Gaussian pure states in quantum mechanics and the symplectic group

Simon, R and Sudarshan, ECG and Mukunda, N (1988) Gaussian pure states in quantum mechanics and the symplectic group. In: Physical Review. A, 37 (8). pp. 3028-3038.

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Abstract

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 x 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:	Journal Article
Publication:	Physical Review. A
Publisher:	American Physical Society
Additional Information:	The copyright of this article belongs to American Physical Society.
Department/Centre:	Division of Physical & Mathematical Sciences > Centre for Theoretical Studies (Ceased to exist at the end of 2003)
Date Deposited:	31 Aug 2004
Last Modified:	19 Sep 2010 04:13
URI:	http://eprints.iisc.ac.in/id/eprint/968

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