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Existence, uniqueness and approximation of the jump-type stochastic Schrodinger equation for two-level systems

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  • Pellegrini, Clément

Abstract

In quantum physics, recent investigations deal with the so-called "stochastic Schrodinger equations" theory. This concerns stochastic differential equations of non-usual-type describing random evolutions of open quantum systems. These equations are often justified with heuristic rules and pose tedious problems in terms of mathematical and physical justifications: notion of solution, existence, uniqueness, etc. In this article, we concentrate on a particular case: the Poisson case. Random Measure theory is used in order to give rigorous sense to such equations. We prove the existence and uniqueness of a solution for the associated stochastic equation. Furthermore, the stochastic model is physically justified by proving that the solution can be obtained as a limit of a concrete discrete time physical model.

Suggested Citation

  • Pellegrini, Clément, 2010. "Existence, uniqueness and approximation of the jump-type stochastic Schrodinger equation for two-level systems," Stochastic Processes and their Applications, Elsevier, vol. 120(9), pages 1722-1747, August.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:9:p:1722-1747
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    References listed on IDEAS

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    1. Sébastien Gleyzes & Stefan Kuhr & Christine Guerlin & Julien Bernu & Samuel Deléglise & Ulrich Busk Hoff & Michel Brune & Jean-Michel Raimond & Serge Haroche, 2007. "Quantum jumps of light recording the birth and death of a photon in a cavity," Nature, Nature, vol. 446(7133), pages 297-300, March.
    2. Nicola Bruti-Liberati & Eckhard Platen, 2005. "On the Strong Approximation of Jump-Diffusion Processes," Research Paper Series 157, Quantitative Finance Research Centre, University of Technology, Sydney.
    3. Belavkin, Viacheslav P., 1992. "Quantum stochastic calculus and quantum nonlinear filtering," Journal of Multivariate Analysis, Elsevier, vol. 42(2), pages 171-201, August.
    4. Barchielli, A. & Holevo, A. S., 1995. "Constructing quantum measurement processes via classical stochastic calculus," Stochastic Processes and their Applications, Elsevier, vol. 58(2), pages 293-317, August.
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    Cited by:

    1. Maryam Siddiqui & Mhamed Eddahbi & Omar Kebiri, 2023. "Numerical Solutions of Stochastic Differential Equations with Jumps and Measurable Drifts," Mathematics, MDPI, vol. 11(17), pages 1-14, August.

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