Stochastic Schrödinger Equations and Memory

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

doi:10.1142/9789814338745_0003

Cited by 3 publications

(11 citation statements)

References 30 publications

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“…The master equation approach is receiving growing attention in the mathematical physics literature, especially in determining standard forms and quantifying the non-Markovianity of the evolution, and its potential application to "quantum biology" problems makes those models a natural candidate for control oriented studies. Recent developments on non-Markovian stochastic dynamics make those a natural object for further research on feedback control [147].…”

Section: A Brief Outlookmentioning

confidence: 99%

Modeling and Control of Quantum Systems: An Introduction

Altafini

Ticozzi²

2012

IEEE Trans. Automat. Contr.

246

214

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Abstract-The scope of this work is to provide a self-contained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems, as well as a brief survey on the main approaches to control synthesis. While part of the existing theory, especially in the open-loop setting, stems directly from classical control theory (most notably geometric control and optimal control), a number of tools specifically tailored for quantum systems have been developed since the 1980s, in order to take into account their distinctive features: the probabilistic nature of atomic-scale physical systems, the effect of dissipation and the irreversible character of the measurements have all proved to be critical in feedbackdesign problems. The relevant dynamical models for both closed and open quantum systems are presented, along with the main results on their controllability and stability. A brief review of several currently available control design methods is meant to provide the interested reader with a roadmap for further studies.

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Section: A Brief Outlookmentioning

confidence: 99%

Modeling and Control of Quantum Systems: An Introduction

Altafini

Ticozzi²

2012

IEEE Trans. Automat. Contr.

246

214

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“…However, this master equation is not closed, because the X(t) and ρ(t) are random and not independent. A closed equation can be obtained by using the Nakajima-Zwanzig method and the generalized master equation one obtains in this way can be the starting point for some approximations [30]. Indeed, the operation of taking the mean is a projection in the space of random trace class operators.…”

Section: Projection Techniques and Closed Master Equations With Memorymentioning

confidence: 99%

Stochastic Schrödinger Equations for Markovian and non-Markovian Cases

Semina

Semin

Petruccione

et al. 2014

Open Syst. Inf. Dyn.

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Abstract. 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. Then, we 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.

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“…The best way to introduce memory in quantum evolutions is to start from a dynamical equation in Hilbert space; this approach automatically guarantees the complete positivity of the evolution of the state (statistical operator) of the system. Moreover, considering the linear version of the SSE allows to construct the instruments related to the continuous monitoring even in the non Markov case [22][23][24]. We shall introduce first several mathematical objects, from the linear SSE (1) to the instruments (14), and later, thanks to these latter, we shall give a consistent physical interpretation of the whole construction.…”

Section: The Stochastic Schrödinger Equation and The Stochastic Mastementioning

confidence: 99%

“…More recently, motivated by the growing interest in non-Markovian evolutions, we have begun to analyse possible physical applications of this theory. Some applications have already been developed for continuously monitored systems, which are affected by coloured noises [23,24]. This paper, instead, will be focused mainly on feedback control.…”

Section: Quantum Trajectories and Controlmentioning

confidence: 99%

“…which is not a closed differential equation when L(t) is stochastic, contrary to the Markov case [9,Section 3.4]. By the projection operator technique a closed integro-differential equation for the a priori state η(t) could be obtained [24] (an evolution equation with memory), but this equation is too involved to be of practical use. Again (1) and ( 3) are an unravelling of a non Markovian evolution.…”

Section: A Priori Statesmentioning

confidence: 99%

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Quantum measurements in continuous time, non-Markovian evolutions and feedback

Barchielli

Gregoratti

2012

Phil. Trans. R. Soc. A.

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In this article, we reconsider a version of quantum trajectory theory based on the stochastic Schrödinger equation with stochastic coefficients, which was mathematically introduced in the 1990s, and we develop it in order to describe the non-Markovian evolution of a quantum system continuously measured and controlled, thanks to a measurement-based feedback. Indeed, realistic descriptions of a feedback loop have to include delay and thus need a non-Markovian theory. The theory allows us to put together non-Markovian evolutions and measurements in continuous time, in agreement with the modern axiomatic formulation of quantum mechanics. To illustrate the possibilities of such a theory, we apply it to a two-level atom stimulated by a laser. We introduce closed loop control too, via the stimulating laser, with the aim of enhancing the 'squeezing' of the emitted light, or other typical quantum properties. Note that here we change the point of view with respect to the usual applications of control theory. In our model, the 'system' is the two-level atom, but we do not want to control its state, to bring the atom to a final target state. Our aim is to control the 'Mandel Q-parameter' and the spectrum of the emitted light; in particular, the spectrum is not a property at a single time, but involves a long interval of times (a Fourier transform of the autocorrelation function of the observed output is needed).Keywords: quantum trajectories; non-Markovian evolutions; closed loop control; squeezing Quantum trajectories and controlStochastic wave function methods for the description of open quantum systems are now widely used [1][2][3][4][5] and are often referred to as quantum trajectory theory. These approaches are very important for numerical simulations and allow the continuous measurement description of detection schemes in quantum optics, namely direct, homodyne and heterodyne photo-detection [6][7][8][9]. In the Markovian case, the stochastic differential equations of the quantum trajectory theory can be interpreted in terms of measurements in continuous time because they can be related to positive operator valued measures and instruments [10][11][12], which are the objects representing observables and state changes in the *Author for correspondence (alberto.barchielli@polimi.it).One contribution of 15 to a Theo Murphy Meeting Issue 'Principles and applications of quantum control engineering'.This journal is

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Stochastic Schrödinger Equations and Memory

Cited by 3 publications

References 30 publications

Modeling and Control of Quantum Systems: An Introduction

Modeling and Control of Quantum Systems: An Introduction

Stochastic Schrödinger Equations for Markovian and non-Markovian Cases

Quantum measurements in continuous time, non-Markovian evolutions and feedback

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