continuous time to be developed for these non-Markovian quantum trajectory models. In this context, the notions of ``instrument,'' ``a priori,'' and ``a posteriori'' states can be introduced. The key point is that by starting from a stochastic equation on the Hilbert space of the system, we are able to respect the complete positivity of the mean dynamics for the statistical operator and the requirements of the axioms of quantum measurement theory. The flexibility of the theory is next illustrated by a concrete physical model of a noisy oscillator where non-Markovian effects come from the random environment, colored noises, randomness in the stimulating light, and delay effects. The statistics of the emitted photons and the heterodyne and homodyne spectra are studied, and we show how these quantities are sensitive to the non-Markovian features of the system dynamics, so that, in principle, the observation and analysis of the fluorescent light could reveal the presence of non-Markovian effects and allow for a measure of the spectra of the noises affecting the system dynamics.
The stochastic Schrödinger equation (SSE) provides a trajectory-level route to simulate the dynamics of open quantum systems with applications ranging from molecular processes to quantum technologies. We study a colored-noise extension of the SSE based on an Ornstein-Uhlenbeck (OU) noise drive and benchmark its ensemble-averaged dynamics against the standard white-noise SSE and against a fluctuating OU random Hamiltonian. When the environment exhibits a finite correlation time, averaging over pure-state trajectories yields master equations that are generally open-form and not of Lindblad type, yet remain positive by construction. By considering the differential of the OU process, we define an effective correlated noise, whose properties we analyze and use to formulate an SSE unraveling of its associated open-form quantum master equation. We show that the averaged dissipator separates into a Lindblad contribution stemming from the white-noise component and additional correlation terms arising from the fluctuations of the OU Hamiltonian. To obtain a practical closed description and physical intuition, we introduce a Redfield-inspired perturbative closure for these correlation terms, providing an effective master equation for the colored SSE. For a two-level system, the resulting dynamics exhibit long-lived coherences, nontrivial stationary (including oscillatory) states, and multi-timescale relaxation, rationalized through the components of a time-dependent Redfield tensor.
Stochastic Schr\"odinger equations for quantum trajectories offer an alternative and sometimes superior approach to the study of open quantum system dynamics. Here we show that recently established convolutionless non-Markovian stochastic Schr\"odinger equations may serve as a powerful tool for the derivation of convolutionless master equations for non-Markovian open quantum systems. The most interesting example is quantum Brownian motion (QBM) of a harmonic oscillator coupled to a heat bath of oscillators, one of the most employed exactly soluble models of open system dynamics. We show explicitly how to establish the direct connection between the exact convolutionless master equation of QBM and the corresponding convolutionless exact stochastic Schr\"odinger equation.