Gaussian pure states in quantum mechanics and the symplectic group

Simon, R. ; Sudarshan, E. C. G. ; Mukunda, N. (1988) Gaussian pure states in quantum mechanics and the symplectic group Physical Review A, 37 (8). pp. 3028-3038. ISSN 1050-2947

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Gaussian pure states of systems with n degrees of freedom and their evolution under quadratic Hamiltonians are studied. The Wigner-Moyal technique together with the symplectic group Sp(2n,openR) is shown to give a convenient framework for handling these problems. By mapping these states to the set of n×n complex symmetric matrices with a positive-definite real part, it is shown that their evolution under quadratic Hamiltonians is compactly described by a matrix generalization of the Mobius transformation; the connection between this result and the "abcd law" of Kogelnik in the context of laser beams is brought out. An equivalent Poisson-bracket description over a special orbit in the Lie algebra of Sp(2n,openR) is derived. Transformation properties of a special class of partially coherent anisotropic Gaussian Schell-model optical fields under the action of Sp(4, openR) first-order systems are worked out as an example, and a generalization of the "abcd law" to the partially coherent case is derived. The relevance of these results to the problem of squeezing in multimode systems is noted.

Item Type:Article
Source:Copyright of this article belongs to American Physical Society.
ID Code:25291
Deposited On:06 Dec 2010 13:37
Last Modified:17 May 2016 08:47

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