arXiv:0805.4002

A Wave Function approach to dissipative processes

Yvan Castin, Jean Dalibard, Klaus Mølmer

Published May 26, 2008• 4 citations

Abstract

Due to growing interest in quantum measurement, control and feedback, we reproduce a manuscript from 1992, presenting a simple physical and mathematical derivation of stochastic differential equations for wave functions of probed quantum systems. V. P. Belavkins seminal quantum filtering theory with similar equations, developed in the 1980es, was not known to the authors at the time of writing of the present manuscript.

DOI: 10.48550/arxiv.0805.4002

Related papers

Quantum filtering and propagation of chaos for open quantum systems, with applications to quantum feedback control and quantum mean-field games

Vassili Nikitich Kolokoltsov

The theory of quantum filtering (of quantum continuous measurements) was developed by V.P. Belavkin about 40 years ago. Since then it attracted attention of numerous investigators including mathematicians, theoretical and experimental physicists. However, the rigorous mathematical theory of the filtering equations for mixed states in basic infinite-dimensional quantum systems remained an open problem, which was resolved by the author recently. This survey paper presents in full the mathematical theory of quantum filtering equations, their rigorous derivation from basic principles, the corresponding law of large number limits (propagation of chaos) and related topics. Applications to feedback control, quantum dynamic and mean-field games are discussed.

arXiv:2607.08507Jul 9, 20260 citations

Stochastic Schrödinger equations

Luc Bouten, Mădălin Guţǎ, Hans Maassen

A derivation of Belavkin's stochastic Schrödinger equations is given using quantum filtering theory. We study an open system in contact with its environment, the electromagnetic field. Continuous observation of the field yields information on the system: it is possible to keep track in real time of the best estimate of the system's quantum state given the observations made. This estimate satisfies a stochastic Schrödinger equation, which can be derived from the quantum stochastic differential equation for the interaction picture evolution of system and field together. Throughout the paper we focus on the basic example of resonance fluorescence.

arXiv:quant-ph/0309205Journal of Physics A Mathematical and GeneralFeb 18, 2004108 citations

Measurement and Filtering of Quantum Diffusion II

V. P. Belavkin

Next article Measurement and Filtering of Quantum Diffusion IIV. P. BelavkinV. P. Belavkinhttps://doi.org/10.1137/1139026PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] A. N. Kolmogorov, Foundations of the Theory of Probability, Chelsea Publishing Company, New York, 1984 Google Scholar[2] John von Neumann, Mathematical foundations of quantum mechanics, Princeton University Press, Princeton, 1955xii+445, N.Y. 16,654a Google Scholar[3] A. S. Holevo, Probability and Stochastic Aspects of Quantum Theory, Princeton University Press, Princeton, N.J., 1955 Google Scholar[4] B. Misra and , E. C. G. Sudarshan, The Zeno's paradox in quantum theory, J. Mathematical Phys., 18 (1977), 756–763 10.1063/1.523304 55:5076 CrossrefGoogle Scholar[5] V. P. Belavkin, A stochastic calculus of quantum input-output processes and quantum nondemolition filteringCurrent problems in mathematics. Newest results, Vol. 36 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989, 29–67, 187 91f:81074 Google Scholar[6] Viacheslav P. Belavkin, Nondemolition measurements, nonlinear filtering and dynamic programming of quantum stochastic processesModeling and control of systems in engineering, quantum mechanics, economics and biosciences (Sophia-Antipolis, 1988), Lecture Notes in Control and Inform. Sci., Vol. 121, Springer, Berlin, 1989, 245–265 1 231 124 0683.93084 CrossrefGoogle Scholar[7] V. P. Belavkin, Optimal quantum filtering of Markov signals, Problemy upravleniya i teoriya inform., 7 (1978), 345–360, (In Russian.) 0397.94001 Google Scholar[8] V. P. Belavkin, A. Blaquiere, Nondemolition measurement and control in quantum dynamical systemsInformation complexity and control in quantum physics (Udine, 1985), CISM Courses and Lectures, Vol. 294, Springer, Vienna, 1987, 311–329 913 955 CrossrefGoogle Scholar[9] V. P. Belavkin, B. Grigelionis, Stochastic posterior equations for quantum nonlinear filteringProbability theory and mathematical statistics, Vol. I (Vilnius, 1989), “Mokslas”, Vilnius, 1990, 91–109, Proceedings of the Fifth Vilnius Conference 93b:81126 0914.60082 Google Scholar[10] V. P. Belavkin, A quantum nonadapted stochastic calculus and nonstationary evolution in Fock scaleQuantum probability & related topics, QP-PQ, VI, World Sci. Publishing, River Edge, NJ, 1991, 137–179 92m:81119 0937.60045 CrossrefGoogle Scholar[11] Viacheslav P. Belavkin, Quantum stochastic calculus and quantum nonlinear filtering, J. Multivariate Anal., 42 (1992), 171–201 10.1016/0047-259X(92)90042-E 93m:81013 0762.60059 CrossrefGoogle Scholar[12] R. L. Hudson and , K. R. Parthasarathy, Quantum Ito's formula and stochastic evolutions, Comm. Math. Phys., 93 (1984), 301–323 86e:46057 0546.60058 CrossrefGoogle Scholar[13] A. S. Holevo, Quantum stochastic calculus, Itogi nauki tekhniki. Modern problems of mathematics series. Newest achievements, 36 (1989), 3–28 Google Scholar[14] V. P. Belavkin, Quantum continual measurements and a posteriori collapse on CCR, Comm. Math. Phys., 146 (1992), 611–635 93j:81006 0757.60057 CrossrefGoogle Scholar[15] V. P. Belavkin, A posterior Schrödinger equation for continuous nondemolition measurement, J. Math. Phys., 31 (1990), 2930–2934 10.1063/1.528946 91m:81020 0718.47035 CrossrefGoogle Scholar[16] V. P. Belavkin and , P. Staszewski, Nondemolition observation of a free quantum particle, Phys. Rev., Ser. A, 45 (1992), 1347–1356 10.1103/PhysRevA.45.1347 CrossrefGoogle Scholar[17] M. P. Evans and , R. L. Hudson, Multidimensional quantum diffusionsQuantum probability and applications, III (Oberwolfach, 1987), Lecture Notes in Math., Vol. 1303, Springer, Berlin, 1988, 69–88 91a:46065 0648.46056 CrossrefGoogle Scholar[18] R. S. Liptser and , A. N. Shiryayev, Statistics of random processes. I, Springer-Verlag, New York, 1977x+394, Berlin 57:14125 R. S. Liptser and , A. N. Shiryayev, Statistics of random processes. II, Springer-Verlag, New York, 1978x+339, Berlin 58:7827 CrossrefGoogle Scholar[19] R. L. Stratonovitch, Conditional Markov Processes and their Application to Optimal Control Theory, Moscow State University, Moscow, 1966 Google Scholar[20] V. P. Belavkin, Chaotic states and stochastic integration in quantum systems, Russian Math. Surveys, 47 (1992), 47–106, (In Russian.) CrossrefGoogle Scholar Next article FiguresRelatedReferencesCited ByDetails Open-system many-body dynamics through interferometric measurements and feedbackPhysical Review A, Vol. 94, No. 5 | 17 November 2016 Cross Ref Continuous Quantum Hypothesis TestingPhysical Review Letters, Vol. 108, No. 17 | 1 April 2012 Cross Ref Volume 39, Issue 3| 1995Theory of Probability & Its Applications History Submitted:08 October 1991Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1139026Article page range:pp. 363-378ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Theory of Probability and Its ApplicationsJan 1, 19952 citations

Filtering and Control in Quantum Optics

Luc Bouten

In this thesis we study continuously observed open quantum systems from the point of view of quantum filtering theory. The conditioned state of the open system obeys a non-commutative or quantum analogue of the Kushner-Stratonovich equation in classical filtering theory. The thesis consists out of four chapters. The first chapter is a brief introduction to quantum probability theory and quantum stochastic calculus. The second chapter is a description of a photon counting experiment in continuous time within the framework of Davies and Srinivas. The third chapter focusses on the derivation of the quantum filtering or Belavkin equation from the quantum stochastic differential equation governing the interaction between the open system and its environment, the electromagnetic field. The fourth chapter shows how the quantum filtering equation can be used to control quantum systems.

arXiv:quant-ph/0410080Oct 11, 200424 citations

Diffusive Dynamics and Continuous Monitoring

Alessio Serafini

In the last chapter, we dealt with the wide class of physical operations that preserve the Gaussian character of the initial state, treating them as discrete transformations. Here, we describe what could loosely be considered the continuous-time version of such transformations. This chapter is intended to provide the reader with the tools to describe a wide range of notable quantum dynamics, with a very broad applicative scope. After a note on the action of linear and quadratic Hamiltonians, whose description follows directly from the theory of Gaussian states presented in Chapter 3 but was not explicitly clarified so far, we shall address systems linearly coupled to a reservoir of modes that are delta-correlated in time (a situation commonly referred to as ‘white noise’), and thus give rise to Markovian dynamics which we shall refer to as ‘diffusive’ for reasons that will be made clear in due time. Our story will commence with a derivation of the Gaussian dynamics resulting from such an environmental coupling, in the form of dynamical equations for the first and second statistical moments. Besides their interest per se , the latter will be instrumental in obtaining an unconventional derivation of the master equations that govern general, not necessarily Gaussian, quantum states under the same conditions. The corresponding quantum Langevin equations for the canonical operators in the Heisenberg picture will also be derived. Next, we will turn our attention to the study of the system's dynamics that result from the measurement of the environment. This is a topic with a longstanding tradition in quantum mechanics, grounded in the formalism of quantum stochastic calculus that emerged within the mathematical physics community as early as the seventies, and then re-introduced as the quantum trajectories approach in quantum optics almost twenty years later. The ‘weak’ POVMs resulting from observing a weakly coupled environment (as opposed to ‘strong’, projective measurements), carried out continuously in time (a prescription customarily subsumed by the term ‘monitoring’), leads to a stochastic dynamics for the system, where the randomness of the measurement outcomes is compactly and most elegantly represented by infinitesimal stochastic increments, giving rise to continuous but not differentiable quantum dynamics. Existing approaches to this subject tend to be quite impervious to the uninitiated. In the quantum optics literature, this is largely due to the technicalities involved in incorporating the discontinuous update of the quantum state prescribed by the von Neumann postulate. More mathematically oriented treatments have instead typically preferred an algebraic approach where the usual difficulty of describing measurements in the Heisenberg picture is compounded by the adoption of a rather abstract hierarchy of filters, in part inspired by classical control theory, to describe different possible updates of the quantum state. This state of affairs is further complicated by the added difficulties that come with the necessary introduction of some form of quantum stochastic calculus. Our approach to monitoring will therefore take a different route. Like in our derivation of diffusion and master equations, we shall keep restricting to linear couplings with a white noise environment, forfeiting full generality This situation is well modelled by the so called input-output formalism, whose physical origin will be discussed in detail in Chapter 9 , which approximates reliably, among other instances, the interaction between internal resonating modes and electromagnetic radiation leaking in and out of an optical cavity. First, we shall introduce stochastic Wiener increments and the Itô rule, which will arise naturally from the identification of the continuum field's modes subject to monitoring. We shall deliberately keep our arsenal of stochastic methods to these bare essentials, sacrificing to some extent both depth and generality, but still providing the reader with a powerful and seminal demonstration of the formalism. We shall then restrict to Gaussian general-dyne detections and apply the exact formalism developed in the previous chapter to the continuous filtering of the system's state conditioned by such measurements' outcomes. Next, as in our derivation of the unconditional master equations, we shall use the Gaussian dynamics as a key to unlock the full stochastic master equations governing general states. Hence, we will build on the quantum theory of filtering in the Gaussian regime to derive dynamical equations for linear feedback control and also apply our theoretical tools to the enhanced production of squeezing. Finally, we will turn to the closely related paradigm of coherent feedback, where environmental modes do not undergo measurements but are fed back to interact coherently with the system, see in detail how it modifies Gaussian diffusive dynamics and assess its potential for control by reconsidering the production of squeezing assisted by interferometric coherent feedback loops.

CRC Press eBooksJun 1, 20230 citations

Poisson and Diffusion Approximation of Stochastic Schrodinger equations with Control

Clément Pellegrini, Université C. Bernard

"Quantum trajectories" are solutions of stochastic differential equations of non-usual type. Such equations are called ``Belavkin'' or ``Stochastic Schrödinger Equations'' and describe random phenomena in continuous measurement theory of Open Quantum System. Many recent investigations deal with the control theory in such model. In this article, stochastic models are mathematically and physically justified as limit of concrete discrete procedures called ``Quantum Repeated Measurements''. In particular, this gives a rigorous justification of the Poisson and diffusion approximation in quantum measurement theory with control. Furthermore we investigate some examples using control in quantum mechanics.

Jan 1, 20090 citations