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Non Markovian Quantum Repeated Interactions and Measurements

Clément Pellegrini, Francesco Petruccione

Published Jan 1, 2009• 21 citations
OpenAlex

Abstract

A non-Markovian model of quantum repeated interactions between a small quantum system and an infinite chain of quantum systems is presented. By adapting and applying usual projection operator techniques in this context, discrete versions of the integro-differential and time-convolutioness Master equations for the reduced system are derived. Next, an intuitive and rigorous description of the indirect quantum measurement principle is developed and a discrete non Markovian stochastic Master equation for the open system is obtained. Finally, the question of unravelling in a particular model of non-Markovian quantum interactions is discussed. 1

Related papers

Non-Markovian quantum repeated interactions and measurements

Clément Pellegrini, F Petruccione

A non-Markovian model of quantum repeated interactions between a small quantum system and an infinite chain of quantum systems is presented. By adapting and applying usual projection operator techniques in this context, discrete versions of the integro-differential and time-convolutionless master equations for the reduced system are derived. Next, an intuitive and rigorous description of the indirect quantum measurement principle is developed and a discrete non-Markovian stochastic master equation for the open system is obtained. Finally, the question of unravelling in a particular model of non-Markovian quantum interactions is discussed.

arXiv:0903.3859•Journal of Physics A Mathematical and Theoretical•Sep 30, 2009•42 citations

Projection Operator Stochastic Equations for Non-Markovian Quantum Systems Under Continuous Measurement-Based Feedback

H. I. Nurdin

Quantum Markov models have been successfully used to accurately model various physical quantum systems in fields such as quantum optics, optomechanics and superconducting circuits and they provide the basis for (measurement-based) quantum feedback control. However, the quantum Markov assumption is a strong one and it is not expected to hold for general quantum systems of interest. The projection operator approach is one approach that has been developed to model non-Markovian quantum systems by considering its embedding in a larger Markovian quantum system, but mainly in the context of quantum master equations for the dynamics of the unmonitored reduced quantum state of a quantum system. This approach was recently adapted for continuously measured non-Markovian quantum systems, which enables open-loop control but did not yet consider the presence of feedback of the stochastic measurement record, deriving non-Markovian SDEs for the evolution of the projected state of the Markovian embedding. This paper generalizes these stochastic equations to the setting of stochastic feedback based on the continuous-measurement record and shows that the equations take the same form but that previously deterministic terms become stochastic ones which depend on the measurement record, as would be intuitively expected. The stochastic equations are obtained for a generalized class of measurements that includes continuous (possibly adaptive) homodyne and photon counting measurements.

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Stochastic Master Equations in Thermal Environment

Stéphane Attal, université Lyon, Clément Pellegrini

Abstract. We derive the stochastic master equations which describe the evolution of open quantum systems in contact with a heat bath and undergoing indirect measurements. These equations are obtained as a limit of a quantum repeated measurement model where we consider a small system in contact with an infinite chain at positive temperature. At zero temperature it is well-known that one obtains stochastic differential equations of jump-diffusion type. At strictly positive temperature, we show that only pure diffusion type equations are relevant. 1.

•Sep 1, 2008•0 citations

Quantum jump unravelings for non-Markovian open system dynamics: a review

Federico Settimo, Jyrki Piilo

Stochastic unravelings provide a useful way to represent open quantum system dynamics in terms of pure state realizations, and have been widely studied both from a fundamental and from a computational point of view. They were initially formulated for Markovian dynamics described by the Gorini-Kossakowski-Sudarshan-Lindblad master equation. However, due to recent technological and experimental development, most physical relevant dynamics present temporal correlations beyond the Markov approximation. Such correlations cause decay rates to turn temporarily negative, thus requiring the generalization of stochastic unravelings from Markovian to non-Markovian scenarios. Indeed, many unraveling techniques have been introduced in this regime, and a comprehensive review of the different jump methods is currently missing. In this work, we provide an overview of widely used quantum jump unraveling techniques for non-Markovian systems and also discuss them in terms of their numerical efficiency, divisibility requirements, Hilbert space extension, and measurement interpretation.

quant-pharXiv:2605.07797•May 11, 2026•0 citations

RETURN TO EQUILIBRIUM FOR SOME STOCHASTIC SCHRÖDINGER EQUATIONS

Stéphane Attal, Clément Pellegrini

The principle of "indirect continuous measurement" in "open quantum system theory" is usually described by non-usual types of stochastic differential equations. These equations are called "stochastic Schrödinger equations" and their solutions are called "quantum trajectories". Physically, they describe the random evolution of the state of a quantum system undergoing indirect quantum measurement (such models are widely used in quantum optics, quantum computing and quantum information theory). In this chapter, we consider a physically realistic discrete-time setup for two-level quantum systems and we present the theory of "discrete quantum trajectories". These discrete trajectories are Markov chains which can be expressed as solutions of "discrete-time" stochastic differential equations". In particular, these equations appear as time discretization of "stochastic Schrödinger equations". Going to the continuous-time limit, we justify the stochastic Schrödinger equations associated to the two-level systems. Within this approach, we obtain two different types of behaviors described either by jump-type or diffusive-type stochastic differential equations. Finally we investigate the large time behavior of the solutions and we prove return to equilibrium properties for the associated physical models.

•Jan 1, 2011•1 citation
A non-Markovian model of quantum repeated interactions between a small quantum system and an infinite chain of quantum systems is presented. By adapting and applying usual projection operator techniques in this context, discrete versions of the integro-differential and time-convolutioness Master equations for the reduced system are derived. Next, an intuitive and rigorous description of the indirect quantum measurement principle is developed and a discrete non Markovian stochastic Master equation for the open system is obtained. Finally, the question of unravelling in a particular model of non-Markovian quantum interactions is discussed. 1
A non-Markovian model of quantum repeated interactions between a small quantum system and an infinite chain of quantum systems is presented. By adapting and applying usual projection operator techniques in this context, discrete versions of the integro-differential and time-convolutionless master equations for the reduced system are derived. Next, an intuitive and rigorous description of the indirect quantum measurement principle is developed and a discrete non-Markovian stochastic master equation for the open system is obtained. Finally, the question of unravelling in a particular model of non-Markovian quantum interactions is discussed.
Quantum Markov models have been successfully used to accurately model various physical quantum systems in fields such as quantum optics, optomechanics and superconducting circuits and they provide the basis for (measurement-based) quantum feedback control. However, the quantum Markov assumption is a strong one and it is not expected to hold for general quantum systems of interest. The projection operator approach is one approach that has been developed to model non-Markovian quantum systems by considering its embedding in a larger Markovian quantum system, but mainly in the context of quantum master equations for the dynamics of the unmonitored reduced quantum state of a quantum system. This approach was recently adapted for continuously measured non-Markovian quantum systems, which enables open-loop control but did not yet consider the presence of feedback of the stochastic measurement record, deriving non-Markovian SDEs for the evolution of the projected state of the Markovian embedding. This paper generalizes these stochastic equations to the setting of stochastic feedback based on the continuous-measurement record and shows that the equations take the same form but that previously deterministic terms become stochastic ones which depend on the measurement record, as would be intuitively expected. The stochastic equations are obtained for a generalized class of measurements that includes continuous (possibly adaptive) homodyne and photon counting measurements.
Abstract. We derive the stochastic master equations which describe the evolution of open quantum systems in contact with a heat bath and undergoing indirect measurements. These equations are obtained as a limit of a quantum repeated measurement model where we consider a small system in contact with an infinite chain at positive temperature. At zero temperature it is well-known that one obtains stochastic differential equations of jump-diffusion type. At strictly positive temperature, we show that only pure diffusion type equations are relevant. 1.
Stochastic unravelings provide a useful way to represent open quantum system dynamics in terms of pure state realizations, and have been widely studied both from a fundamental and from a computational point of view. They were initially formulated for Markovian dynamics described by the Gorini-Kossakowski-Sudarshan-Lindblad master equation. However, due to recent technological and experimental development, most physical relevant dynamics present temporal correlations beyond the Markov approximation. Such correlations cause decay rates to turn temporarily negative, thus requiring the generalization of stochastic unravelings from Markovian to non-Markovian scenarios. Indeed, many unraveling techniques have been introduced in this regime, and a comprehensive review of the different jump methods is currently missing. In this work, we provide an overview of widely used quantum jump unraveling techniques for non-Markovian systems and also discuss them in terms of their numerical efficiency, divisibility requirements, Hilbert space extension, and measurement interpretation.
The principle of "indirect continuous measurement" in "open quantum system theory" is usually described by non-usual types of stochastic differential equations. These equations are called "stochastic Schrödinger equations" and their solutions are called "quantum trajectories". Physically, they describe the random evolution of the state of a quantum system undergoing indirect quantum measurement (such models are widely used in quantum optics, quantum computing and quantum information theory). In this chapter, we consider a physically realistic discrete-time setup for two-level quantum systems and we present the theory of "discrete quantum trajectories". These discrete trajectories are Markov chains which can be expressed as solutions of "discrete-time" stochastic differential equations". In particular, these equations appear as time discretization of "stochastic Schrödinger equations". Going to the continuous-time limit, we justify the stochastic Schrödinger equations associated to the two-level systems. Within this approach, we obtain two different types of behaviors described either by jump-type or diffusive-type stochastic differential equations. Finally we investigate the large time behavior of the solutions and we prove return to equilibrium properties for the associated physical models.