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. 30283038.

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Abstract
Gaussian pure states of systems with n degrees of freedom and their evolution under quadratic Hamiltonians are studied. The WignerMoyal 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 positivedefinite 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 Poissonbracket 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 Schellmodel optical fields under the action of Sp(4, openR) firstorder 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 

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) 
Depositing User:  jobish pachat 
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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