Mathematics – Probability
Scientific paper
2006-06-05
SIAM Review 51, pp. 239-316, 2009
Mathematics
Probability
76 pages, 12 figures. A PDF file with high resolution figures can be found at http://minty.caltech.edu/papers.php
Scientific paper
10.1137/060671504
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 finite-dimensional 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.
Bouten Luc
Handel Ramon van
James Matthew R.
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