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A Discrete Invitation to Quantum Filtering and Feedback Control
, 2009
"... The engineering and control of devices at the quantum mechanical level—such as those consisting of small numbers of atoms and photons—is a delicate business. The fundamental uncertainty that is inherently present at this scale manifests itself in the unavoidable presence of noise, making this a nov ..."
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Cited by 25 (4 self)
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The engineering and control of devices at the quantum mechanical level—such as those consisting of small numbers of atoms and photons—is a delicate business. The fundamental uncertainty that is inherently present at this scale manifests itself in the unavoidable presence of noise, making this a novel field of application for stochastic estimation and control theory. In this expository paper we demonstrate estimation and feedback control of quantum mechanical systems in what is essentially a noncommutative version of the binomial model that is popular in mathematical finance. The model is extremely rich and allows a full development of the theory while remaining completely within the setting of finitedimensional Hilbert spaces (thus avoiding the technical complications of the continuous theory). We introduce discretized models of an atom in interaction with the electromagnetic field, obtain filtering equations for photon counting and homodyne detection, and solve a stochastic control problem using dynamic programming and Lyapunov function methods.
Modeling and Control of Quantum Systems: An introduction
"... The scope of this work is to provide a selfcontained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems, as well ..."
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Cited by 22 (7 self)
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The scope of this work is to provide a selfcontained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems, as well
Control of Quantum Systems Despite Feedback Delay
, 2007
"... A quantum control scenario follows the formalism of continuous quantum measurement and quantum state estimation. In quantum feedback control implementations the state estimation is delayed, which is known to degrade the resulting control performance. As yet there are no theoretical means to take suc ..."
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A quantum control scenario follows the formalism of continuous quantum measurement and quantum state estimation. In quantum feedback control implementations the state estimation is delayed, which is known to degrade the resulting control performance. As yet there are no theoretical means to take such estimation delays into consideration. In this paper, we formulate control problems for general quantum systems taking feedback delays into explicit account. Then we derive a globally stabilizing control law for quantum spin systems.
Quantummeasurements in continuous time, nonMarkovian evolutions and feedback
, 2012
"... 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 ’90s, and we develop it in order to describe the non Markovian evolution of a quantum system continuously measu ..."
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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 ’90s, 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 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 twolevel atom stimulated by a laser. We introduce closed loop control too, via the stimulating laser, with the aim to enhance 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 twolevel 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 Qparameter” and the spectrum of the emitted light; in particular the spectrum is not a property at a
Applications of quantum stochastic processes in quantum optics
"... These lecture notes provide an introduction to quantum filtering and its applications in quantum optics. We start with a brief introduction to quantum probability, focusing on the spectral theorem. Then we introduce the conditional expectation and quantum stochastic calculus. In the last part of the ..."
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These lecture notes provide an introduction to quantum filtering and its applications in quantum optics. We start with a brief introduction to quantum probability, focusing on the spectral theorem. Then we introduce the conditional expectation and quantum stochastic calculus. In the last part of the notes we discuss the filtering problem. 1 Quantum probability In quantum theory observables are represented by selfadjoint operators on a Hilbert space. When an observable is being measured, we randomly obtain a measurement result. That is, the result of the measurement is given by a random variable on some classical probability space. In this section we will investigate how we can pass from a selfadjoint operator to a classical random variable. Along the way we will setup a generalised theory of probability, called noncommutative or quantum probability, that is rich enough to contain quantum mechanics. 1.1 The spectral theorem In these notes we take as a part of its definition that a Hilbert space is separable. The following theorem shows that a selfadjoint operator S on a Hilbert space can be identified with a random
Quantum filter processes driven by Markovian white noises have classical versions
, 903
"... We study quantum filters that are driven by basic quantum noises and construct classical versions. Our approach is based on exploiting the quantum markovian component of the observation and measurement processes of the filters. This approach leads in a natural way the classical versions for a class ..."
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We study quantum filters that are driven by basic quantum noises and construct classical versions. Our approach is based on exploiting the quantum markovian component of the observation and measurement processes of the filters. This approach leads in a natural way the classical versions for a class of quantum filters. We consider quantum white noises derived from Wiener and Poisson processes that drive the signal and measurement processes and derive the recursive filtering equations using classical machinery. 1
École doctorale nO432: SMI Sciences des Métiers de l’ingénieur
"... pour obtenir le grade de docteur délivré par l’École nationale supérieure des mines de Paris Spécialité « Mathématiques et Automatique » présentée et soutenue publiquement par Hadis AMINI le 27 septembre 2012 Stabilisation des systèmes quantiques à temps discrets et stabilité des filtres quantiques ..."
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pour obtenir le grade de docteur délivré par l’École nationale supérieure des mines de Paris Spécialité « Mathématiques et Automatique » présentée et soutenue publiquement par Hadis AMINI le 27 septembre 2012 Stabilisation des systèmes quantiques à temps discrets et stabilité des filtres quantiques à temps continus Directeur de thèse: Pierre ROUCHON Coencadrant de la thèse: Mazyar MIRRAHIMI