Gaussian pure states in quantum mechanics and the symplectic group

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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×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 Möbius 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.