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Applications of Quantum Stochastic Processes in Quantum Optics

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Quantum Potential Theory

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1954))

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Abstract

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.

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References

  1. L. Accardi, A. Frigerio, and Y. Lu. The weak coupling limit as a quantum functional central limit. Commun. Math. Phys., 131:537–570, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  2. A. Barchielli. Direct and heterodyne detection and other applications of quantum stochastic calculus to quantum optics. Quantum Opt., 2:423–441, 1990.

    Article  MathSciNet  Google Scholar 

  3. V. P. Belavkin. Nondemolition stochastic calculus in Fock space and nonlinear filtering and control in quantum systems. In R. Guelerak and W. Karwowski, editors, Proceedings XXIV Karpacz winter school, Stochastic methods in mathematics and physics, pages 310–324. World Scientific, Singapore, 1988.

    Google Scholar 

  4. V. P. Belavkin. Quantum stochastic calculus and quantum nonlinear filtering. J. Multivar. Anal., 42:171–201, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  5. L. Bouten, J. Stockton, G. Sarma, and H. Mabuchi. Scattering of polarized laser light by an atomic gas in free space: a quantum stochastic differential equation approach. Phys. Rev. A, 75:052111, 2007.

    Article  Google Scholar 

  6. L. M. Bouten, M. I. Guţă, and H. Maassen. Stochastic Schrödinger equations. J. Phys. A, 37:3189–3209, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  7. L. M. Bouten and R. Van Handel. On the separation principle of quantum control, 2005. math-ph/0511021.

    Google Scholar 

  8. L. M. Bouten and R. Van Handel. Quantum filtering: a reference probability approach, 2005. math-ph/0508006.

    Google Scholar 

  9. L. M. Bouten, R. Van Handel, and M. James. An introduction to quantum filtering. SIAM J. Control Optim., 46:2199–2241, 2007.

    Article  MathSciNet  Google Scholar 

  10. C. Cohen Tannoudji, J. Dupont Roc, and G. Grynberg. Photons and Atoms: Introduction to Quantum Electrodynamics. Wiley, 1989.

    Google Scholar 

  11. J. Derezinski and W. De Roeck. Extended weak coupling limit for Pauli-Fierz operators, 2006. math-ph/0610054.

    Google Scholar 

  12. T. Duncan. Evaluation of likelihood functions. Information and Control, (13):62–74, 1968.

    Google Scholar 

  13. F. Fagnola. On quantum stochastic differential equations with unbounded coefficients. Probab. Th. Rel. Fields, 86:501–516, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  14. J. Gough. Quantum flows as Markovian limit of emission, absorption and scattering interactions. Commun. Math. Phys., 254:489–512, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  15. A. Holevo. Quantum stochastic calculus. J. Soviet Math., 56:2609–2624, 1991. Translation of Itogi Nauki i Tekhniki, ser. sovr. prob. mat. 36, 3–28, 1990.

    Article  Google Scholar 

  16. R. L. Hudson and K. R. Parthasarathy. Quantum Itô’s formula and stochastic evolutions. Commun. Math. Phys., 93:301–323, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  17. R. V. Kadison and J. R. Ringrose. Fundamentals of the Theory of Operator Algebras, volume I. Academic Press, San Diego, 1983.

    MATH  Google Scholar 

  18. G. Lindblad. On the generators of quantum dynamical semigroups. Commun. Math. Phys., 48:119–130, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  19. H. Maassen. Quantum probability applied to the damped harmonic oscillator. In S. Attal and J. Lindsay, editors, Quantum Probability Communications, volume XII, pages 23–58. World Scientific, Singapore, 2003.

    Google Scholar 

  20. R. Mortensen. Optimal Control of Continuous-Time Stochastic Systems. PhD thesis, Univ. California, Berkeley, 1966.

    Google Scholar 

  21. K. R. Parthasarathy. An Introduction to Quantum Stochastic Calculus. Birkhäuser, Basel, 1992.

    MATH  Google Scholar 

  22. D. Petz. An Invitation to the Algebra of Canonical Commutation Relations. Leuven University Press, Leuven, 1990.

    MATH  Google Scholar 

  23. M. Reed and B. Simon. Functional Analysis, volume 1 of Methods of Modern Mathematical Physics. Elsevier, 1980.

    Google Scholar 

  24. I. Segal. Tensor algebras over Hilbert spaces. I. Trans. Am. Math. Soc., 81:106–134, 1956.

    Article  MATH  Google Scholar 

  25. J. K. Stockton. Continuous quantum measurement of cold alkali-atom spins. PhD thesis, California Institute of Technology, 2006.

    Google Scholar 

  26. M. Takesaki. Conditional expectations in von Neumann algebras. J. Funct. Anal., 9:306–321, 1971.

    Article  MathSciNet  Google Scholar 

  27. J. von Neumann. Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren. Math. Ann., 102:370–427, 1929.

    Article  MATH  Google Scholar 

  28. D. Williams. Probability with Martingales. Cambridge University Press, Cambridge, 1991.

    MATH  Google Scholar 

  29. M. Zakai. On the optimal filtering of diffusion processes. Z. Wahrsch. Verw. Geb., 11:230–243, 1969.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Caltech Physical Measurement and Control, MC 266-33, 1200 E California Blvd, Pasadena, CA, 91125, USA

    Lue Bouten

Authors
  1. Lue Bouten
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Editor information

Editors and Affiliations

  1. Département de mathématiques de Besancon, Université de Franche-Comté, 16, route de Gray, 25030, Besancon cedex, France

    Uwe Franz

  2. Graduate School of Information Sciences, Tohoku University, 6-3-09 Aramaki-Aza-Aoba, Aoba-ku, Sendai, 980-8579, Japan

    Uwe Franz

  3. Institut für Mathematik und Informatik, Ernst-Moritz-Arndt-Universität Greifswald, Jahnstrasse 15a, 17487, Greifswald, Germany

    Michael Schürmann

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Bouten, L. (2008). Applications of Quantum Stochastic Processes in Quantum Optics. In: Franz, U., Schürmann, M. (eds) Quantum Potential Theory. Lecture Notes in Mathematics, vol 1954. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69365-9_6

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