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arXiv:0903.3859 · Journal of Physics A Mathematical and Theoretical

Non-Markovian quantum repeated interactions and measurements

Clément Pellegrini, F Petruccione

Published Sep 30, 2009• 42 citations
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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-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.

DOI
10.1088/1751-8113/42/42/425304

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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.
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
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.
A generalization of the stochastic wave-function method to quantum master equations which are not in Lindblad form is developed. The proposed stochastic unraveling is based on a description of the reduced system in a doubled Hilbert space and it is shown that this method is capable of simulating quantum master equations with negative transition rates. Non-Markovian effects in the reduced systems dynamics can be treated within this approach by employing the time-convolutionless projection operator technique. This ansatz yields a systematic perturbative expansion of the reduced systems dynamics in the coupling strength. Several examples such as the damped Jaynes-Cummings model and the spontaneous decay of a two-level system into a photonic band gap are discussed. The power as well as the limitations of the method are demonstrated.
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.